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Computing the Eclipse: Astronomy in the Wolfram Language

Par : Mark Long
29 mars 2024 à 19:30

When Exactly Will the Eclipse Happen? A Multimillennium Tale of Computation

Computing the Eclipse: Astronomy in the Wolfram Language

Basic Eclipse Computation

It’s taken millennia to get to the point where it’s possible to accurately compute eclipses. But now—as a tiny part of making “everything in the world” computable—computation about eclipses is just a built-in feature of the Wolfram Language.

The core function is SolarEclipse. By default, SolarEclipse tells us the time of the next solar eclipse from now:

It can also tell us the next solar eclipse from any time more than 10,000 years in the past and future:

By default, SolarEclipse tells us about all solar eclipses, including partial ones. But we can request only total eclipses (or annular ones, etc.):

SolarEclipse immediately lets you compute nearly 100 properties of an eclipse. The most basic property is the type of eclipse; this tells us that the next eclipse from now will be total:

And here are the types of all eclipses for the rest of the decade:

This gives a timeline of these eclipses; notice that most eclipses are separated by 5 or 6 months, but there’s one pair (in 2029) that is just a month apart:

Another basic property of an eclipse is its magnitude: how much of the diameter of the disk of the Sun is covered by the Moon. The next eclipse is total, so the magnitude is greater than 1:

Looking at the sequence of eclipses for the remainder of the decade, we see which ones are total and which are not:

Here’s the path of totality for the upcoming total eclipse:

And here’s where the eclipse is partial:

This is easier to understand with a different geo projection:

There are all sorts of special points and lines associated with these regions. This gives the “point of maximum eclipse” (i.e. essentially the place where the eclipse lasts longest):

And this gives the precise time (converted to my current time zone) of the maximum eclipse:

This gives the time of maximum eclipse in the time zone of the point of maximum eclipse:

At the point of maximum eclipse, this gives the duration of the umbra (i.e. the time of totality):

And here’s a map of where on the Earth the umbra is at the time of maximum eclipse:

Zooming out to a range of 500 miles makes it easier to tell where this is:

This shows the position of the umbra every minute for the hour after the time of maximum eclipse:

Here’s a summary map of the eclipse, including for example (in red) the points of first and last contact of the penumbra:

Instead of looking at what amounts to the shadow of the Moon on the Earth, we can ask what we’d see in the sky. During a total eclipse the Moon will completely cover the Sun. But here’s what happens just 15 minutes before the time of maximum eclipse:

Patterns of Eclipses

Here’s a map of the path of totality for all the total eclipses for the next 50 years:

Over the course of 500 years there are lots of total eclipses:

Although it’s not terribly obvious there, there’s actually a lot of regularity in these paths. In particular, as we discussed previously, similar eclipses occur in “saros series”, separated by a time of about 1 saros, or roughly 18 years. Here are the paths of eclipses that appear for the 10 saroses after the next eclipse:

Each successive eclipse in the saros series is systematically about 120° to the west of the previous one, and a little south (or north, depending on the series). The series continues like this until the eclipse paths hit one of the poles, at which point the series ends:

Any given eclipse is in a saros series. The next eclipse is in series 139 (the numbering scheme for these series was set in 1955—with series 0 chosen, quite arbitrarily, to be the one that starts just after 3000 BC):

There are 71 eclipses in this saros series

running from the year 1501 to the year 2763 (a span of 1263 years):

Not all these eclipses are total, however. But if we plot the magnitudes of all the eclipses, we see that the partial ones appear only at the ends of the saros series:

If we look at the next few eclipses, we’ll see that they are part of all sorts of different saros series:

Here are dates for eclipses in a sequence of different saros series:

Right now there are 40 saros series active:

Any given eclipse could be specified by its “index number” within its saros series. But in the mid-20th century it was realized that there’s a more convenient and robust way to label eclipses, using a combination of “saros number” and what’s called “inex number”.

As we discussed previously, for an eclipse to occur, a new moon must happen at a time when the Moon is close to the plane of the ecliptic. The average time between new moons is the so-called synodic month:

Meanwhile, the average time between the Moon’s crossings of the plane of the ecliptic is half the so-called draconic month:

Given a particular eclipse, the time before an approximate “repeat eclipse” will correspond to a coincidence between an integer multiple of the synodic month, and of half the draconic month. And to figure out when these coincidences will occur is essentially a question of number theory.

Let’s compute the continued fraction expansion:

From this we can derive a sequence of rational approximations:

These approximations get progressively better:

The 8th one is 484/223—and this corresponds to the saros cycle, which reflects the close similarity of 223 synodic months and 242 draconic months:

But now let’s look at the 9th rational approximation: 777/358. This reflects the coincidence:

And now this coincidence defines another cycle—which is the inex cycle. There are lots of other cycles one can identify—but all the common ones can be expressed as linear combinations of the saros and inex cycles.

We saw above how eclipses occur in saros series. But we now see that they will also occur in inex series. And a convenient way to specify an eclipse is to say what saros series and what inex series it appears in. The saros and inex series are numbered according to when they started, with the 0th saros series by convention being the one that spans:

With this setup, the April 2024 eclipse can then be specified by its saros and inex numbers:

But how does this fit in with other eclipses? Here’s a plot of the inex and saros numbers of all eclipses between 1000 AD and 3000 AD (with the April 2024 eclipse indicated in red):

Each saros series shows up as a vertical line of eclipses, and each inex series as a horizontal line. The finite overall date range for the picture leads to the diagonal cutoffs on each side.

For each saros, inex number pair there’ll be some kind of eclipse. But most of the eclipses won’t be total. Here’s where total eclipses occur in the saros, inex plane:

And here’s a complete color-coded “panorama” of different kinds of eclipses:

(Note that—as we discussed above—the eclipses at the ends of saros series are partial.)

By the way, when we’re discussing eclipses in the future, there’s a subtlety to mention. By computing the overall motion of the Earth and Moon we can work out when there’ll be an eclipse, and where in space the cone of shadow associated with it will be. But it’s a different question where geographically that shadow will land on the Earth because that depends on the rotation of the Earth—which isn’t precisely predictable, as illustrated by the somewhat haphazard way in which leap seconds have had to be added to align “universal” UTC time with time based on the rotation of the Earth and the length of a day. Is it a large effect? Over the past 500 years the discrepancy has been about 176 seconds; over the next 500 years it could easily be as much as 1000 seconds. Meanwhile, for an eclipse on the equator each second corresponds to a change of 0.29 miles in where the eclipse will be total (the change is less farther from the equator). In what we’ve shown here we’ve made some standard assumptions about how the rotation of the Earth is slowing down—but ultimately this isn’t predictable, making the locations of eclipses 500 years from now uncertain in the east-west direction by as much as several hundred miles.

The Eclipse from First Principles

The function SolarEclipse in the Wolfram Language immediately tells us when eclipses occur. But we can also deduce this information from other “lower-level” functions. In particular, we know that an eclipse (of at least some kind) occurs if the angular separation between the Sun and the Moon in the sky is smaller than the total of their apparent radii (or about 0.5°). The angular separation depends on where you are on the Earth. Let’s pick a location from which we know an eclipse will be visible:

Now let’s plot angular separation for each hour over the course of the next year:

The minima occur once per lunar month, near the time of the new moons. Changing the plot range, we can see that these minima are different in different (lunar) months:

Let’s look at early April in more detail, now sampling every minute:

And what we see here is that the angular separation goes to zero—reflecting the fact that there’s a total eclipse. What are those little glitches? They’re a consequence of the fact that the apparent positions of the Sun and Moon change when they’re close to the horizon because of refraction in the atmosphere. Not including refraction gives a smoother curve:

We’ve been looking at the angular separation between the centers of the disks of the Sun and Moon in the sky. But what about the actual positions in the sky? Here’s the astro position of the Sun (in local horizon coordinates) at the time and place of maximum eclipse:

And here now is the almost-exactly-equal result for the Moon:

What will it actually look like in the sky? Here’s now a graphic showing the disks of the Sun and Moon 15 minutes before the time of maximum eclipse:

Here’s a sequence of images 15 minutes apart:

Analyzing Data from the 2017 Eclipse

I saw the 2017 eclipse from a rather scenic spot near Jackson, Wyoming:

Click to enlarge

Specifically (according to Wolfram|Alpha accessed through my phone) I was at geo position 43.5125°N 110.6506°W—at an elevation of 7526 ft:

Click to enlarge

As a check, here’s the expected (ground) elevation at that lat-lon—definitely within bounds, particularly considering I was holding the phone about 5 feet off the ground, etc.:

Defining my location as

the predicted arrival time of the eclipse at that location was then:

And indeed that’s exactly what our precisioneclipse.com site told me at the time:

Click to enlarge

What would happen at the “moment of totality”? Here’s where the shadow of the Moon was predicted to be (where I was standing is indicated by a red dot; the whole picture is 100 miles across):

Fifteen seconds earlier, the shadow of the Moon would have just “crossed” the row of mountains I could see:

Looking (in exaggerated 3D) at the terrain, here’s the “umbral cone” at the moment of totality for me:

And it was moving at a speed

which is effectively a vector difference of the rotation speed of the Earth

corrected for latitude

and the orbital velocity of the Moon:

Shown at 5-second intervals for 30 seconds, here’s how the edge of the umbra was moving just before it reached me:

That day, I had brought some not-very-high-tech “equipment” to record the eclipse:

Click to enlarge

The main video I captured—which was from the iPad—is now stored for posterity in the Wolfram Data Repository:

It was 11 minutes long. Sampling frames from it we get:

To get more of a sense of the eclipse, we can pick out the center column of the video, and arrange it in time:

Notice that the “band of totality” appears slightly further to the left at the top of the image—reflecting the fact that totality reaches the mountains in the distance (at the top of the image) slightly earlier than it reaches the “foreground” at the bottom of the image.

To be more quantitative, we can measure the mean intensity of the bottom part of the image as a function of time (where here we’ve aligned with the timestamp that records the start time of the video):

We see in this “light curve” a big dip during the period of totality. There’s also a little dip earlier from someone walking in front of the camera. And we also see lots of little glitches that we’ll discuss later.

But what should we expect this light curve to look like? Well, we can predict what the obscuration of the Sun around the time of totality should be—and but for the 20% effect of “limb darkening”, this should give the intensity of sunlight:

Rescaling values somewhat, we can compare the curves:

And zooming in on the minute around the beginning of totality, we see:

And, yes indeed, the observed onset of totality seems to agree basically to the second with our prediction!

But even though there’s this agreement, the overall shapes of the observed and predicted light curves definitely aren’t the same. And, yes, this is a story of experimental science. And, arguably, of a mistake I made—of using consumer electronics, optimized for consumer purposes, to make a quantitative scientific measurement. You see, as I now realize, an iPad by default always tries to “get a good picture” by maintaining the brightness of an image independent of overall light level. And while for “consumer purposes” that’s usually the right thing to do, it definitely confuses things if one’s trying to measure the light curve for an eclipse.

And indeed if we look at our “measured light curve” it’s very flat until the period of totality. In other words, the iPad succeeded in maintaining the same image brightness until there just wasn’t enough light at all, at which point the image “faded to black”. (At the end of totality, the iPad gradually realized “yes, there’s more light now”, and the measured light curve slowly climbs back up.)

But what’s with all the glitches we see? They’re already visible in our “video time collage” above. And one thought might be that they’re an actual eclipse phenomenon—perhaps associated with the “shadow bands” that I did indeed see just before this eclipse. But the characteristic shimmering associated with such shadow bands—while very difficult to capture on video, perhaps because actual images aren’t being formed—happens much faster than the glitches we’re seeing.

Looking in a bit more detail, we see that there are upward glitches in the period before totality, and downward ones after. Zooming in on a couple of minutes of the “before” period and a couple of minutes of the “after” period, we see:

And, yes, we can validate that this is an “instrumental phenomenon” by doing a simple experiment—using the very same iPad as in 2017—and continuously sliding a piece of cardboard in front of a light and then capturing video and measuring the light curve for this in-my-basement-style “model eclipse”:

We see the very same kind of glitches as from the actual eclipse video. And presumably in both cases they’re reflections of the automatic exposure control system used by the iPad. (If the individual frames of the video had EXIF metadata, we might be able to see that explicitly, but as it is, there’s only EXIF data for the whole video.) I don’t know in detail how this iPad’s exposure control system works, but what we’re seeing is a kind of overshooting-and-correction that’s very common in all sorts of control systems. If we knew in advance everything that happens in the video, then maybe we could avoid the glitches. But if we’re going to try to maintain light level on an ongoing basis during the recording of the video (perhaps by adjusting the gamma correction that determines how raw sensor values are translated to pixel values), then control theory most likely implies that glitches are inevitable.

My First Eclipse

So many years later, I still remember it well. I was 6 years old (almost 7), walking the couple of blocks to school (yes, on my own, which kids in England in those days did). It happened when I was walking under a tree (and, yes, I still remember exactly where). There were lots of dappled patches of light on the ground. And something looked strange about them. And suddenly I realized what it was—and I still have an image of it in my mind today. All the patches of light had the same bite taken out of them. And it didn’t take me long to look up at the Sun, and see that it too had a bite taken out of it.

Being already something of a science enthusiast, I’d heard of eclipses, and realized this must be one. I arrived at school a few minutes later, and regaled the other kids with my “discovery”. But despite the obviousness (at least to me) of what was going on, I wasn’t widely believed. And, yes, that was a very educational experience. But that’s a different story

So what was that eclipse? Well, it was the one in May 1966:

It was a partial eclipse—visible from England:

The geo location of my “discovery tree” was 51.7636° N 1.2558° W. So now we can compute the magnitude of the eclipse there that morning:

I believe school started at 9am, so what I saw was an eclipse with rough magnitude:

The Sun (and Moon) were about 50° above the horizon:

And the Moon was poking into the disk of the Sun:

Thirty minutes later the Moon had poked a little further into the disk of the Sun. But after a bit more than a hour, the whole Sun was back, and the eclipse was over:

And it would be 25 years before I’d see another eclipse—though this time a total one.

By the way, this is me back at the time of my first eclipse—captured in a long-before-those-were-popular “selfie”, taken with a film camera and manual focus, and, it seems, a lot of concentration (and, yes, I think I still make the same strange expression when I’m concentrating hard today):

Click to enlarge

How to Think Computationally about AI, the Universe and Everything

27 octobre 2023 à 21:47

Transcript of a talk at TED AI on October 17, 2023, in San Francisco

Human language. Mathematics. Logic. These are all ways to formalize the world. And in our century there’s a new and yet more powerful one: computation.

And for nearly 50 years I’ve had the great privilege of building an ever taller tower of science and technology based on that idea of computation. And today I want to tell you some of what that’s led to.

There’s a lot to talk about—so I’m going to go quickly… sometimes with just a sentence summarizing what I’ve written a whole book about.

You know, I last gave a TED talk thirteen years ago—in February 2010—soon after Wolfram|Alpha launched.

TED Talk 2010

And I ended that talk with a question: is computation ultimately what’s underneath everything in our universe?

I gave myself a decade to find out. And actually it could have needed a century. But in April 2020—just after the decade mark—we were thrilled to be able to announce what seems to be the ultimate “machine code” of the universe.

Wolfram Physics Project

And, yes, it’s computational. So computation isn’t just a possible formalization; it’s the ultimate one for our universe.

It all starts from the idea that space—like matter—is made of discrete elements. And that the structure of space and everything in it is just defined by the network of relations between these elements—that we might call atoms of space. It’s very elegant—but deeply abstract.

But here’s a humanized representation:

A version of the very beginning of the universe. And what we’re seeing here is the emergence of space and everything in it by the successive application of very simple computational rules. And, remember, those dots are not atoms in any existing space. They’re atoms of space—that are getting put together to make space. And, yes, if we kept going long enough, we could build our whole universe this way.

Eons later here’s a chunk of space with two little black holes, that eventually merge, radiating ripples of gravitational radiation:

And remember—all this is built from pure computation. But like fluid mechanics emerging from molecules, what emerges here is spacetime—and Einstein’s equations for gravity. Though there are deviations that we just might be able to detect. Like that the dimensionality of space won’t always be precisely 3.

And there’s something else. Our computational rules can inevitably be applied in many ways, each defining a different thread of time—a different path of history—that can branch and merge:

But as observers embedded in this universe, we’re branching and merging too. And it turns out that quantum mechanics emerges as the story of how branching minds perceive a branching universe.

The little pink lines here show the structure of what we call branchial space—the space of quantum branches. And one of the stunningly beautiful things—at least for a physicist like me—is that the same phenomenon that in physical space gives us gravity, in branchial space gives us quantum mechanics.

In the history of science so far, I think we can identify four broad paradigms for making models of the world—that can be distinguished by how they deal with time.

4 paradigms

In antiquity—and in plenty of areas of science even today—it’s all about “what things are made of”, and time doesn’t really enter. But in the 1600s came the idea of modeling things with mathematical formulas—in which time enters, but basically just as a coordinate value.

Then in the 1980s—and this is something in which I was deeply involved—came the idea of making models by starting with simple computational rules and then just letting them run:

Can one predict what will happen? No, there’s what I call computational irreducibility: in effect the passage of time corresponds to an irreducible computation that we have to run to know how it will turn out.

But now there’s something even more: in our Physics Project things become multicomputational, with many threads of time, that can only be knitted together by an observer.

It’s a new paradigm—that actually seems to unlock things not only in fundamental physics, but also in the foundations of mathematics and computer science, and possibly in areas like biology and economics too.

You know, I talked about building up the universe by repeatedly applying a computational rule. But how is that rule picked? Well, actually, it isn’t. Because all possible rules are used. And we’re building up what I call the ruliad: the deeply abstract but unique object that is the entangled limit of all possible computational processes. Here’s a tiny fragment of it shown in terms of Turing machines:

OK, so the ruliad is everything. And we as observers are necessarily part of it. In the ruliad as a whole, everything computationally possible can happen. But observers like us can just sample specific slices of the ruliad.

And there are two crucial facts about us. First, we’re computationally bounded—our minds are limited. And second, we believe we’re persistent in time—even though we’re made of different atoms of space at every moment.

So then here’s the big result. What observers with those characteristics perceive in the ruliad necessarily follows certain laws. And those laws turn out to be precisely the three key theories of 20th-century physics: general relativity, quantum mechanics, and statistical mechanics and the Second Law.

It’s because we’re observers like us that we perceive the laws of physics we do.

We can think of different minds as being at different places in rulial space. Human minds who think alike are nearby. Animals further away. And further out we get to alien minds where it’s hard to make a translation.

How can we get intuition for all this? We can use generative AI to take what amounts to an incredibly tiny slice of the ruliad—aligned with images we humans have produced.

We can think of this as a place in the ruliad described using the concept of a cat in a party hat:

Zooming out, we see what we might call “cat island”. But pretty soon we’re in interconcept space. Occasionally things will look familiar, but mostly we’ll see things we humans don’t have words for.

In physical space we explore more of the universe by sending out spacecraft. In rulial space we explore more by expanding our concepts and our paradigms.

We can get a sense of what’s out there by sampling possible rules—doing what I call ruliology:

Even with incredibly simple rules there’s incredible richness. But the issue is that most of it doesn’t yet connect with things we humans understand or care about. It’s like when we look at the natural world and only gradually realize we can use features of it for technology. Even after everything our civilization has achieved, we’re just at the very, very beginning of exploring rulial space.

But what about AIs? Just like we can do ruliology, AIs can in principle go out and explore rulial space. But left to their own devices, they’ll mostly be doing things we humans don’t connect with, or care about.

The big achievements of AI in recent times have been about making systems that are closely aligned with us humans. We train LLMs on billions of webpages so they can produce text that’s typical of what we humans write. And, yes, the fact that this works is undoubtedly telling us some deep scientific things about the semantic grammar of language—and generalizations of things like logic—that perhaps we should have known centuries ago.

You know, for much of human history we were kind of like LLMs, figuring things out by matching patterns in our minds. But then came more systematic formalization—and eventually computation. And with that we got a whole other level of power—to create truly new things, and in effect to go wherever we want in the ruliad.

But the challenge is to do that in a way that connects with what we humans—and our AIs—understand.

And in fact I’ve devoted a large part of my life to building that bridge. It’s all been about creating a language for expressing ourselves computationally: a language for computational thinking.

The goal is to formalize what we know about the world—in computational terms. To have computational ways to represent cities and chemicals and movies and formulas—and our knowledge about them.

It’s been a vast undertaking—that’s spanned more than four decades of my life. It’s something very unique and different. But I’m happy to report that in what has been Mathematica and is now the Wolfram Language I think we have now firmly succeeded in creating a truly full-scale computational language.

In effect, every one of the functions here can be thought of as formalizing—and encapsulating in computational terms—some facet of the intellectual achievements of our civilization:

It’s the most concentrated form of intellectual expression I know: finding the essence of everything and coherently expressing it in the design of our computational language. For me personally it’s been an amazing journey, year after year building the tower of ideas and technology that’s needed—and nowadays sharing that process with the world on open livestreams.

A few centuries ago the development of mathematical notation, and what amounts to the “language of mathematics”, gave a systematic way to express math—and made possible algebra, and calculus, and ultimately all of modern mathematical science. And computational language now provides a similar path—letting us ultimately create a “computational X” for all imaginable fields X.

We’ve seen the growth of computer science—CS. But computational language opens up something ultimately much bigger and broader: CX. For 70 years we’ve had programming languages—which are about telling computers in their terms what to do. But computational language is about something intellectually much bigger: it’s about taking everything we can think about and operationalizing it in computational terms.

You know, I built the Wolfram Language first and foremost because I wanted to use it myself. And now when I use it, I feel like it’s giving me a superpower:

I just have to imagine something in computational terms and then the language almost magically lets me bring it into reality, see its consequences and then build on them. And, yes, that’s the superpower that’s let me do things like our Physics Project.

And over the past 35 years it’s been my great privilege to share this superpower with many other people—and by doing so to have enabled such an incredible number of advances across so many fields. It’s a wonderful thing to see people—researchers, CEOs, kids—using our language to fluently think in computational terms, crispening up their own thinking and then in effect automatically calling in computational superpowers.

And now it’s not just people who can do that. AIs can use our computational language as a tool too. Yes, to get their facts straight, but even more importantly, to compute new facts. There are already some integrations of our technology into LLMs—and there’s a lot more you’ll be seeing soon. And, you know, when it comes to building new things, a very powerful emerging workflow is basically to start by telling the LLM roughly what you want, then have it try to express that in precise Wolfram Language. Then—and this is a critical feature of our computational language compared to a programming language—you as a human can “read the code”. And if it does what you want, you can use it as a dependable component to build on.

OK, but let’s say we use more and more AI—and more and more computation. What’s the world going to be like? From the Industrial Revolution on, we’ve been used to doing engineering where we can in effect “see how the gears mesh” to “understand” how things work. But computational irreducibility now shows that won’t always be possible. We won’t always be able to make a simple human—or, say, mathematical—narrative to explain or predict what a system will do.

And, yes, this is science in effect eating itself from the inside. From all the successes of mathematical science we’ve come to believe that somehow—if only we could find them—there’d be formulas to predict everything. But now computational irreducibility shows that isn’t true. And that in effect to find out what a system will do, we have to go through the same irreducible computational steps as the system itself.

Yes, it’s a weakness of science. But it’s also why the passage of time is significant—and meaningful. We can’t just jump ahead and get the answer; we have to “live the steps”.

It’s going to be a great societal dilemma of the future. If we let our AIs achieve their full computational potential, they’ll have lots of computational irreducibility, and we won’t be able to predict what they’ll do. But if we put constraints on them to make them predictable, we’ll limit what they can do for us.

So what will it feel like if our world is full of computational irreducibility? Well, it’s really nothing new—because that’s the story with much of nature. And what’s happened there is that we’ve found ways to operate within nature—even though nature can still surprise us.

And so it will be with the AIs. We might give them a constitution, but there will always be consequences we can’t predict. Of course, even figuring out societally what we want from the AIs is hard. Maybe we need a promptocracy where people write prompts instead of just voting. But basically every control-the-outcome scheme seems full of both political philosophy and computational irreducibility gotchas.

You know, if we look at the whole arc of human history, the one thing that’s systematically changed is that more and more gets automated. And LLMs just gave us a dramatic and unexpected example of that. So does that mean that in the end we humans will have nothing to do? Well, if you look at history, what seems to happen is that when one thing gets automated away, it opens up lots of new things to do. And as economies develop, the pie chart of occupations seems to get more and more fragmented.

And now we’re back to the ruliad. Because at a foundational level what’s happening is that automation is opening up more directions to go in the ruliad. And there’s no abstract way to choose between them. It’s just a question of what we humans want—and it requires humans “doing work” to define that.

A society of AIs untethered by human input would effectively go off and explore the whole ruliad. But most of what they’d do would seem to us random and pointless. Much like now most of nature doesn’t seem like it’s “achieving a purpose”.

One used to imagine that to build things that are useful to us, we’d have to do it step by step. But AI and the whole phenomenon of computation tell us that really what we need is more just to define what we want. Then computation, AI, automation can make it happen.

And, yes, I think the key to defining in a clear way what we want is computational language. You know—even after 35 years—for many people the Wolfram Language is still an artifact from the future. If your job is to program it seems like a cheat: how come you can do in an hour what would usually take a week? But it can also be daunting, because having dashed off that one thing, you now have to conceptualize the next thing. Of course, it’s great for CEOs and CTOs and intellectual leaders who are ready to race onto the next thing. And indeed it’s impressively popular in that set.

In a sense, what’s happening is that Wolfram Language shifts from concentrating on mechanics to concentrating on conceptualization. And the key to that conceptualization is broad computational thinking. So how can one learn to do that? It’s not really a story of CS. It’s really a story of CX. And as a kind of education, it’s more like liberal arts than STEM. It’s part of a trend that when you automate technical execution, what becomes important is not figuring out how to do things—but what to do. And that’s more a story of broad knowledge and general thinking than any kind of narrow specialization.

You know, there’s an unexpected human-centeredness to all of this. We might have thought that with the advance of science and technology, the particulars of us humans would become ever less relevant. But we’ve discovered that that’s not true. And that in fact everything—even our physics—depends on how we humans happen to have sampled the ruliad.

Before our Physics Project we didn’t know if our universe really was computational. But now it’s pretty clear that it is. And from that we’re inexorably led to the ruliad—with all its vastness, so hugely greater than all the physical space in our universe.

So where will we go in the ruliad? Computational language is what lets us chart our path. It lets us humans define our goals and our journeys. And what’s amazing is that all the power and depth of what’s out there in the ruliad is accessible to everyone. One just has to learn to harness those computational superpowers. Which starts here. Our portal to the ruliad:

Games and Puzzles as Multicomputational Systems

Par : Mark Long
8 juin 2022 à 22:17

Games and Puzzles as Multicomputational Systems

Humanizing Multicomputational Processes

Multicomputation is one of the core ideas of the Wolfram Physics Project—and in particular is at the heart of our emerging understanding of quantum mechanics. But how can one get an intuition for what is initially the rather abstract idea of multicomputation? A good approach, I believe, is to see it in action in familiar systems and situations. And I explore here what seems like a particularly good example: games and puzzles.

One might not imagine that something as everyday as well-known games and puzzles would have any connection to the formalism for something like quantum mechanics. But the idea of multicomputation provides a link. And indeed one can view the very possibility of being able to have “interesting” games and puzzles as being related to a core phenomenon of multicomputation: multicomputational irreducibility.

In an ordinary computational system each state of the system has a unique successor, and ultimately there is a single thread of time that defines a process of computation. But in a multicomputational system the key idea is that states can have multiple successors—and tracing their behavior defines a whole multiway graph of branching and merging threads of time. And the point is that this is directly related to how one can think about typical games and puzzles.

Given a particular state of a game or puzzle, a player must typically decide what to do next. And where the idea of multicomputation comes in is that there are usually several choices that they can make. In any particular instance of the game, they’ll make a particular choice. But the point of the multicomputational paradigm is to look globally at the consequences of all possible choices—and to produce a multiway graph that represents them.

The notion of making what we call a multiway graph has actually existed—usually under the name of “game graphs”—for games and puzzles for a bit more than a hundred years. But with the multicomputational paradigm there are now some more general concepts that can be applied to these constructs. And in turn understanding the relation to games and puzzles has the potential to provide a new level of intuition and familiarity about multiway graphs.

My particular goal here is to investigate—fairly systematically—a sequence of well-known games and puzzles using the general methods we’ve been developing for studying multicomputational systems. As is typical in investigations that connect with everyday things, we’ll encounter all sorts of specific details. And while these may not immediately seem relevant to larger-scale discussions, they are important in our effort to provide a realistic and relatable picture of actual games and puzzles—and in allowing the connections we make with multicomputation to be on a solid foundation.

It’s worth mentioning that the possibility of relating games and puzzles to physics is basically something that wouldn’t make sense without our Physics Project. For games and puzzles are normally at some fundamental level discrete—especially in the way that they involve discrete branching of possibilities. And if one assumes physics is fundamentally continuous, there’s no reason to expect a connection. But a key idea of our Physics Project is that the physical world is at the lowest level discrete—like games and puzzles. And what’s more, our Physics Project posits that physics—like games and puzzles—has discrete possibilities to explore.

At the outset each of the games and puzzles I discuss here may seem rather different in their structure and operation. But what we’ll see is that when viewed in a multicomputational way, there is remarkable—and almost monotonous—uniformity across our different examples. I won’t comment too much on the significance of what we see until the end, when I’ll begin to discuss how various important multicomputational phenomena may play out in the context of games and puzzles. And how the very difficulty of conceptualizing multicomputation in straightforward human terms is what fundamentally leads to the engaging character of games and puzzles.

Tic-Tac-Toe

Consider a simplified version of tic-tac-toe (AKA “noughts and crosses”) played on a 2×2 board. Assume X plays first. Then one can represent the possible moves by the graph:

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On the next turn one gets:

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So far this graph is a simple tree. But if we play another turn we’ll see that different branches can merge, and “playing until the board is full” we get a multiway graph—or “game graph”—of the form:

&#10005

Every path through this graph represents a possible complete game:

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In our setup so far, the total number of board configurations that can ever be reached in any game (i.e. the total number of nodes in the graph) is 35, while the total number of possible complete games (i.e. the number of possible paths from the root of the graph) is 24.

If one renders the graph in 3D one can see that it has a very regular structure:

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And now if we define “winning” 2×2 tic-tac-toe as having two identical elements in a horizontal row, then we can annotate the multiway graph to indicate wins—removing cases where the “game is already over”:

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Much of the core structure of the multiway graph is actually already evident even in the seemingly trivial case of “one-player tic-tac-toe”, in which one is simply progressively filling in squares on the board:

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But what makes this not completely trivial is the existence of distinct paths that lead to equivalent states. Rendered differently the graph (which has 24 = 16 nodes and 4! = 24 “game paths”) has an obvious 4D hypercube form (where now we have dropped the explicit X’s in each cell):

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For a 3×3 board the graph is a 9D hypercube with 29 = 512 nodes and 9! = 362880 “game paths”, or in “move-layered” form:

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This basic structure is already visible in “1-player 1D tic-tac-toe” in which the multiway graph for a “length-n” board just corresponds to an n-dimensional hypercube:

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The total number of distinct board configurations in this case is just 2n, and the number of distinct “games” is n!. At move t the number of distinct board configurations (i.e. states) is Binomial[n, t].

With 2 players the graphs become slightly more complicated:

&#10005

The total number of states in these graphs is

&#10005

which is asymptotically . (Note that for n = 4 the result is the same as for the 2×2 board discussed above.) At move t the number of distinct states is given by

&#10005

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OK, so what about standard 2-player 3×3 tic-tac-toe? Its multiway graph begins:

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After 2 steps (i.e. one move by X and one by O) the graph is still a tree (with the initial state now at the center):

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After 3 steps there is starting to be merging:

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And continuing for all 9 moves the full layered graph—with 6046 states—is:

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At the level of this graph, the results are exactly the same as for a 2-player 1D version with a total of 9 squares. But for actual 2D 3×3 tic-tac-toe there is an additional element to the story: the concept of winning a game, and thereby terminating it. With the usual rules, a game is considered won when a player gets a horizontal, vertical or diagonal line of three squares, as in for example:

&#10005

Whenever a “win state” such as these is reached, the game is considered over, so that subsequent states in the multiway graph are pruned, and what was previously a 6046-node graph becomes a 5478-node graph

&#10005

with examples of the 568 pruned states including (where the “win” that terminated the game is marked):

&#10005

Wins can occur at different steps: anywhere from 5 to 9. The total numbers of distinct wins are as follows

&#10005

(yielding 626 wins at any step for X and 316 for O).

One can’t explicitly tell that a game has ended in a draw until every square has been filled in—and there are ultimately only 16 final “draw configurations” that can be reached:

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We can annotate the full (“game-over-pruned”) multiway graph, indicating win and draw states:

&#10005

To study this further, let’s start by looking at a subgraph that includes only “end games” starting with a board that already has 4 squares filled in:

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We see here that from our initial board

&#10005

it’s possible to get a final win for both X and O:

&#10005

But in many of these cases the outcome is already basically determined a step or more before the actual win occurs—in the sense that unless a given player “makes a mistake” they will always be able to force a win.

So, for example, if it is X’s turn and the state is

&#10005

then X is guaranteed to win if they play as follows:

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We can represent the “pre-forcing” of wins by coloring subgraphs (or in effect “light cones”) in the multiway graph:

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At the very beginning of the game, when X makes the first move, nothing is yet forced. But after just one move, it’s already possible to get to configurations where X can always force a win:

&#10005

Starting from a state obtained after 1 step, we can see that after 2 steps there are configurations where O can force a win:

&#10005

Going to more moves leads to more “forced-win” configurations:

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Annotating the whole multiway graph we get:

&#10005

We can think of this graph as a representation of the “solution” to the game: given any state the coloring in the graph tells us which player can force a win from that state, and the graph defines what moves they can make to do so.

Here’s a summary of possible game states at each move:

&#10005

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Here we’re just counting the number of possible states of various kinds at each step. But is there a way to think about these states as somehow being laid out in “game state space”? Branchial graphs provide a potential way to do this. The basic branchial graph at a particular step is obtained by joining pairs of states that share a common ancestor on the step before. For the case of 2-player 2×2 tic-tac-toe the branchial graphs we get on successive steps are as follows:

&#10005

Things get more complicated for ordinary 3×3 tic-tac-toe. But since the multiway graph for the first two steps is a pure tree, the branchial graphs at these steps still have a rather trivial structure:

&#10005

In general the number of connected components on successive steps is as follows

&#10005

and these are broken down across different graph structures as follows:

&#10005

Here in more detail are the forms of some typical components of branchial graphs achieved at particular steps:

&#10005

Within the branchial graph at a particular step, there can be different numbers of wins in different components:

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It’s notable that the wins are quite broadly distributed across branchial graphs. And this is in a sense why tic-tac-toe is not (more) trivial. If just by knowing what component of the branchial graph one was in one could immediately know the outcome, there would be even less “suspense” in the game. But with broad distribution across branchial space, “knowing roughly where you are” doesn’t help much in determining whether you’re going to win.

So far we’ve always been talking about what states can be reached, but not “how often” they’re reached. Imagine that rather than playing a specific game, we instead at each step just make every possible move with equal probability. The setup for tic-tac-toe is symmetrical enough that for most of the game the probability of every possible configuration at a given step is equal. But as soon as there start to be “wins”, and there is a “cone” of “game-over-pruned” states, then the remaining states no longer have equal probabilities.

For standard 3×3 tic-tac-toe this happens after 7 moves, where there are two classes of states, that occur with slightly different probabilities:

&#10005

At the end of the game, there are several classes of final states with different probabilities:

&#10005

And what this means for the probabilities of different outcomes of the game is as follows:

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Not surprisingly, the player who plays first has an advantage in winning. Perhaps more surprising is that in this kind of “strategyless” play, ties are comparatively uncommon—even though if one player actively tries to block the other, they often force a tie.

We’ve looked at “classic tic-tac-toe” and a few specific variants. But there are ultimately all sorts of possible variants. And a convenient general way to represent the “board” for any tic-tac-toe-like game is just to give a “flattened” list of values—with 0 representing a blank position, and i representing a symbol added by player i.

In standard “2D representation” one might have a board like

&#10005

which in flattened form would be:

&#10005

Typical winning patterns can then be represented

&#10005

where in each case we have framed the relevant “winning symbols”, and then given their positions in the flattened list. In ordinary tic-tac-toe it’s clear that the positions of “winning symbols” must always form an arithmetic progression. And it seems as if a good way to generalize tic-tac-toe is always to define a win for i to be associated with the presence of i symbols at positions that form an arithmetic progression of a certain length s. For ordinary tic-tac-toe s = 3, but for generalizations it could have other values.

Consider now the case of a length-5 list (i.e. 5-position “1D board”). The complete multiway graph is as follows, with “winning states” that contain arithmetic progressions of length s = 3 highlighted:

&#10005

In a more symmetrical rendering this is:

&#10005

Here’s the analogous result for a 7-position board, and 2 players:

&#10005

For each size of board n, we can compute the total number of winning states for any given player, as well as the total number of states altogether. The result when winning is based on arithmetic progressions of length 3 (i.e. s = 3) is:

&#10005

The 2-player n = 9 (= 3×3) case here is similar to ordinary tic-tac-toe, but not the same. In particular, states like

&#10005

are considered wins for X in the flattened setup, but not in ordinary tic-tac-toe.

If we increase the length of progression needed in order to declare a win, say to s = 4, we get:

&#10005

The total number of game states is unchanged, but—as expected—there are “fewer ways to win”.

But let’s say we have boards that are completely filled in. For small board sizes there may well not be an arithmetic progression of positions for any player—so that the game has to be considered a tie—as we see in this n = 5 case:

&#10005

But it is a result related to Ramsey theory that it turns out that for n ≥ 9, it’s inevitable that there will be an “arithmetic progression win” for at least one of the players—so that there is never a tie—as these examples illustrate:

&#10005

 

Walks and Their Multiway Graphs

A game like tic-tac-toe effectively involves at each step moving to one of several possible new board configurations—which we can think of as being at different “places” in “game state space”. But what if instead of board configurations we just consider our states to be positions on a lattice such as

&#10005

and then we look at possible walks, that at each step can, in this case, go one unit in any of 4 directions?

Starting at a particular point, the multiway graph after 1 step is just

&#10005

where we have laid out this graph so that the “states” are placed at their geometrical positions on the lattice.

After 2 steps we get:

&#10005

And in general the structure of the multiway graph just “recapitulates” the structure of the lattice:

&#10005

We can think of the paths in the multiway graph as representing all possible random walks of a certain length in the lattice. We can lay the graph out in 3D, with the vertical position representing the first step at which a given point can be reached:

&#10005

We can also lay out the graph more like we laid out multiway graphs for tic-tac-toe:

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One feature of these “random-walk” multiway graphs is that they contain loops, that record the possibility of “returning to places one’s already been”. And this is different from what happens for example in tic-tac-toe, in which at each step one is just adding an element to the board, and it’s never possible to go back.

But we can set up a similar “never-go-back rule” for walks, by considering “self-avoiding walks” in which any point that’s been visited can never be visited again. Let’s consider first the very trivial lattice:

&#10005

Now indicate the “current place we’ve reached” by a red dot, and the places we’ve visited before by blue dots—and start from one corner:

&#10005

There are only two possible walks here, one going clockwise, the other counterclockwise. Allowing one to start in each possible position yields a slightly more complicated multiway graph:

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With a 2×3 grid we get

&#10005

while with a 3×3 grid we get:

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Starting in the center, and with a different layout for the multiway graph, we get:

&#10005

Note the presence of large “holes”, in which paths on each side basically “get to the same place” in “opposite ways”. Note that of the 2304 possible ways to have 1 red dot and up to 8 blue ones, this actual multiway graph reaches only 57. (Starting from the corner reaches 75 and from all possible initial positions 438.)

With a 4×4 lattice (starting the walker in the corner) the multiway graph has the form

&#10005

or in an alternative layout

&#10005

where now 1677 states out of 524,288 are eventually visited, and the number of new states visited at each step (i.e. the number of nodes in successive layers in the graph) is:

&#10005

For a 5×5 grid 89,961 states are reached, distributed across steps according to:

&#10005

(For a grid with n vertices, there are a total of n 2n–1 possible states, but the number actually reached is always much smaller.)

In talking about walks, an obvious question to ask is about mazes. Consider the maze:

&#10005

As far as traversing this maze is concerned, it is equivalent to “walking” on the graph

&#10005

which in another embedding is just

&#10005

But just as before, the multiway graph that represents all possible walks essentially just “recapitulates” this graph. And that means that “solving” the maze can in a sense equally be thought of as finding a path directly in the maze graph, or in the multiway graph:

&#10005

 

The Icosian Game & Some Relatives

Our discussion of self-avoiding walks turns out to be immediately related to the “Icosian game” of William Rowan Hamilton from 1857 (which is somewhat related to the early computer game Hunt the Wumpus):

The Icosian game

The object of the “game” (or, more properly, puzzle) is to find a path (yes, a Hamiltonian path) around the icosahedron graph that visits every node (and returns back to where it started from). And once again we can construct a multiway graph that represents all possible sequences of “moves” in the game.

Let’s start with the simpler case of an underlying tetrahedron graph:

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From this we get the multiway graph:

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The “combined multiway graph” from all possible starting positions on the tetrahedron graph gives a truncated cuboctahedron multiway graph:

&#10005

And following this graph we see that from any initial state it’s always possible to reach a state where every node in the tetrahedron graph has been visited. In fact, because the tetrahedron graph is a complete graph it’s even guaranteed that the last node in the sequence will be “adjacent” to the starting node—so that one has formed a Hamiltonian cycle and solved the puzzle.

Things are less trivial for the cube graph:

&#10005

The multiway graph (starting from a particular state) in this case is:

&#10005

Now there are 13 configurations where no further moves are possible:

&#10005

In some of these, one’s effectively “boxed in” with no adjacent node to visit. In others, all the nodes have been filled in. But only 3 ultimately achieve a true Hamiltonian cycle that ends adjacent to the starting node:

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It turns out that one can reach each of these states through 4 distinct paths from the root of the multiway graph. An example of such a path is:

&#10005

We can summarize this path as a Hamiltonian circuit of the original cube graph:

&#10005

In the multiway graph, the 12 “winning paths” are

&#10005

In a different rendering this becomes

&#10005

and keeping only “winning paths” the subgraph of the multiway graph has the symmetrical form:

&#10005

The actual Hamiltonian circuits through the underlying cube graph corresponding to these winning paths are:

&#10005

For the dodecahedral graph (i.e. the original Icosian game), the multiway graph is larger and more complicated. It begins

&#10005

and has its first merge after 11 steps (and 529 in all), and ends up with a total of 11,093 nodes—of which 2446 are “end states” where no further move is possible. This shows the number of end (below) and non-end (above) states at each successive step:

&#10005

The successive fractions of “on-track-to-succeed” states are as follows, indicating that the puzzle is in a sense harder at the beginning than at the end:

&#10005

There are 13 “end states” which fill in every position of the underlying dodecahedral graph, with 3 of these corresponding to Hamiltonian cycles:

&#10005

The total number of paths from the root of the multiway graph leading to end states (in effect the total number of ways to try to solve the puzzle) is 3120. Of these, 60 lead to the 3 Hamiltonian cycle end states. An example of one of these “winning paths” is:

&#10005

Examples of underlying Hamiltonian cycles corresponding to each of the 3 Hamiltonian cycle end states are:

&#10005

And this now shows all 60 paths through the multiway graph that reach Hamiltonian cycle end states—and thus correspond to solutions to the puzzle:

&#10005

In effect, solving the puzzle consists in successfully finding these paths out of all the possibilities in the multiway graph. In practice, though—much as in theorem-proving, for example—there are considerably more efficient ways to find “winning paths” than to look directly at all possibilities in the multiway graph (e.g. FindHamiltonianCycle in Wolfram Language). But for our purpose of understanding games and puzzles in a multicomputational framework, it’s useful to see how solutions to this puzzle lay out in the multiway graph.

The Icosian game from Hamilton was what launched the idea of Hamiltonian cycles on graphs. But already in 1736 Leonhard Euler had discussed what are now called Eulerian cycles in connection with the puzzle of the Bridges of Königsberg. In modern terms, we can state the puzzle as the problem of finding a path that visits once and only once all the edges in the graph (in which the “double bridges” from the original puzzle have been disambiguated by extra nodes):

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We can create a multiway graph that represents all possible paths starting from a particular vertex:

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But now we see that the end states here are

&#10005

and since none of them have visited every edge, there is no Eulerian circuit here. To completely resolve the puzzle we need to make a multiway graph in which we start from all possible underlying vertices. The result is a disconnected multiway graph whose end states again never visit every edge in the underlying graph (as one can tell from the fact that the number of “levels” in each subgraph is less than 10):

&#10005

 

The Geography Game

In the Geography Game one has a collection of words (say place names) and then one attempts to “string the words together”, with the last letter of one word being the same as the first letter of the next. The game typically ends when nobody can come up with a word that works and hasn’t been used before.

Usually in practice the game is played with multiple players. But one can perfectly well consider a version with just one player. And as an example let’s take our “words” to be the abbreviations for the states in the US. Then we can make a graph of what can follow what:

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Let’s at first ignore the question of whether a state has “already been used”. Then, starting, say, from Massachusetts (MA), we can construct the beginning of a multiway graph that gives us all possible sequences:

&#10005

After 10 steps the graph is

&#10005

or in a different rendering:

&#10005

This shows the total number of paths as a function of length through this graph, assuming one doesn’t allow any state to be repeated:

&#10005

The maximum length of path is 23—and there are 256 such paths, 88 ending with TX and 168 ending with AZ. A few sample such paths are

&#10005

and all these paths can be represented by what amounts to a finite state machine:

&#10005

By the way, the starting state that leads to the longest path is MN—which achieves length 24 in 2336 different ways, with possible endings being AZ, DE, KY and TX. A few samples are:

&#10005

Drawing these paths in the first few steps of the multiway graph starting from MN we get:

&#10005

 

Groups and (Simplified) Rubik’s Cubes

We’ve talked about puzzles that effectively involve walks on graphs. A particularly famous example of a puzzle that can be thought about in this way is the Rubik’s Cube. The graph in question is then the Cayley graph for the group formed by the transformations that can be applied to the cube.

As a very simple analog, we can consider the symmetry group of the square, D4, based on the operations of reflection and 90° rotation. We generate the Cayley graph just like a multiway graph: by applying each operation at each step. And in this example the Cayley graph is:

&#10005

This graph is small enough that it is straightforward to see how to get from any configuration to any other. But while this Cayley graph has 8 nodes and maximum path length from any one node to any other of 3, the Cayley graph for the Rubik’s Cube has nodes and a maximum shortest path length of 20.

To get some sense of the structure of an object like this, we can consider the very simplified case of a “2×2×2 cube”—colored only on its corners—in which each face can be rotated by 90°:

&#10005

The first step in the multiway graph—starting from the configuration above—is then (note that the edges in the graph are not directed, since the underlying transformations are always reversible):

&#10005

Going another step gives:

&#10005

The complete multiway graph (which is also the Cayley graph for the group—which turns out to be S8—generated by the transformations) has 8! = 40,320 nodes (and 483,840 edges). Starting from a state (i.e. node in the Cayley graph) the number of new states reached at successive steps is:

&#10005

The maximum shortest paths in the graph consist of 8 steps; an example is:

&#10005

Between these particular two endpoints there are actually 3216 “geodesic” paths—which spread out quite far in the multiway graph

&#10005

Picking out only geodesic paths we see there are many ways to get from one configuration of the cube to one of its “antipodes”:

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Peg Solitaire

Whereas something like tic-tac-toe involves progressively filling in a board, a large class of puzzles that have been used since at least the 1600s involve basically removing pegs from a board. The typical rules involve pegs being able to jump over a single intermediate peg into a hole, with the intermediate peg then being removed. The goal is to end up with just a single peg on the board.

Here’s a very simple example based on a T arrangement of pegs:

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In this case, there’s only one way to “solve the puzzle”. But in general there’s a multiway graph:

&#10005

A more complicated example is the “Tricky Triangle” (AKA the “Cracker Barrel puzzle”). Its multiway graph begins:

&#10005

After another couple of steps it becomes:

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There are a total of 3016 states in the final multiway graph, of which 118 are “dead-end” configurations from which no further moves are possible. The “earliest” of these dead-end configurations are:

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There are just 4 “winning states” that can be reached, and the “end games” that lead to them are:

&#10005

Starting from the initial configuration, the number of possible states reached at each step is given as follows, where the states that can lead to winning configurations is shown in yellow:

&#10005

This shows the complete multiway graph, with “winning paths” highlighted:

&#10005

At successive steps, the fraction of states that can lead to a winning state is as follows:

&#10005

The branchial graphs are highly connected, implying that in a sense the puzzle remains “well mixed” and “unpredictable” until the very end:

&#10005

 

Checkers

Peg solitaire is a one-player “game”. Checkers (AKA draughts) is a two-player game with a somewhat similar setup. “Black” and “red” pieces move diagonally in different directions on a board, “taking” each other by jumping over when they are adjacent.

Let’s consider the rather minimal example of a 4×4 board. The basic set of possible moves for “black” and “red” is defined by the graphs (note that a 4×4 board is too small to support “multiple jumps”):

&#10005

With this setup we can immediately start to generate a multiway graph, based on alternating black and red moves:

&#10005

With the rules as defined so far, the full 161-node multiway graph is:

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It’s not completely clear what it means to “win” in this simple 4×4 case. But one possibility is to say that it happens when the other player can’t do anything at their next move. This corresponds to “dead ends” in the multiway graph. There are 26 of these, of which only 3 occur when it is red’s move next, and the rest all occur when it is black’s move:

&#10005

As before, any particular checkers game corresponds to a path in the multiway graph from the root to one of these end states. If we look at branchial graphs in this case, we find that they have many disconnected pieces, indicating that there are many largely independent “game paths” for this simple game—so there is not much “mixing” of outcomes:

&#10005

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The rules we’ve used so far don’t account for what amounts to the second level of rules for checkers: the fact that when a piece reaches the other side of the board it becomes a “king” that’s allowed to move backwards as well as forwards. Even with a single piece and single player this already generates a multiway graph—notably now with loops:

&#10005

or in an alternative layout (with explicitly undirected edges):

&#10005

With two pieces (and two players taking turns) the “kings” multiway graph begins:

&#10005

With this initial configuration, but without backward motion, the whole multiway graph is just:

&#10005

The full “kings” multiway graph in this case also only has 62 nodes—but includes all sorts of loops (though with this few pieces and black playing first it’s inevitable that any win will be for black):

&#10005

What about the ordinary + kings multiway graph from our original initial conditions? The combined graph has 161 nodes from the “pre-king” phase, and 4302 from the “post-king” phase—giving the final form:

&#10005

 

(Very Simplified) Go

The full game of Go is sophisticated and its multiway graph in any realistic case is far too big for us to generate at all explicitly (though one can certainly wonder if there are meaningful “continuum limit” results). However, to get some flavor of Go we can consider a vastly simplified version in which black and white “stones” are progressively placed on nodes of a graph, and the game is considered “won” if one player has successfully surrounded a connected collection of the other player’s stones.

Imagine that we start with a blank “board” consisting of a 2×2 square of positions, then on a sequence of “turns” add black and white stones in all possible ways. The resulting multiway graph is:

&#10005

Every state that has no successor here is a win for either black or white. The “black wins” (with the surrounded stone highlighted) are

&#10005

while the “white wins” are:

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At this level what we have is basically equivalent to 2×2 tic-tac-toe, albeit with a “diagonal” win condition. With a 3×2 “board”, the first two steps in the multiway graph are:

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The final multiway graph is:

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The graph has 235 nodes, of which 24 are wins for white, and 34 for black:

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The successive branchial graphs in this case are (with wins for black and white indicated):

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For a 3×3 “board” the multiway graph has 5172 states, with 604 being wins for white and 684 being wins for black.

Nim

As another example of a simple game, we’ll now consider Nim. In Nim, there are k piles of objects, and at each step p players alternate in removing as many objects as they want from whatever single pile they choose. The loser of the game is the player who is forced to have 0 objects in all the piles.

Starting off with 2 piles each containing 2 objects, one can construct a multiway graph for the game:

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With 3 piles this becomes:

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These graphs show all the different possible moves that relate different configurations of the piles. However, they do not indicate which player moves when. Adding this we get in the 22 case

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and in the 222 case:

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Even though these graphs look somewhat complicated, it turns out there is a very straightforward criterion for when a particular state has the property that its “opponent” can force a lose: just take the list of numbers and see if Apply[BitXor, list] is 0. Highlighting when this occurs we get:

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It turns that for Nim, the sequence of branchial graphs we get have a rather regular structure. In the 22 case, with the same highlighting as before, we get:

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In the 222 case the sequence of branchial graphs becomes:

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Here are results for some other cases:

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Sliding Block Puzzles

They go under many names—with many different kinds of theming. But many puzzles are ultimately sliding block puzzles. A simple example might ask to go from

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by sliding blocks into the empty (darker) square. A solution to this is:

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One can use a multiway graph to represent all possible transformations:

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(Note that only 12 of the 4! = 24 possible configurations of the blocks appear here; a configuration like cannot be reached.)

Since blocks can always be “slid both ways” every edge in a sliding-block-puzzle multiway graph has an inverse—so going forward we’ll just draw these multiway graphs as undirected.

Here are some simple cases:

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With a 3×2 board, things quickly get more complicated:

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Rendered in 3D this becomes:

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When all the blocks are distinct, one tends to get multiway graphs with a kind of spherical structure:

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(Note that in the first three cases here, it’s possible to reach all 30, 120, 360 conceivable arrangements of the blocks, while in the last case one can only reach “even permutations” of the blocks, or 360 of the 720 conceivable arrangements.)

This shows how one gets from to :

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With many identical blocks one tends to build up a simple lattice:

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Making one block different basically just “adds decoration”:

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As the number of “1” and “2” blocks becomes closer to equal, the structure fills in:

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Adding a third type of block rapidly leads to a very complicated structure:

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This summarizes a few of the graphs obtained:

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Towers of Hanoi, etc.

Another well-known puzzle is the Towers of Hanoi. And once again we can construct a multiway graph for it. Starting with all disks on the left peg the first step in the multiway graph is:

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Going two steps we get:

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The complete multiway graph is then (showing undirected edges in place of pairs of directed edges):

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It is rather easy to see how the recursive structure of this multiway graph builds up. Here’s the “base case” of 2 disks (and 3 pegs):

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And as each disk is added, the number of nodes in the multiway graph increases by a factor of 3—yielding for example with 4 disks (and still 3 pegs):

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With 4 pegs, things at first look more complicated, even with 2 disks:

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In a 3D rendering, more structure begins to emerge:

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And here are the results for 3, 4 and 5 disks—with the “points of the ears” corresponding to states where all the disks are on a single peg:

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With 3 pegs, the shortest “solution to the puzzle”—of moving all disks from one peg to another—goes along the “side” of the multiway graph, and for n pegs is of length 2n – 1:

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With 4 pegs, there is no longer a unique “geodesic path”:

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(And the sequence of path lengths for successive numbers of pegs is

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or a little below for a large number of pegs n.)

What about branchial graphs? For the standard 3-disk 3-peg case we have

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where successive “time slices” are assumed to be obtained by looking at successive vertical levels in the rendering of the multiway graph above.

For 4 disks one essentially gets “more of the same”:

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With 4 pegs things become slightly more complicated:

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And the trend continues for 5 pegs:

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Multicomputational Implications & Interpretation

We’ve now gone through many examples of games and puzzles. And in each case we’ve explored the multiway graphs that encapsulate the whole spectrum of their possible behavior. So what do we conclude? The most obvious point is that when games and puzzles seem to us difficult—and potentially “interesting”—it’s some kind of reflection of apparent complexity in the multiway graph. Or, put another way, it’s when we find the multiway graph somehow “difficult to decode” that we get a rich and engaging game or puzzle.

In any particular instance of playing a game we’re basically following a specific path (that in analogy to physics we can call a “timelike path”) through the multiway graph (or “game graph”) for the game. And at some level we might just make the global statement that the game graph represents all such paths. But what the multicomputational paradigm suggests is that there are also more local statements that we can usefully make. In particular, at every step along a timelike path we can look “transversally” in the multiway graph, and see the “instantaneous branchial graph” that represents the “entanglement” of our path with “nearby paths”.

Figuring out “what move to make next” is then in a sense about deciding in “what direction” in branchial space to go. And what makes a game difficult is that we can’t readily predict what happens as we “travel through branchial space”. There’s a certain analogy here to the concept of computational irreducibility. Going from one state to another along some timelike path, computational irreducibility implies that even though we may know the underlying rules, we can’t readily predict their consequences—because it can require an irreducible amount of computation to figure out what their consequences will be after many steps.

Predicting “across branchial space” is a related, but slightly different phenomenon, that one can describe as “multicomputational irreducibility”. It’s not about the difficulty of working out a particular path of computation, but instead about the difficulty of seeing how many entangled paths interact.

When one plays a game, it’s common to talk about “how many moves ahead one can see”. And in our terms here, this is basically about asking how “far out in branchial space” we can readily get. As computationally bounded entities, we have a certain “reach” in branchial space. And the game is “difficult for us” if that reach isn’t sufficient to be able to get to something like a “winning position”.

There’s another point here, though. What counts as “winning” in a game is typically reaching some particular places or regions in the multiway graph. But the definition of these places or regions is typically something very computationally bounded (“just see if there’s a line of X’s”, etc.). It’s a certain “observation” of the system, that extracts just a particular (computationally bounded) sampling of the complete state. And then what’s key is that this sampling doesn’t manage to “decode the multicomputational irreducibility”.

There’s an analogy here to thermodynamics. The fact that in thermodynamics we perceive “heat” and “entropy increase” is a consequence of the fact that our (coarse-grained) measurements can’t “decode” the computationally irreducible process that leads to the particular states generated in the system. Similarly, the fact we perceive it to be “hard to figure out how to win a game” is a consequence of the fact that our criterion for winning isn’t able to “look inside the playing of the game” and “decode what’s going on” to the point where it’s in effect just selecting one particular, straightforward path. Instead it’s a question of going through the multicomputationally irreducible process of playing the game, and in effect “seeing where it lands” relative to the observation of winning.

There’s also an analogy here to quantum mechanics. Tracing through many possible paths of playing a game is like following many threads of history in quantum mechanics, and the criterion of winning is like a quantum measurement that selects certain threads. In our Physics Project we imagine that we as observers are extended in branchial space, “knitting together” different threads of history through our belief in our own single thread of experience. In games, the analog of our belief in a single thread of experience is presumably in effect that “all that matters is who wins or loses; it doesn’t matter how the game is played inside”.

To make a closer analogy with quantum mechanics one can start thinking about combining different chunks of “multiway game play”, and trying to work out a calculus for how those chunks fit together.

The games we’ve discussed here are all in a sense pure “games of skill”. But in games where there’s also an element of chance we can think of this as causing what is otherwise a single path in the multiway graph to “fuzz out” into a bundle of paths, and what is otherwise a single point in branchial space to become a whole extended region.

In studying different specific games and puzzles, we’ve often had to look at rather simplified cases in order to get multiway graphs of manageable size. But if we look at very large multiway graphs, are there perhaps overall regularities that will emerge? Is there potentially some kind of “continuum limit” for game graphs?

It’ll almost inevitably be the case that if we look in “enough detail” we’ll see all sorts of multicomputational irreducibility in action. But in our Physics Project—and indeed in the multicomputational paradigm in general—a key issue is that relevant observers don’t see that level of detail. And much like the emergence of thermodynamics or the gas laws from underlying molecular dynamics, the very existence of underlying computational irreducibility inevitably leads to simple laws for what the observer can perceive.

So what is the analog of “the observer” for a game? For at least some purposes it can be thought of as basically the “win” criteria. So now the question arises: if we look only at these criteria, can we derive the analog of “laws of physics”, insensitive to all the multicomputationally irreducible details underneath?

There’s much more to figure out about this, but perhaps one place to start is to look at the large-scale structure of branchial space—and the multiway graph—in various games. And one basic impression in many different games is that—while the character of branchial graphs may change between “different stages” in the game—across a single branchial graph there tends to be a certain uniformity. If one looks at the details there may be plenty of multicomputational irreducibility. But at some kind of “perceptible level” different parts of the graph may seem similar. And this suggests that the “local impression of the game” will tend to be similar at a particular stage even when quite different moves have been made, that take one to quite different parts of the “game space” defined by the branchial graph.

But while there may be similarity between different parts of the branchial graph, what we’ve seen is that in some games and puzzles the branchial graph breaks up into multiple disconnected regions. And what this reflects is the presence of distinct “conserved sectors” in a game—regions of game space that players can get into, but are then stuck with (at least for a certain time), much as in spacetime event horizons can prevent transport between different regions of physical space.

Another (related) effect that we notice in some games and puzzles but not others is large “holes” in the multiway graph: places where between two points in the graph there are multiple “distant” paths. When the multiway graph is densely connected, there’ll typically always be a way to “fix any mistake” by rerouting through nearby paths. But when there is a hole it is a sign that one can end up getting “committed” to one course of action rather than another, and it will be many steps before it’s possible to get to the same place as the other course of action would have reached.

If we assume that at some level all we ultimately “observe” in the multiway graph is the kind of coarse-graining that corresponds to assessing winning or losing then inevitably we’ll be dealing with a distribution of possible paths. Without “holes” these paths can be close together, and may seem obviously similar. But when there’s a hole there can be different paths that are far apart. And the fact there can be distant paths that are “part of the same distribution” can then potentially be thought of as something like a quantum superposition effect.

Are there analogs of general relativity and the path integral in games? To formulate this with clarity we’d have to define more carefully the character of “game space”. Presumably there’ll be the analog of a causal graph. And presumably there’ll also be an analog of energy in game space, associated with the “density of activity” at different places in game space. Then the analog of the phenomenon of gravity will be something like that the best game plays (i.e. the geodesic paths through the game graph) will tend to be deflected by the presence of high densities of activity. In other words, if there are lots of things to do when a game is in a certain state, good game plays will tend to be “pulled towards that state”. And at some level this isn’t surprising: when there’s high density of activity in the game graph, there will tend to be more options about what to do, so it’s more likely that one will be able to “do a good game play” if one goes through there.

So far we didn’t explicitly talk about strategies for games. But in our multicomputational framework a strategy has a fairly definite interpretation: it is a place in rulial space, where in effect one’s assuming a certain set of rules about how to construct the multiway graph. In other words, given a strategy one is choosing some edges in the multiway graph (or some possible events in the associated multiway causal graph), and dropping others.

In general it can be hard to talk about the “space of possible strategies”—because it’s like talking about the “space of possible programs”. But this is precisely what rulial space lets us talk about. What exact “geometry” the “space of strategies” has will depend on how we choose to coordinatize rulial space. But once again there will tend to be a certain level of coarse-graining achieved by looking only at the kinds of things one discusses in game theory—and at this level we can expect that all sorts of standard “structural” game-theoretic results will generically hold.

Personal Notes

Even as a kid I was never particularly into playing games or doing puzzles. And maybe it’s a sign I was always a bit too much of a scientist. Because just picking specific moves always seemed to me too arbitrary. To get my interest I needed a bigger picture, more of a coherent intellectual story. But now, in a sense, that’s just what the multicomputational approach to games and puzzles that I discuss here is bringing to us. Yes, it’s very “humanizing” to be able think about making particular moves. But the multicomputational approach immediately gives one a coherent global view that, at least to me, is intellectually much more satisfying.

The explorations I’ve discussed here can be thought of as originating from a single note in A New Kind of Science. In Chapter 5 of A New Kind of Science I had a section where I first introduced multiway systems. And as the very last note for that section I discussed “Game systems”:

Game systems
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I did the research for this in the 1990s—and indeed I now find a notebook from 1998 about tic-tac-toe with some of the same results derived here

Tic-tac-toe ListPlot
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together with a curious-looking graphical representation of the tic-tac-toe game graph:

Tic-tac-toe game graph
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But back at that time I didn’t conclude much from the game graphs I generated; they just seemed large and complicated. Twenty years passed and I didn’t think much more about this. But then in 2017 my son Christopher was playing with a puzzle called Rush Hour:

Rush Hour game

And perhaps in a sign of familial tendency he decided to construct its game graph—coming up with what to me seemed like a very surprising result:

Rush Hour game graph
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At the time I didn’t try to understand the structure one has here—but I still “filed this away” as evidence that game graphs can have “visible large-scale structure”.

A couple of years later—in late 2019—our Physics Project was underway and we’d realized that there are deep relations between quantum mechanics and multiway graphs. Quantum mechanics had always seemed like something mysterious—the abstract result of pure mathematical formalism. But seeing the connection to multiway systems began to suggest that one might actually be able to “understand quantum mechanics” as something that could “mechanically arise” from some concrete underlying structure.

I started to think about finding ways to explain quantum mechanics at an intuitive level. And for that I needed a familiar analogy: something everyday that one could connect to multiway systems. I immediately thought about games. And in September 2020 I decided to take a look at games to explore this analogy in more detail. I quickly analyzed games like tic-tac-toe and Nim—as well as simple sliding block puzzles and the Towers of Hanoi. But I wanted to explore more games and puzzles. And I had other projects to do, so the multicomputational analysis of games and puzzles got set aside. The Towers of Hanoi reappeared earlier this year, when I used it as an example of generating a proof-like multiway graph, in connection with my study of the physicalization of metamathematics. And finally, a few weeks ago I decided it was time to write down what I knew so far about games and puzzles—and produce what’s here.

Thanks

Thanks to Brad Klee and Ed Pegg for extensive help in the final stages of the analysis given here—as well as to Christopher Wolfram for inspiration in 2017, and help in 2020.

Will AIs Take All Our Jobs and End Human History—or Not? Well, It’s Complicated…

16 mars 2023 à 02:41

The Shock of ChatGPT

Just a few months ago writing an original essay seemed like something only a human could do. But then ChatGPT burst onto the scene. And suddenly we realized that an AI could write a passable human-like essay. So now it’s natural to wonder: How far will this go? What will AIs be able to do? And how will we humans fit in?

My goal here is to explore some of the science, technology—and philosophy—of what we can expect from AIs. I should say at the outset that this is a subject fraught with both intellectual and practical difficulty. And all I’ll be able to do here is give a snapshot of my current thinking—which will inevitably be incomplete—not least because, as I’ll discuss, trying to predict how history in an area like this will unfold is something that runs straight into an issue of basic science: the phenomenon of computational irreducibility.

But let’s start off by talking about that particularly dramatic example of AI that’s just arrived on the scene: ChatGPT. So what is ChatGPT? Ultimately, it’s a computational system for generating text that’s been set up to follow the patterns defined by human-written text from billions of webpages, millions of books, etc. Give it a textual prompt and it’ll continue in a way that’s somehow typical of what it’s seen us humans write.

The results (which ultimately rely on all sorts of specific engineering) are remarkably “human like”. And what makes this work is that whenever ChatGPT has to “extrapolate” beyond anything it’s explicitly seen from us humans it does so in ways that seem similar to what we as humans might do.

Inside ChatGPT is something that’s actually computationally probably quite similar to a brain—with millions of simple elements (“neurons”) forming a “neural net” with billions of connections that have been “tweaked” through a progressive process of training until they successfully reproduce the patterns of human-written text seen on all those webpages, etc. Even without training the neural net would still produce some kind of text. But the key point is that it won’t be text that we humans consider meaningful. To get such text we need to build on all that “human context” defined by the webpages and other materials we humans have written. The “raw computational system” will just do “raw computation”; to get something aligned with us humans requires leveraging the detailed human history captured by all those pages on the web, etc.

But so what do we get in the end? Well, it’s text that basically reads like it was written by a human. In the past we might have thought that human language was somehow a uniquely human thing to produce. But now we’ve got an AI doing it. So what’s left for us humans? Well, somewhere things have got to get started: in the case of text, there’s got to be a prompt specified that tells the AI “what direction to go in”. And this is the kind of thing we’ll see over and over again. Given a defined “goal”, an AI can automatically work towards achieving it. But it ultimately takes something beyond the raw computational system of the AI to define what us humans would consider a meaningful goal. And that’s where we humans come in.

What does this mean at a practical, everyday level? Typically we use ChatGPT by telling it—using text—what we basically want. And then it’ll fill in a whole essay’s worth of text talking about it. We can think of this interaction as corresponding to a kind of “linguistic user interface” (that we might dub a “LUI”). In a graphical user interface (GUI) there’s core content that’s being rendered (and input) through some potentially elaborate graphical presentation. In the LUI provided by ChatGPT there’s instead core content that’s being rendered (and input) through a textual (“linguistic”) presentation.

You might jot down a few “bullet points”. And in their raw form someone else would probably have a hard time understanding them. But through the LUI provided by ChatGPT those bullet points can be turned into an “essay” that can be generally understood—because it’s based on the “shared context” defined by everything from the billions of webpages, etc. on which ChatGPT has been trained.

There’s something about this that might seem rather unnerving. In the past, if you saw a custom-written essay you’d reasonably be able to conclude that a certain irreducible human effort was spent in producing it. But with ChatGPT this is no longer true. Turning things into essays is now “free” and automated. “Essayification” is no longer evidence of human effort.

Of course, it’s hardly the first time there’s been a development like this. Back when I was a kid, for example, seeing that a document had been typeset was basically evidence that someone had gone to the considerable effort of printing it on printing press. But then came desktop publishing, and it became basically free to make any document be elaborately typeset.

And in a longer view, this kind of thing is basically a constant trend in history: what once took human effort eventually becomes automated and “free to do” through technology. There’s a direct analog of this in the realm of ideas: that with time higher and higher levels of abstraction are developed, that subsume what were formerly laborious details and specifics.

Will this end? Will we eventually have automated everything? Discovered everything? Invented everything? At some level, we now know that the answer is a resounding no. Because one of the consequences of the phenomenon of computational irreducibility is that there’ll always be more computations to do—that can’t in the end be reduced by any finite amount of automation, discovery or invention.

Ultimately, though, this will be a more subtle story. Because while there may always be more computations to do, it could still be that we as humans don’t care about them. And that somehow everything we care about can successfully be automated—say by AIs—leaving “nothing more for us to do”.

Untangling this issue will be at the heart of questions about how we fit into the AI future. And in what follows we’ll see over and over again that what might at first essentially seem like practical matters of technology quickly get enmeshed with deep questions of science and philosophy.

Intuition from the Computational Universe

I’ve already mentioned computational irreducibility a couple of times. And it turns out that this is part of a circle of rather deep—and at first surprising—ideas that I believe are crucial to thinking about the AI future.

Most of our existing intuition about “machinery” and “automation” comes from a kind of “clockwork” view of engineering—in which we specifically build systems component by component to achieve objectives we want. And it’s the same with most software: we write it line by line to specifically do—step by step—whatever it is we want. And we expect that if we want our machinery—or software—to do complex things then the underlying structure of the machinery or software must somehow be correspondingly complex.

So when I started exploring the whole computational universe of possible programs in the early 1980s it was a big surprise to discover that things work quite differently there. And indeed even tiny programs—that effectively just apply very simple rules repeatedly—can generate great complexity. In our usual practice of engineering we haven’t seen this, because we’ve always specifically picked programs (or other structures) where we can readily foresee how they’ll behave, so that we can explicitly set them up to do what we want. But out in the computational universe it’s very common to see programs that just “intrinsically generate” great complexity, without us ever having to explicitly “put it in”.

And having discovered this, we realize that there’s actually a big example that’s been around forever: the natural world. And indeed it increasingly seems as if the “secret” that nature uses to make the complexity it so often shows is exactly to operate according to the rules of simple programs. (For about three centuries it seemed as if mathematical equations were the ultimate way to describe the natural world—but in the past few decades, and particularly poignantly with our recent Physics Project, it’s become clear that simple programs are in general a more powerful approach.)

How does all this relate to technology? Well, technology is about taking what’s out there in the world, and harnessing it for human purposes. And there’s a fundamental tradeoff here. There may be some system out in nature that does amazingly complex things. But the question is whether we can “slice off” certain particular things that we humans happen to find useful. A donkey has all sorts of complex things going on inside. But at some point it was discovered that we can use it “technologically” to do the rather simple thing of pulling a cart.

And when it comes to programs out in the computational universe it’s extremely common to see ones that do amazingly complex things. But the question is whether we can find some aspect of those things that’s useful to us. Maybe the program is good at making pseudorandomness. Or distributedly determining consensus. Or maybe it’s just doing its complex thing, and we don’t yet know any “human purpose” that this achieves.

One of the notable features of a system like ChatGPT is that it isn’t constructed in an “understand-every-step” traditional engineering way. Instead one basically just starts from a “raw computational system” (in the case of ChatGPT, a neural net), then progressively tweaks it until its behavior aligns with the “human-relevant” examples one has. And this alignment is what makes the system “technologically useful”—to us humans.

Underneath, though, it’s still a computational system, with all the potential “wildness” that implies. And free from the “technological objective” of “human-relevant alignment” the system might do all sorts of sophisticated things. But they might not be things that (at least at this time in history) we care about. Even though some putative alien (or our future selves) might.

OK, but let’s come back to the “raw computation” side of things. There’s something very different about computation from all other kinds of “mechanisms” we’ve seen before. We might have a cart that can move forward. And we might have a stapler that can put staples in things. But carts and staplers do very different things; there’s no equivalence between them. But for computational systems (at least ones that don’t just always behave in obviously simple ways) there’s my Principle of Computational Equivalence—which implies that all these systems are in a sense equivalent in the kinds of computations they can do.

This equivalence has many consequences. One of them is that one can expect to make something equally computationally sophisticated out of all sorts of different kinds of things—whether brain tissue or electronics, or some system in nature. And this is effectively where computational irreducibility comes from.

One might think that given, say, some computational system based on a simple program it would always be possible for us—with our sophisticated brains, mathematics, computers, etc.—to “jump ahead” and figure out what the system will do before it’s gone through all the steps to do it. But the Principle of Computational Equivalence implies that this won’t in general be possible—because the system itself can be as computationally sophisticated as our brains, mathematics, computers, etc. are. So this means that the system will be computationally irreducible: the only way to find out what it does is effectively just to go through the same whole computational process that it does.

There’s a prevailing impression that science will always eventually be able do better than this: that it’ll be able to make “predictions” that allow us to work out what will happen without having to trace through each step. And indeed over the past three centuries there’s been lots of success in doing this, mainly by using mathematical equations. But ultimately it turns out that this has only been possible because science has ended up concentrating on particular systems where these methods work (and then these systems have been used for engineering). But the reality is that many systems show computational irreducibility. And in the phenomenon of computational irreducibility science is in effect “deriving its own limitedness”.

Contrary to traditional intuition, try as we might, in many systems we’ll never be able find “formulas” (or other “shortcuts”) that describe what’s going to happen in the systems—because the systems are simply computationally irreducible. And, yes, this represents a limitation on science, and on knowledge in general. But while at first this might seem like a bad thing, there’s also something fundamentally satisfying about it. Because if everything were computationally reducible, we could always “jump ahead” and find out what will happen in the end, say in our lives. But computational irreducibility implies that in general we can’t do that—so that in some sense “something irreducible is being achieved” by the passage of time.

There are a great many consequences of computational irreducibility. Some—that I have particularly explored recently—are in the domain of basic science (for example, establishing core laws of physics as we perceive them from the interplay of computational irreducibility and our computational limitations as observers). But computational irreducibility is also central in thinking about the AI future—and in fact I increasingly feel that it adds the single most important intellectual element needed to make sense of many of the most important questions about the potential roles of AIs and humans in the future.

For example, from our traditional experience with engineering we’re used to the idea that to find out why something happened in a particular way we can just “look inside” a machine or program and “see what it did”. But when there’s computational irreducibility, that won’t work. Yes, we could “look inside” and see, say, a few steps. But computational irreducibility implies that to find out what happened, we’d have to trace through all the steps. We can’t expect to find a “simple human narrative” that “says why something happened”.

But having said this, one feature of computational irreducibility is that within any computationally irreducible systems there must always be (ultimately, infinitely many) “pockets of computational reducibility” to be found. So for example, even though we can’t say in general what will happen, we’ll always be able to identify specific features that we can predict. (“The leftmost cell will always be black”, etc.) And as we’ll discuss later we can potentially think of technological (as well as scientific) progress as being intimately tied to the discovery of these “pockets of reducibility”. And in effect the existence of infinitely many such pockets is the reason that “there’ll always be inventions and discoveries to be made”.

Another consequence of computational irreducibility has to do with trying to ensure things about the behavior of a system. Let’s say one wants to set up an AI so it’ll “never do anything bad”. One might imagine that one could just come up with particular rules that ensure this. But as soon as the behavior of the system (or its environment) is computationally irreducible one will never be able to guarantee what will happen in the system. Yes, there may be particular computationally reducible features one can be sure about. But in general computational irreducibility implies that there’ll always be a “possibility of surprise” or the potential for “unintended consequences”. And the only way to systematically avoid this is to make the system not computationally irreducible—which means it can’t make use of the full power of computation.

“AIs Will Never Be Able to Do That”

We humans like to feel special, and feel as if there’s something “fundamentally unique” about us. Five centuries ago we thought we lived at the center of the universe. Now we just tend to think that there’s something about our intellectual capabilities that’s fundamentally unique and beyond anything else. But the progress of AI—and things like ChatGPT—keep on giving us more and more evidence that that’s not the case. And indeed my Principle of Computational Equivalence says something even more extreme: that at a fundamental computational level there’s just nothing fundamentally special about us at all—and that in fact we’re computationally just equivalent to lots of systems in nature, and even to simple programs.

This broad equivalence is important in being able to make very general scientific statements (like the existence of computational irreducibility). But it also highlights how significant our specifics—our particular history, biology, etc.—are. It’s very much like with ChatGPT. We can have a generic (untrained) neural net with the same structure as ChatGPT, that can do certain “raw computation”. But what makes ChatGPT interesting—at least to us—is that it’s been trained with the “human specifics” described on billions of webpages, etc. In other words, for both us and ChatGPT there’s nothing computationally “generally special”. But there is something “specifically special”—and it’s the particular history we’ve had, particular knowledge our civilization has accumulated, etc.

There’s a curious analogy here to our physical place in the universe. There’s a certain uniformity to the universe, which means there’s nothing “generally special” about our physical location. But at least to us there’s still something “specifically special” about it, because it’s only here that we have our particular planet, etc. At a deeper level, ideas based on our Physics Project have led to the concept of the ruliad: the unique object that is the entangled limit of all possible computational processes. And we can then view our whole experience as “observers of the universe” as consisting of sampling the ruliad at a particular place.

It’s a bit abstract (and a long story, which I won’t go into in any detail here), but we can think of different possible observers as being both at different places in physical space, and at different places in rulial space—giving them different “points of view” about what happens in the universe. Human minds are in effect concentrated in a particular region of physical space (mostly on this planet) and a particular region of rulial space. And in rulial space different human minds—with their different experiences and thus different ways of thinking about the universe—are in slightly different places. Animal minds might be fairly close in rulial space. But other computational systems (like, say, the weather, which is sometimes said to “have a mind of its own”) are further away—as putative aliens might also be.

So what about AIs? It depends what we mean by “AIs”. If we’re talking about computational systems that are set up to do “human-like things” then that means they’ll be close to us in rulial space. But insofar as “an AI” is an arbitrary computational system it can be anywhere in rulial space, and it can do anything that’s computationally possible—which is far broader than what we humans can do, or even think about. (As we’ll talk about later, as our intellectual paradigms—and ways of observing things—expand, the region of rulial space in which we humans operate will correspondingly expand.)

But, OK, just how “general” are the computations that we humans (and the AIs that follow us) are doing? We don’t know enough about the brain to be sure. But if we look at artificial neural net systems—like ChatGPT—we can potentially get some sense. And in fact the computations really don’t seem to be that “general”. In most neural net systems data that’s given as input just “ripples once through the system” to produce output. It’s not like in a computational system like a Turing machine where there can be arbitrary “recirculation of data”. And indeed without such “arbitrary recirculation” the computation is necessarily quite “shallow” and can’t ultimately show computational irreducibility.

It’s a bit of a technical point, but one can ask whether ChatGPT, with its “re-feeding of text produced so far” can in fact achieve arbitrary (“universal”) computation. And I suspect that in some formal sense it can (or at least a sufficiently expanded analog of it can)—though by producing an extremely verbose piece of text that for example in effect lists successive (self-delimiting) states of a Turing machine tape, and in which finding “the answer” to a computation will take a bit of effort. But—as I’ve discussed elsewhere—in practice ChatGPT is presumably almost exclusively doing “quite shallow” computation.

It’s an interesting feature of the history of practical computing that what one might consider “deep pure computations” (say in mathematics or science) were done for decades before “shallow human-like computations” became feasible. And the basic reason for this is that for “human-like computations” (like recognizing images or generating text) one needs to capture lots of “human context”, which requires having lots of “human-generated data” and the computational resources to store and process it.

And, by the way, brains also seem to specialize in fundamentally shallow computations. And to do the kind of deeper computations that allow one to take advantage of more of what’s out there in the computational universe, one has to turn to computers. As we’ve discussed, there’s plenty out in the computational universe that we humans don’t (yet) care about: we just consider it “raw computation”, that doesn’t seem to be “achieving human purposes”. But as a practical matter it’s important to make a bridge between the things we humans do care about and think about, and what’s possible in the computational universe. And in a sense that’s at the core of the project I’ve put so much effort into in the Wolfram Language of creating a full-scale computational language that describes in computational terms the things we think about, and experience in the world.

OK, people have been saying for years: “It’s nice that computers can do A and B, but only humans can do X”. What X is supposed to be has changed—and narrowed—over the years. And ChatGPT provides us with a major unexpected new example of something more that computers can do.

So what’s left? People might say: “Computers can never show creativity or originality”. But—perhaps disappointingly—that’s surprisingly easy to get, and indeed just a bit of randomness “seeding” a computation can often do a pretty good job, as we saw years ago with our WolframTones music-generation system, and as we see today with ChatGPT’s writing. People might also say: “Computers can never show emotions”. But before we had a good way to generate human language we wouldn’t really have been able to tell. And now it already works pretty well to ask ChatGPT to write “happily”, “sadly”, etc. (In their raw form emotions in both humans and other animals are presumably associated with rather simple “global variables” like neurotransmitter concentrations.)

In the past people might have said: “Computers can never show judgement”. But by now there are endless examples of machine learning systems that do well at reproducing human judgement in lots of domains. People might also say: “Computers don’t show common sense”. And by this they typically mean that in a particular situation a computer might locally give an answer, but there’s a global reason why that answer doesn’t make sense, that the computer “doesn’t notice”, but a person would.

So how does ChatGPT do on this? Not too badly. In plenty of cases it correctly recognizes that “that’s not what I’ve typically read”. But, yes, it makes mistakes. Some of them have to do with it not being able to do—purely with its neural net—even slightly “deeper”computations. (And, yes, that’s something that can often be fixed by it calling Wolfram|Alpha as a tool.) But in other cases the problem seems to be that it can’t quite connect different domains well enough.

It’s perfectly capable of doing simple (“SAT-style”) analogies. But when it comes to larger-scale ones it doesn’t manage them. My guess, though, is that it won’t take much scaling up before it starts to be able to make what seem like very impressive analogies (that most of us humans would never even be able to make)—at which point it’ll probably successfully show broader “common sense”.

But so what’s left that humans can do, and AIs can’t? There’s—almost by definition—one fundamental thing: define what we would consider goals for what to do. We’ll talk more about this later. But for now we can note that any computational system, once “set in motion”, will just follow its rules and do what it does. But what “direction should it be pointed in”? That’s something that has to come from “outside the system”.

So how does it work for us humans? Well, our goals are in effect defined by the whole web of history—both from biological evolution and from our cultural development—in which we are embedded. But ultimately the only way to truly participate in that web of history is to be part of it.

Of course, we can imagine technologically emulating every “relevant” aspect of a brain—and indeed things like the success of ChatGPT may suggest that that’s easier to do than we might have thought. But that won’t be enough. To participate in the “human web of history” (as we’ll discuss later) we’ll have to emulate other aspects of “being human”—like moving around, being mortal, etc. And, yes, if we make an “artificial human” we can expect it (by definition) to show all the features of us humans.

But while we’re still talking about AIs as—for example—“running on computers” or “being purely digital” then, at least as far as we’re concerned, they’ll have to “get their goals from outside”. One day (as we’ll discuss) there will no doubt be some kind of “civilization of AIs”—which will form its own web of history. But at this point there’s no reason to think that we’ll still be able to describe what’s going on in terms of goals that we recognize. In effect the AIs will at that point have left our domain of rulial space. And—as we’ll discuss—they’ll be operating more like the kind of systems we see in nature, where we can tell there’s computation going on, but we can’t describe it, except rather anthropomorphically, in terms of human goals and purposes.

Will There Be Anything Left for the Humans to Do?

It’s been an issue that’s been raised—with varying degrees of urgency—for centuries: with the advance of automation (and now AI), will there eventually be nothing left for humans to do? Back in the early days of our species, there was lots of hard work of hunting and gathering to do, just to survive. But at least in the developed parts of the world, that kind of work is now at best a distant historical memory.

And yet at each stage in history—at least so far—there always seem to be other kinds of work that keep people busy. But there’s a pattern that increasingly seems to repeat. Technology in some way or another enables some new occupation. And eventually that occupation becomes widespread, and lots of people do it. But then there’s a technological advance, and the occupation gets automated—and people aren’t needed to do it anymore. But now there’s a new level of technology, that enables new occupations. And the cycle continues.

A century ago the increasingly widespread use of telephones meant that more and more people worked as switchboard operators. But then telephone switching was automated—and those switchboard operators weren’t needed anymore. But with automated switching there could be huge development of telecommunications infrastructure, opening up all sorts of new types of jobs, that in aggregate employ vastly more people than were ever switchboard operators.

Something somewhat similar happened with accounting clerks. Before there were computers, one needed to have people laboriously tallying up numbers. But with computers, that was all automated away. But with that automation came the ability to do more complex financial computations—which allowed for more complex financial transactions, more complex regulations, etc., which in turn led to all sorts of new types of jobs.

And across a whole range of industries, it’s been the same kind of story. Automation obsoletes some jobs, but enables others. There’s quite often a gap in time, and a change in the skills that are needed. But at least so far there always seems to have been a broad frontier of jobs that have been made possible—but haven’t yet been automated.

Will this at some point end? Will there come a time when everything we humans want (or at least need) is delivered automatically? Well, of course, that depends on what we want, and whether, for example, that evolves with what technology has made possible. But could we just decide that “enough is enough”; let’s stop here, and just let everything be automated?

I don’t think so. And the reason is ultimately because of computational irreducibility. We try to get the world to be “just so”, say set up so we’re “predictably comfortable”. Well, the problem is that there’s inevitably computational irreducibility in the way things develop—not just in nature, but in things like societal dynamics too. And that means that things won’t stay “just so”. There’ll always be something unpredictable that happens; something that the automation doesn’t cover.

At first we humans might just say “we don’t care about that”. But in time computational irreducibility will affect everything. So if there’s anything at all we care about (including, for example, not going extinct), we’ll eventually have to do something—and go beyond whatever automation was already set up.

It’s easy to find practical examples. We might think that when computers and people are all connected in a seamless automated network, there’d be nothing more to do. But what about the “unintended consequence” of computer security issues? What might have seemed like a case where “technology finished things” quickly creates a new kind of job for people to do. And at some level, computational irreducibility implies that things like this must always happen. There must always be a “frontier”. At least if there’s anything at all we want to preserve (like not going extinct).

But let’s come back to the situation here and now with AI. ChatGPT just automated all sorts of text-related tasks. It used to take lots of effort—and people—to write customized reports, letters, etc. But (at least so long as one’s dealing with situations where one doesn’t need 100% “correctness”) ChatGPT just automated a lot of that, so people aren’t needed for it anymore. But what will this mean? Well, it means that there’ll be a lot more customized reports, letters, etc. that can be produced. And that will lead to new kinds of jobs—managing, analyzing, validating etc. all that mass-customized text. Not to mention the need for prompt engineers (a job category that just didn’t exist until a few months ago), and what amount to AI wranglers, AI psychologists, etc.

But let’s talk about today’s “frontier” of jobs that haven’t been “automated away”. There’s one category that in many ways seems surprising to still be “with us”: jobs that involve lots of mechanical manipulation, like construction, fulfillment, food preparation, etc. But there’s a missing piece of technology here: there isn’t yet good general-purpose robotics (as there is general-purpose computing), and we humans still have the edge in dexterity, mechanical adaptability, etc. But I’m quite sure that in time—and perhaps quite suddenly—the necessary technology will be developed (and, yes, I have ideas about how to do it). And this will mean that most of today’s “mechanical manipulation” jobs will be “automated away”—and won’t need people to do them.

But then, just as in our other examples, this will mean that mechanical manipulation will become much easier and cheaper to do, and more of it will be done. Houses might routinely be built and dismantled. Products might routinely be picked up from wherever they’ve ended up, and redistributed. Vastly more ornate “food constructions” might become the norm. And each of these things—and many more—will open up new jobs.

But will every job that exists in the world today “on the frontier” eventually be automated? What about jobs where it seems like a large part of the value is just “having a human be there”? Jobs like flying a plane where one wants the “commitment” of the pilot being there in the plane. Caregiver jobs where one wants the “connection” of a human being there. Sales or education jobs where one wants “human persuasion” or “human encouragement”. Today one might think “only a human can make one feel that way”. But that’s typically based on the way the job is done now. And maybe there’ll be different ways found that allow the essence of the task to be automated, almost inevitably opening up new tasks to be done.

For example, something that in the past needed “human persuasion” might be “automated” by something like gamification—but then more of it can be done, with new needs for design, analytics, management, etc.

We’ve been talking about “jobs”. And that term immediately brings to mind wages, economics, etc. And, yes, plenty of what people do (at least in the world as it is today) is driven by issues of economics. But plenty is also not. There are things we “just want to do”—as a “social matter”, for “entertainment”, for “personal satisfaction”, etc.

Why do we want to do these things? Some of it seems intrinsic to our biological nature. Some of it seems determined by the “cultural environment” in which we find ourselves. Why might one walk on a treadmill? In today’s world one might explain that it’s good for health, lifespan, etc. But a few centuries ago, without modern scientific understanding, and with a different view of the significance of life and death, that explanation really wouldn’t work.

What drives such changes in our view of what we “want to do”, or “should do”? Some seems to be driven by the pure “dynamics of society”, presumably with its own computational irreducibility. But some has to do with our ways of interacting with the world—both the increasing automation delivered by the advance of technology, and the increasing abstraction delivered by the advance of knowledge.

And there seem to be similar “cycles” seen here as in the kinds of things we consider to be “occupations” or “jobs”. For a while something is hard to do, and serves as a good “pastime”. But then it gets “too easy” (“everybody now knows how to win at game X”, etc.), and something at a “higher level” takes its place.

About our “base” biologically driven motivations it doesn’t seem like anything has really changed in the course of human history. But there are certainly technological developments that could have an effect in the future. Effective human immortality, for example, would change many aspects of our motivation structure. As would things like the ability to implant memories or, for that matter, implant motivations.

For now, there’s a certain element of what we want to do that’s “anchored” by our biological nature. But at some point we’ll surely be able to emulate with a computer at least the essence of what our brains are doing (and indeed the success of things like ChatGPT makes it seems like the moment when that will happen is closer at hand than we might have thought). And at that point we’ll have the possibility of what amount to “disembodied human souls”.

To us today it’s very hard to imagine what the “motivations” of such a “disembodied soul” might be. Looked at “from the outside” we might “see the soul” doing things that “don’t make much sense” to us. But it’s like asking what someone from a thousand years ago would think about many of our activities today. These activities make sense to us today because we’re embedded in our whole “current framework”. But without that framework they don’t make sense. And so it will be for the “disembodied soul”. To us, what it does may not make sense. But to it, with its “current framework”, it will.

Could we “learn how to make sense of it”? There’s likely to be a certain barrier of computational irreducibility: in effect the only way to “understand the soul of the future” is to retrace its steps to get to where it is. So from our vantage point today, we’re separated by a certain “irreducible distance”, in effect in rulial space.

But could there be some science of the future that will at least tell us general things about how such “souls” behave? Even when there’s computational irreducibility we know that there will always be pockets of computational reducibility—and thus features of behavior that are predictable. But will those features be “interesting”, say from our vantage point today? Maybe some of them will be. Maybe they’ll show us some kind of metapsychology of souls. But inevitably they can only go so far. Because in order for those souls to even experience the passage of time there has to be computational irreducibility. If too much of what happens is too predictable, it’s as if “nothing is happening”—or at least nothing “meaningful”.

And, yes, this is all tied up with questions about “free will”. Even when there’s a disembodied soul that’s operating according to some completely deterministic underlying program, computational irreducibility means its behavior can still “seem free”—because nothing can “outrun it” and say what it’s going to be. And the “inner experience” of the disembodied soul can be significant: it’s “intrinsically defining its future”, not just “having its future defined for it”.

One might have assumed that once everything is just “visibly operating” as “mere computation” it would necessarily be “soulless” and “meaningless”. But computational irreducibility is what breaks out of this, and what allows there to be something irreducible and “meaningful” achieved. And it’s the same phenomenon whether one’s talking about our life now in the physical universe, or a future “disembodied” computational existence. Or in other words, even if absolutely everything—even our very existence—has been “automated by computation”, that doesn’t mean we can’t have a perfectly good “inner experience” of meaningful existence.

Generalized Economics and the Concept of Progress

If we look at human history—or, for that matter, the history of life on Earth—there’s a certain pervasive sense that there’s some kind of “progress” happening. But what fundamentally is this “progress”? One can view it as the process of things being done at a progressively “higher level”, so that in effect “more of what’s important” can happen with a given effort. This idea of “going to a higher level” takes many forms—but they’re all fundamentally about eliding details below, and being able to operate purely in terms of the “things one cares about”.

In technology, this shows up as automation, in which what used to take lots of detailed steps gets packaged into something that can be done “with the push of a button”. In science—and the intellectual realm in general—it shows up as abstraction, where what used to involve lots of specific details gets packaged into something that can be talked about “purely collectively”. And in biology it shows up as some structure (ribosome, cell, wing, etc.) that can be treated as a “modular unit”.

That it’s possible to “do things at a higher level” is a reflection of being able to find “pockets of computational reducibility”. And—as we mentioned above—the fact that (given underlying computational irreducibility) there are necessarily an infinite number of such pockets means that “progress can always go on forever”.

When it comes to human affairs we tend to value such progress highly, because (at least for now) we live finite lives, and insofar as we “want more to happen”, “progress” makes that possible. It’s certainly not self-evident that having more happen is “good”; one might just “want a quiet life”. But there is one constraint that in a sense originates from the deep foundations of biology.

If something doesn’t exist, then nothing can ever “happen to it”. So in biology, if one’s going to have anything “happen” with organisms, they’d better not be extinct. But the physical environment in which biological organisms exist is finite, with many resources that are finite. And given organisms with finite lives, there’s an inevitability to the process of biological evolution, and to the “competition” for resources between organisms.

Will there eventually be an “ultimate winning organism”? Well, no, there can’t be—because of computational irreducibility. There’ll in a sense always be more to explore in the computational universe—more “raw computational material for possible organisms”. And given any “fitness criterion” (like—in a Turing machine analog—“living longer before halting”) there’ll always be a way to “do better” with it.

One might still wonder, however, whether perhaps biological evolution—with its underlying process of random genetic mutation—could “get stuck” and never be able to discover some “way to do better”. And indeed simple models of evolution might give one the intuition that this would happen. But actual evolution seems more like deep learning with a large neural net—where one’s effectively operating in an extremely high-dimensional space where there’s typically always a “way to get there from here”, at least given enough time.

But, OK, so from our history of biological evolution there’s a certain built-in sense of “competition for scarce resources”. And this sense of competition has (so far) also carried over to human affairs. And indeed it’s the basic driver for most of the processes of economics.

But what if resources aren’t “scarce” anymore? What if progress—in the form of automation, or AI—makes it easy to “get anything one wants”? We might imagine robots building everything, AIs figuring everything out, etc. But there are still things that are inevitably scarce. There’s only so much real estate. Only one thing can be “the first ___”. And, in the end, if we have finite lives, we only have so much time.

Still, the more efficient—or high level—the things we do (or have) are, the more we’ll be able to get done in the time we have. And it seems as if what we perceive as “economic value” is intimately connected with “making things higher level”. A finished phone is “worth more” than its raw materials. An organization is “worth more” than its separate parts. But what if we could have “infinite automation”? Then in a sense there’d be “infinite economic value everywhere”, and one might imagine there’d be “no competition left”.

But once again computational irreducibility stands in the way. Because it tells us there’ll never be “infinite automation”, just as there’ll never be an ultimate winning biological organism. There’ll always be “more to explore” in the computational universe, and different paths to follow.

What will this look like in practice? Presumably it’ll lead to all sorts of diversity. So that, for example, a chart of “what the components of an economy are” will become more and more fragmented; it won’t just be “the single winning economic activity is ___”.

There is one potential wrinkle in this picture of unending progress. What if nobody cares? What if the innovations and discoveries just don’t matter, say to us humans? And, yes, there is of course plenty in the world that at any given time in history we don’t care about. That piece of silicon we’ve been able to pick out? It’s just part of a rock. Well, until we start making microprocessors out of it.

But as we’ve discussed, as soon as we’re “operating at some level of abstraction” computational irreducibility makes it inevitable that we’ll eventually be exposed to things that “require going beyond that level”.

But then—critically—there will be choices. There will be different paths to explore (or “mine”) in the computational universe—in the end infinitely many of them. And whatever the computational resources of AIs etc. might be, they’ll never be able to explore all of them. So something—or someone—will have to make a choice of which ones to take.

Given a particular set of things one cares about at a particular point, one might successfully be able to automate all of them. But computational irreducibility implies there will always be a “frontier”, where choices have to be made. And there’s no “right answer”; no “theoretically derivable” conclusion. Instead, if we humans are involved, this is where we get to define what’s going to happen.

How will we do that? Well, ultimately it’ll be based on our history—biological, cultural, etc. We’ll get to use all that irreducible computation that went into getting us to where we are to define what to do next. In a sense it’ll be something that goes “through us”, and that uses what we are. It’s the place where—even when there’s automation all around—there’s still always something us humans can “meaningfully” do.

How Can We Tell the AIs What to Do?

Let’s say we want an AI (or any computational system) to do a particular thing. We might think we could just set up its rules (or “program it”) to do that thing. And indeed for certain kinds of tasks that works just fine. But the deeper the use we make of computation, the more we’re going to run into computational irreducibility, and the less we’ll be able to know how to set up particular rules to achieve what we want.

And then, of course, there’s the question of defining what “we want” in the first place. Yes, we could have specific rules that say what particular pattern of bits should occur at a particular point in a computation. But that probably won’t have much to do with the kind of overall “human-level” objective that we typically care about. And indeed for any objective we can even reasonably define, we’d better be able to coherently “form a thought” about it. Or, in effect, we’d better have some “human-level narrative” to describe it.

But how can we represent such a narrative? Well, we have natural language—probably the single most important innovation in the history of our species. And what natural language fundamentally does is to allow us to talk about things at a “human level”. It’s made of words that we can think of as representing “human-level packets of meaning”. And so, for example, the word “chair” represents the human-level concept of a chair. It’s not referring to some particular arrangement of atoms. Instead, it’s referring to any arrangement of atoms that we can usefully conflate into the single human-level concept of a chair, and from which we can deduce things like the fact that we can expect to sit on it, etc.

So, OK, when we’re “talking to an AI” can we expect to just say what we want using natural language? We can definitely get a certain distance—and indeed ChatGPT helps us get further than ever before. But as we try to make things more precise we run into trouble, and the language we need rapidly becomes increasingly ornate, as in the “legalese” of complex legal documents. So what can we do? If we’re going to keep things at the level of “human thoughts” we can’t “reach down” into all the computational details. But yet we want a precise definition of how what we might say can be implemented in terms of those computational details.

Well, there’s a way to deal with this, and it’s one that I’ve personally devoted many decades to: it’s the idea of computational language. When we think about programming languages, they’re things that operate solely at the level of computational details, defining in more or less the native terms of a computer what the computer should do. But the point of a true computational language (and, yes, in the world today the Wolfram Language is the sole example) is to do something different: to define a precise way of talking in computational terms about things in the world (whether concretely countries or minerals, or abstractly computational or mathematical structures).

Out in the computational universe, there’s immense diversity in the “raw computation” that can happen. But there’s only a thin sliver of it that we humans (at least currently) care about and think about. And we can view computational language as defining a bridge between the things we think about and what’s computationally possible. The functions in our computational language (7000 or so of them in the Wolfram Language) are in effect like words in a human language—but now they have a precise grounding in the “bedrock” of explicit computation. And the point is to design the computational language so it’s convenient for us humans to think and express ourselves in (like a vastly expanded analog of mathematical notation), but so it can also be precisely implemented in practice on a computer.

Given a piece of natural language it’s often possible to give a precise, computational interpretation of it—in computational language. And indeed this is exactly what happens in Wolfram|Alpha. Give a piece of natural language and the Wolfram|Alpha NLU system will try to find an interpretation of it as computational language. And from this interpretation, it’s then up to the Wolfram Language to do the computation that’s specified, and give back the results—and potentially synthesize natural language to express them.

As a practical matter, this setup is useful not only for humans, but also for AIs—like ChatGPT. Given a system that produces natural language, the Wolfram|Alpha NLU system can “catch” natural language it is “thrown”, and interpret it as computational language that precisely specifies a potentially irreducible computation to do.

With both natural language and computational language one’s basically “directly saying what one wants”. But an alternative approach—more aligned with machine learning—is just to give examples, and (implicitly or explicitly) say “follow these”. Inevitably there has to be some underlying model for how to do that following—typically in practice just defined by “what a neural net with a certain architecture will do”. But will the result be “right”? Well, the result will be whatever the neural net gives. But typically we’ll tend to consider it “right” if it’s somehow consistent with what we humans would have concluded. And in practice this often seems to happen, presumably because the actual architecture of our brains is somehow similar enough to the architecture of the neural nets we’re using.

But what if we want to “know for sure” what’s going to happen—or, for example, that some particular “mistake” can never be made? Well then we’re presumably thrust back into computational irreducibility, with the result that there’s no way to know, for example, whether a particular set of training examples can lead to a system that’s capable of doing (or not doing) some particular thing.

OK, but let’s say we’re setting up some AI system, and we want to make sure it “doesn’t do anything bad”. There are several levels of issues here. The first is to decide what we mean by “anything bad”. And, as we’ll discuss below, that in itself is very hard. But even if we could abstractly figure this out, how should we actually express it? We could give examples—but then the AI will inevitably have to “extrapolate” from them, in ways we can’t predict. Or we could describe what we want in computational language. It might be difficult to cover “every case” (as it is in present-day human laws, or complex contracts). But at least we as humans can read what we’re specifying. Though even in this case, there’s an issue of computational irreducibility: that given the specification it won’t be possible to work out all its consequences.

What does all this mean? In essence it’s just a reflection of the fact that as soon as there’s “serious computation” (i.e. irreducible computation) involved, one isn’t going to be immediately able to say what will happen. (And in a sense that’s inevitable, because if one could say, it would mean the computation wasn’t in fact irreducible.) So, yes, we can try to “tell AIs what to do”. But it’ll be like many systems in nature (or, for that matter, people): you can set them on a path, but you can’t know for sure what will happen; you just have to wait and see.

A World Run by AIs

In the world today, there are already plenty of things that are being done by AIs. And, as we’ve discussed, there’ll surely be more in the future. But who’s “in charge”? Are we telling the AIs what to do, or are they telling us? Today it’s at best a mixture: AIs suggest content for us (for example from the web), and in general make all sorts of recommendations about what we should do. And no doubt in the future those recommendations will be even more extensive and tightly coupled to us: we’ll be recording everything we do, processing it with AI, and continually annotating with recommendations—say through augmented reality—everything we see. And in some sense things might even go beyond “recommendations”. If we have direct neural interfaces, then we might be making our brains just “decide” they want to do things, so that in some sense we become pure “puppets of the AI”.

And beyond “personal recommendations” there’s also the question of AIs running the systems we use, or in fact running the whole infrastructure of our civilization. Today we ultimately expect people to make large-scale decisions for our world—often operating in systems of rules defined by laws, and perhaps aided by computation, and even what one might call AI. But there may well come a time when it seems as if AIs could just “do a better job than humans”, say at running a central bank or waging a war.

One might ask how one would ever know if the AI would “do a better job”. Well, one could try tests, and run examples. But once again one’s faced with computational irreducibility. Yes, the particular tests one tries might work fine. But one can’t ultimately predict everything that could happen. What will the AI do if there’s suddenly a never-before-seen seismic event? We basically won’t know until it happens.

But can we be sure the AI won’t do anything “crazy”? Could we—with some definition of “crazy”—effectively “prove a theorem” that the AI can never do that? For any realistically nontrivial definition of crazy we’ll again run into computational irreducibility—and this won’t be possible.

Of course, if we’ve put a person (or even a group of people) “in charge” there’s also no way to “prove” that they won’t do anything “crazy”—and history shows that people in charge quite often have done things that, at least in retrospect, we consider “crazy”. But even though at some level there’s no more certainty about what people will do than about what AIs might do, we still get a certain comfort when people are in charge if we think that “we’re in it together”, and that if something goes wrong those people will also “feel the effects”.

But still, it seems inevitable that lots of decisions and actions in the world will be taken directly by AIs. Perhaps it’ll be because this will be cheaper. Perhaps the results (based on tests) will be better. Or perhaps, for example, things will just have to be done too quickly and in numbers too large for us humans to be in the loop.

But, OK, if a lot of what happens in our world is happening through AIs, and the AIs are effectively doing irreducible computations, what will this be like? We’ll be in a situation where things are “just happening” and we don’t quite know why. But in a sense we’ve very much been in this situation before. Because it’s what happens all the time in our interaction with nature.

Processes in nature—like, for example, the weather—can be thought of as corresponding to computations. And much of the time there’ll be irreducibility in those computations. So we won’t be able to readily predict them. Yes, we can do natural science to figure out some aspects of what’s going to happen. But it’ll inevitably be limited.

And so we can expect it to be with the “AI infrastructure” of the world. Things are happening in it—as they are in the weather—that we can’t readily predict. We’ll be able to say some things—though perhaps in ways that are closer to psychology or social science than to traditional exact science. But there’ll be surprises—like maybe some strange AI analog of a hurricane or an ice age. And in the end all we’ll really be able to do is to try to build up our human civilization so that such things “don’t fundamentally matter” to it.

In a sense the picture we have is that in time there’ll be a whole “civilization of AIs” operating—like nature—in ways that we can’t readily understand. And like with nature, we’ll coexist with it.

But at least at first we might think there’s an important difference between nature and AIs. Because we imagine that we don’t “pick our natural laws”—yet insofar as we’re the ones building the AIs we imagine we can “pick their laws”. But both parts of this aren’t quite right. Because in fact one of the implications of our Physics Project is precisely that the laws of nature that we perceive are the way they are because we are observers who are the way we are. And on the AI side, computational irreducibility implies that we can’t expect to be able to determine the final behavior of the AIs just from knowing the underlying laws we gave them.

But what will the “emergent laws” of the AIs be? Well, just like in physics, it’ll depend on how we “sample” the behavior of the AIs. If we look down at the level of individual bits, it’ll be like looking at molecular dynamics (or the behavior of atoms of space). But typically we won’t do this. And just like in physics, we’ll operate as computationally bounded observers—measuring only certain aggregated features of an underlying computationally irreducible process. But what will the “overall laws of AIs” be like? Maybe they’ll show close analogies to physics. Or maybe they’ll seem more like psychological theories (superegos for AIs?). But we can expect them in many ways to be like large-scale laws of nature of the kind we know.

Still, there’s one more difference between at least our interaction with nature and with AIs. Because we have in effect been “co-evolving” with nature for billions of years—yet AIs are “new on the scene”. And through our co-evolution with nature we’ve developed all sorts of structural, sensory and cognitive features that allow us to “interact successfully” with nature. But with AIs we don’t have these. So what does this mean?

Well, our ways of interacting with nature can be thought of as leveraging pockets of computational reducibility that exist in natural processes—to make things seem at least somewhat predictable to us. But without having found such pockets for AIs, we’re likely to be faced with much more “raw computational irreducibility”—and thus much more unpredictability. It’s been a conceit of modern times that—particularly with the help of science—we’ve been able to make more and more of our world predictable to us, though in practice a large part of what’s led to this is the way we’ve built and controlled the environment in which we live, and the things we choose to do.

But for the new “AI world”, we’re effectively starting from scratch. And to make things predictable in that world may be partly a matter of some new science, but perhaps more importantly a matter of choosing how we set up our “way of life” around the AIs there. (And, yes, if there’s lots of unpredictability we may be back to more ancient points of view about the importance of fate—or we may view AIs as a bit like the Olympians of Greek mythology, duking it out among themselves and sometimes having an effect on mortals.)

Governance in an AI World

Let’s say the world is effectively being run by AIs, but let’s assume that we humans have at least some control over what they do. Then what principles should we have them follow? And what, for example, should their “ethics” be?

Well, the first thing to say is that there’s no ultimate, theoretical “right answer” to this. There are many ethical and other principles that AIs could follow. And it’s basically just a choice which ones should be followed.

When we talk about “principles” and “ethics” we tend to think more in terms of constraints on behavior than in terms of rules for generating behavior. And that means we’re dealing with something more like mathematical axioms, where we ask things like what theorems are true according to those axioms, and what are not. And that means there can be issues like whether the axioms are consistent—and whether they’re complete, in the sense that they can “determine the ethics of anything”. But now, once again, we’re face to face with computational irreducibility, here in the form of Gödel’s theorem and its generalizations.

And what this means is that it’s in general undecidable whether any given set of principles is inconsistent, or incomplete. One might “ask an ethical question”, and find that there’s a “proof chain” of unbounded length to determine what the answer to that question is within one’s specified ethical system, or whether there is even a consistent answer.

One might imagine that somehow one could add axioms to “patch up” whatever issues there are. But Gödel’s theorem basically says that it’ll never work. It’s the same story as so often with computational irreducibility: there’ll always be “new situations” that can arise, that in this case can’t be captured by a finite set of axioms.

OK, but let’s imagine we’re picking a collection of principles for AIs. What criteria could we use to do it? One might be that these principles won’t inexorably lead to a simple state—like one where the AIs are extinct, or have to keep looping doing the same thing forever. And there may be cases where one can readily see that some set of principles will lead to such outcomes. But most of the time, computational irreducibility (here in the form of things like the halting problem) will once again get in the way, and one won’t be able to tell what will happen, or successfully pick “viable principles” this way.

So this means that there are going to be a wide range of principles that we could in theory pick. But presumably what we’ll want is to pick ones that make AIs give us humans some sort of “good time”, whatever that might mean.

And a minimal idea might be to get AIs just to observe what we humans do, and then somehow imitate this. But most people wouldn’t consider this the right thing. They’d point out all the “bad” things people do. And they’d perhaps say “let’s have the AIs follow not what we actually do, but what we aspire to do”.

But where should we get these aspirations from? Different people, and different cultures, can have very different aspirations—with very different resulting principles. So whose should we pick? And, yes, there are pitifully few—if any—principles that we truly find in common everywhere. (Though, for example, the major religions all tend to share things like respect for human life, the Golden Rule, etc.)

But do we in fact have to pick one set of principles? Maybe some AIs can have some principles, and some can have others. Maybe it should be like different countries, or different online communities: different principles for different groups or in different places.

Right now that doesn’t seem plausible, because technological and commercial forces have tended to make it seem as if powerful AIs always have to be centralized. But I expect that this is just a feature of the present time, and not something intrinsic to any “human-like” AI.

So could everyone (and maybe every organization) have “their own AI” with its own principles? For some purposes this might work OK. But there are many situations where AIs (or people) can’t really act independently, and where there have to be “collective decisions” made.

Why is this? In some cases it’s because everyone is in the same physical environment. In other cases it’s because if there’s to be social cohesion—of the kind needed to support even something like a language that’s useful for communication—then there has to be certain conceptual alignment.

It’s worth pointing out, though, that at some level having a “collective conclusion” is effectively just a way of introducing certain computational reducibility to make it “easier to see what to do”. And potentially it can be avoided if one has enough computation capability. For example, one might assume that there has to be a collective conclusion about which side of the road cars should drive on. But that wouldn’t be true if every car had the computation capability to just compute a trajectory that would for example optimally weave around other cars using both sides of the road.

But if we humans are going to be in the loop, we presumably need a certain amount of computational reducibility to make our world sufficiently comprehensible to us that we can operate in it. So that means there’ll be collective—“societal”—decisions to make. We might want to just tell the AIs to “make everything as good as it can be for us”. But inevitably there will be tradeoffs. Making a collective decision one way might be really good for 99% of people, but really bad for 1%; making it the other way might be pretty good for 60%, but pretty bad for 40%. So what should the AI do?

And, of course, this is a classic problem of political philosophy, and there’s no “right answer”. And in reality the setup won’t be as clean as this. It may be fairly easy to work out some immediate effects of different courses of action. But inevitably one will eventually run into computational irreducibility—and “unintended consequences”—and so one won’t be able to say with certainty what the ultimate effects (good or bad) will be.

But, OK, so how should one actually make collective decisions? There’s no perfect answer, but in the world today, democracy in one form or another is usually viewed as the best option. So how might AI affect democracy—and perhaps improve on it? Let’s assume first that “humans are still in charge”, so that it’s ultimately their preferences that matter. (And let’s also assume that humans are more or less in their “current form”: unique and unreplicable discrete entities that believe they have independent minds.)

The basic setup for current democracy is computationally quite simple: discrete votes (or perhaps rankings) are given (sometimes with weights of various kinds), and then numerical totals are used to determine the winner (or winners). And with past technology this was pretty much all that could be done. But now there are some new elements. Imagine not casting discrete votes, but instead using computational language to write a computational essay to describe one’s preferences. Or imagine having a conversation with a linguistically enabled AI that can draw out and debate one’s preferences, and eventually summarize them in some kind of feature vector. Then imagine feeding computational essays or feature vectors from all “voters” to some AI that “works out the best thing to do”.

Well, there are still the same political philosophy issues. It’s not like 60% of people voted for A and 40% for B, so one chose A. It’s much more nuanced. But one still won’t be able to make everyone happy all the time, and one has to have some base principles to know what to do about that.

And there’s a higher-order problem in having an AI “rebalance” collective decisions all the time based on everything it knows about people’s detailed preferences (and perhaps their actions too): for many purposes—like us being able to “keep track of what’s going on”—it’s important to maintain consistency over time. But, yes, one could deal with this by having the AI somehow also weigh consistency in figuring out what to do.

But while there are no doubt ways in which AI can “tune up” democracy, AI doesn’t seem—in and of itself—to deliver any fundamentally new solution for making collective decisions, and for governance in general.

And indeed, in the end things always seem to come down to needing some fundamental set of principles about how one wants things to be. Yes, AIs can be the ones to implement these principles. But there are many possibilities for what the principles could be. And—at least if we humans are “in charge”—we’re the ones who are going to have to come up with them.

Or, in other words, we need to come up with some kind of “AI constitution”. Presumably this constitution should basically be written in precise computational language (and, yes, we’re trying to make it possible for the Wolfram Language to be used), but inevitably (as yet another consequence of computational irreducibility) there’ll be “fuzzy” definitions and distinctions, that will rely on things like examples, “interpolated” by systems like neural nets. Maybe when such a constitution is created, there’ll be multiple “renderings” of it, which can all be applied whenever the constitution is used, with some mechanism for picking the “overall conclusion”. (And, yes, there’s potentially a certain “observer-dependent” multicomputational character to this.)

But whatever its detailed mechanisms, what should the AI constitution say? Different people and groups of people will definitely come to different conclusions about it. And presumably—just as there are different countries, etc. today with different systems of laws—there’ll be different groups that want to adopt different AI constitutions. (And, yes, the same issues about collective decision making apply again when those AI constitutions have to interact.)

But given an AI constitution, one has a base on which AIs can make decisions. And on top of this one imagines a huge network of computational contracts that are autonomously executed, essentially to “run the world”.

And this is perhaps one of those classic “what could possibly go wrong?” moments. An AI constitution has been agreed on, and now everything is being run efficiently and autonomously by AIs that are following it. Well, once again, computational irreducibility rears its head. Because however carefully the AI constitution is drafted, computational irreducibility implies that one won’t be able to foresee all its consequences: “unexpected” things will always happen—and some of them will undoubtedly be things “one doesn’t like”.

In human legal systems there’s always a mechanism for adding “patches”—filling in laws or precedents that cover new situations that have come up. But if everything is being autonomously run by AIs there’s no room for that. Yes, we as humans might characterize “bad things that happen” as “bugs” that could be fixed by adding a patch. But the AI is just supposed to be operating—essentially axiomatically—according to its constitution, so it has no way to “see that it’s a bug”.

Similar to what we discussed above, there’s an interesting analogy here with human law versus natural law. Human law is something we define and can modify. Natural law is something the universe just provides us (notwithstanding the issues about observers discussed above). And by “setting an AI constitution and letting it run” we’re basically forcing ourselves into a situation where the “civilization of the AIs” is some “independent stratum” in the world, that we essentially have to take as it is, and adapt to.

Of course, one might wonder if the AI constitution could “automatically evolve”, say based on what’s actually seen to happen in the world. But one quickly returns to the exact same issues of computational irreducibility, where one can’t predict whether the evolution will be “right”, etc.

So far, we’ve assumed that in some sense “humans are in charge”. But at some level that’s an issue for the AI constitution to define. It’ll have to define whether AIs have “independent rights”—just like humans (and, in many legal systems, some other entities too). Closely related to the question of independent rights for AIs is whether an AI can be considered autonomously “responsible for its actions”—or whether such responsibility must always ultimately rest with the (presumably human) creator or “programmer” of the AI.

Once again, computational irreducibility has something to say. Because it implies that the behavior of the AI can go “irreducibly beyond” what its programmer defined. And in the end (as we discussed above) this is the same basic mechanism that allows us humans to effectively have “free will” even when we’re ultimately operating according to deterministic underlying natural laws. So if we’re going to claim that we humans have free will, and can be “responsible for our actions” (as opposed to having our actions always “dictated by underlying laws”) then we’d better claim the same for AIs.

So just as a human builds up something irreducible and irreplaceable in the course of their life, so can an AI. As a practical matter, though, AIs can presumably be backed up, copied, etc.—which isn’t (yet) possible for humans. So somehow their individual instances don’t seem as valuable, even if the “last copy” might still be valuable. As humans, we might want to say “those AIs are something inferior; they shouldn’t have rights”. But things are going to get more entangled. Imagine a bot that no longer has an identifiable owner but that’s successfully befriending people (say on social media), and paying for its underlying operation from donations, ads, etc. Can we reasonably delete that bot? We might argue that “the bot can feel no pain”—but that’s not true of its human friends. But what if the bot starts doing “bad” things? Well, then we’ll need some form of “bot justice”—and pretty soon we’ll find ourselves building a whole human-like legal structure for the AIs.

So Will It End Badly?

OK, so AIs will learn what they can from us humans, then they’ll fundamentally just be running as autonomous computational systems—much like nature runs as an autonomous computational system—sometimes “interacting with us”. What will they “do to us”? Well, what does nature “do to us”? In a kind of animistic way, we might attribute intentions to nature, but ultimately it’s just “following its rules” and doing what it does. And so it will be with AIs. Yes, we might think we can set things up to determine what the AIs will do. But in the end—insofar as the AIs are really making use of what’s possible in the computational universe—there’ll inevitably be computational irreducibility, and we won’t be able to foresee what will happen, or what consequences it will have.

So will the dynamics of AIs in fact have “bad” effects—like, for example, wiping us out? Well, it’s perfectly possible nature could wipe us out too. But one has the feeling that—extraterrestrial “accidents” aside—the natural world around us is at some level enough in some kind of “equilibrium” that nothing too dramatic will happen. But AIs are something new. So maybe they’ll be different.

And one possibility might be that AIs could “improve themselves” to produce a single “apex intelligence” that would in a sense dominate everything else. But here we can see computational irreducibility as coming to the rescue. Because it implies that there can never be a “best at everything” computational system. It’s a core result of the emerging field of metabiology: that whatever “achievement” you specify, there’ll always be a computational system somewhere out there in the computational universe that will exceed it. (A simple example is that there’s always a Turing machine that can be found that will exceed any upper bound you specify on the time it takes to halt.)

So what this means is that there’ll inevitably be a whole “ecosystem” of AIs—with no single winner. Of course, while that might be an inevitable final outcome, it might not be what happens in the shorter term. And indeed the current tendency to centralize AI systems has a certain danger of AI behavior becoming “unstabilized” relative to what it would be with a whole ecosystem of “AIs in equilibrium”.

And in this situation there’s another potential concern as well. We humans are the product of a long struggle for life played out over the course of the history of biological evolution. And insofar as AIs inherit our attributes we might expect them to inherit a certain “drive to win”—perhaps also against us. And perhaps this is where the AI constitution becomes important: to define a “contract” that supersedes what AIs might “naturally” inherit from effectively observing our behavior. Eventually we can expect the AIs to “independently reach equilibrium”. But in the meantime, the AI constitution can help break their connection with our “competitive” history of biological evolution.

Preparing for an AI World

We’ve talked quite a bit about the ultimate future course of AIs, and their relation to us humans. But what about the short term? How today can we prepare for the growing capabilities and uses of AIs?

As has been true throughout history, people who use tools tend to do better than those who don’t. Yes, you can go on doing by direct human effort what has now been successfully automated, but except in rare cases you’ll increasingly be left behind. And what’s now emerging is an extremely powerful combination of tools: neural-net-style AI for “immediate human-like tasks”, along with computational language for deeper access to the computational universe and computational knowledge.

So what should people do with this? The highest leverage will come from figuring out new possibilities—things that weren’t possible before but have now “come into range” as a result of new capabilities. And as we discussed above, this is a place where we humans are inevitably central contributors—because we’re the ones who must define what we consider has value for us.

So what does this mean for education? What’s worth learning now that so much has been automated? I think the fundamental answer is how to think as broadly and deeply as possible—calling on as much knowledge and as many paradigms as possible, and particularly making use of the computational paradigm, and ways of thinking about things that directly connect with what computation can help with.

In the course of human history a lot of knowledge has been accumulated. But as ways of thinking have advanced, it’s become unnecessary to learn directly that knowledge in all its detail: instead one can learn things at a higher level, abstracting out many of the specific details. But in the past few decades something fundamentally new has come on the scene: computers and the things they enable.

For the first time in history, it’s become realistic to truly automate intellectual tasks. The leverage this provides is completely unprecedented. And we’re only just starting to come to terms with what it means for what and how we should learn. But with all this new power there’s a tendency to think something must be lost. Surely it must still be worth learning all those intricate details—that people in the past worked so hard to figure out—of how to do some mathematical calculation, even though Mathematica has been able to do it automatically for more than a third of a century?

And, yes, at the right time it can be interesting to learn those details. But in the effort to understand and best make use of the intellectual achievements of our civilization, it makes much more sense to leverage the automation we have, and treat those calculations just as “building blocks” that can be put together in “finished form” to do whatever it is we want to do.

One might think this kind of leveraging of automation would just be important for “practical purposes”, and for applying knowledge in the real world. But actually—as I have personally found repeatedly to great benefit over the decades—it’s also crucial at a conceptual level. Because it’s only through automation that one can get enough examples and experience that one’s able to develop the intuition needed to reach a higher level of understanding.

Confronted with the rapidly growing amount of knowledge in the world there’s been a tremendous tendency to assume that people must inevitably become more and more specialized. But with increasing success in the automation of intellectual tasks—and what we might broadly call AI—it becomes clear there’s an alternative: to make more and more use of this automation, so people can operate at a higher level, “integrating” rather than specializing.

And in a sense this is the way to make the best use of our human capabilities: to let us concentrate on setting the “strategy” of what we want to do—delegating the details of how to do it to automated systems that can do it better than us. But, by the way, the very fact that there’s an AI that knows how to do something will no doubt make it easier for humans to learn how to do it too. Because—although we don’t yet have the complete story—it seems inevitable that with modern techniques AIs will be able to successfully “learn how people learn”, and effectively present things an AI “knows” in just the right way for any given person to absorb.

So what should people actually learn? Learn how to use tools to do things. But also learn what things are out there to do—and learn facts to anchor how you think about those things. A lot of education today is about answering questions. But for the future—with AI in the picture—what’s likely to be more important is to learn how to ask questions, and how to figure out what questions are worth asking. Or, in effect, how to lay out an “intellectual strategy” for what to do.

And to be successful at this, what’s going to be important is breadth of knowledge—and clarity of thinking. And when it comes to clarity of thinking, there’s again something new in modern times: the concept of computational thinking. In the past we’ve had things like logic, and mathematics, as ways to structure thinking. But now we have something new: computation.

Does that mean everyone should “learn to program” in some traditional programming language? No. Traditional programming languages are about telling computers what to do in their terms. And, yes, lots of humans do this today. But it’s something that’s fundamentally ripe for direct automation (as examples with ChatGPT already show). And what’s important for the long term is something different. It’s to use the computational paradigm as a structured way to think not about the operation of computers, but about both things in the world and abstract things.

And crucial to this is having a computational language: a language for expressing things using the computational paradigm. It’s perfectly possible to express simple “everyday things” in plain, unstructured natural language. But to build any kind of serious “conceptual tower” one needs something more structured. And that’s what computational language is about.

One can see a rough historical analog in the development of mathematics and mathematical thinking. Up until about half a millennium ago, mathematics basically had to be expressed in natural language. But then came mathematical notation—and from it a more streamlined approach to mathematical thinking, that eventually made possible all the various mathematical sciences. And it’s now the same kind of thing with computational language and the computational paradigm. Except that it’s a much broader story, in which for basically every field or occupation “X” there’s a “computational X” that’s emerging.

In a sense the point of computational language (and all my efforts in the development of the Wolfram Language) is to be able to let people get “as automatically as possible” to computational X—and to let people express themselves using the full power of the computational paradigm.

Something like ChatGPT provides “human-like AI” in effect by piecing together existing human material (like billions of words of human-written text). But computational language lets one tap directly into computation—and gives the ability to do fundamentally new things, that immediately leverage our human capabilities for defining intellectual strategy.

And, yes, while traditional programming is likely to be largely obsoleted by AI, computational language is something that provides a permanent bridge between human thinking and the computational universe: a channel in which the automation is already done in the very design (and implementation) of the language—leaving in a sense an interface directly suitable for humans to learn, and to use as a basis to extend their thinking.

But, OK, what about the future of discovery? Will AIs take over from us humans in, for example, “doing science”? I, for one, have used computation (and many things one might think of as AI) as a tool for scientific discovery for nearly half a century. And, yes, many of my discoveries have in effect been “made by computer”. But science is ultimately about connecting things to human understanding. And so far it’s taken a human to knit what the computer finds into the whole web of human intellectual history.

One can certainly imagine, though, that an AI—even one rather like ChatGPT—could be quite successful in taking a “raw computational discovery” and “explaining” how it might relate to existing human knowledge. One could also imagine that the AI would be successful at identifying what aspects of some system in the world could be picked out to describe in some formal way. But—as is typical for the process of modeling in general—a key step is to decide “what one cares about”, and in effect in what direction to go in extending one’s science. And this—like so much else—is inevitably tied into the specifics of the goals we humans set ourselves.

In the emerging AI world there are plenty of specific skills that won’t make sense for (most) humans to learn—just as today the advance of automation has obsoleted many skills from the past. But—as we’ve discussed—we can expect there to “be a place” for humans. And what’s most important for us humans to learn is in effect how to pick “where next to go”—and where, out of all the infinite possibilities in the computational universe, we should take human civilization.

Afterword: Looking at Some Actual Data

OK, so we’ve talked quite a bit about what might happen in the future. But what about actual data from the past? For example, what’s been the actual history of the evolution of jobs? Conveniently, in the US, the Census Bureau has records of people’s occupations going back to 1850. Of course, many job titles have changed since then. Switchmen (on railroads), chainmen (in surveying) and sextons (in churches) aren’t really things anymore. And telemarketers, aircraft pilots and web developers weren’t things in 1850. But with a bit of effort, it’s possible to more or less match things up—at least if one aggregates into large enough categories.

So here are pie charts of different job categories at 50-year intervals:

And, yes, in 1850 the US was firmly an agricultural economy, with just over half of all jobs being in agriculture. But as agriculture got more efficient—with the introduction of machinery, irrigation, better seeds, fertilizers, etc.—the fraction dropped dramatically, to just a few percent today.

After agriculture, the next biggest category back in 1850 was construction (along with other real-estate-related jobs, mainly maintenance). And this is a category that for a century and a half hasn’t changed much in size (at least so far), presumably because, even though there’s been greater automation, this has just allowed buildings to be more complex.

Looking at the pie charts above, we can see a clear trend towards greater diversification in jobs (and indeed the same thing is seen in the development of other economies around the world). It’s an old theory in economics that increasing specialization is related to economic growth, but from our point of view here, we might say that the very possibility of a more complex economy, with more niches and jobs, is a reflection of the inevitable presence of computational irreducibility, and the complex web of pockets of computational reducibility that it implies.

Beyond the overall distribution of job categories, we can also look at trends in individual categories over time—with each one in a sense providing a certain window onto history:

One can definitely see cases where the number of jobs decreases as a result of automation. And this happens not only in areas like agriculture and mining, but also for example in finance (fewer clerks and bank tellers), as well as in sales and retail (online shopping). Sometimes—as in the case of manufacturing—there’s a decrease of jobs partly because of automation, and partly because the jobs move out of the US (mainly to countries with lower labor costs).

There are cases—like military jobs—where there are clear “exogenous” effects. And then there are cases like transportation+logistics where there’s a steady increase for more than half a century as technology spreads and infrastructure gets built up—but then things “saturate”, presumably at least partly as a result of increased automation. It’s a somewhat similar story with what I’ve called “technical operations”—with more “tending to technology” needed as technology becomes more widespread.

Another clear trend is an increase in job categories associated with the world becoming an “organizationally more complicated place”. Thus we see increases in management, as well as administration, government, finance and sales (which all have recent decreases as a result of computerization). And there’s also a (somewhat recent) increase in legal.

Other areas with increases include healthcare, engineering, science and education—where “more is known and there’s more to do” (as well as there being increased organizational complexity). And then there’s entertainment, and food+hospitality, with increases that one might attribute to people leading (and wanting) “more complex lives”. And, of course, there’s information technology which takes off from nothing in the mid-1950s (and which had to be rather awkwardly grafted into the data we’re using here).

So what can we conclude? The data seems quite well aligned with what we discussed in more general terms above. Well-developed areas get automated and need to employ fewer people. But technology also opens up new areas, which employ additional people. And—as we might expect from computational irreducibility—things generally get progressively more complicated, with additional knowledge and organizational structure opening up more “frontiers” where people are needed. But even though there are sometimes “sudden inventions”, it still always seems to take decades (or effectively a generation) for there to be any dramatic change in the number of jobs. (The few sharp changes visible in the plots seem mostly to be associated with specific economic events, and—often related—changes in government policies.)

But in addition to the different jobs that get done, there’s also the question of how individual people spend their time each day. And—while it certainly doesn’t live up to my own (rather extreme) level of personal analytics—there’s a certain amount of data on this that’s been collected over the years (by getting time diaries from randomly sampled people) in the American Heritage Time Use Study. So here, for example, are plots based on this survey for how the amount of time spent on different broad activities has varied over the decades (the main line shows the mean—in hours—for each activity; the shaded areas indicate successive deciles):

And, yes, people are spending more time on “media & computing”, some mixture of watching TV, playing videogames, etc. Housework, at least for women, takes less time, presumably mostly as a result of automation (appliances, etc.). (“Leisure” is basically “hanging out” as well as hobbies and social, cultural, sporting events, etc.; “Civic” includes volunteer, religious, etc. activities.)

If one looks specifically at people who are doing paid work

one notices several things. First, the average number of hours worked hasn’t changed much in half a century, though the distribution has broadened somewhat. For people doing paid work, media & computing hasn’t increased significantly, at least since the 1980s. One category in which there is systematic increase (though the total time still isn’t very large) is exercise.

What about people who—for one reason or another—aren’t doing paid work? Here are corresponding results in this case:

Not so much increase in exercise (though the total times are larger to begin with), but now a significant increase in media & computing, with the average recently reaching nearly 6 hours per day for men—perhaps as a reflection of “more of life going online”.

But looking at all these results on time use, I think the main conclusion that over the past half century, the ways people (at least in the US) spend their time have remained rather stable—even as we’ve gone from a world with almost no computers to a world in which there are more computers than people.

Wolfram|Alpha as the Way to Bring Computational Knowledge Superpowers to ChatGPT

9 janvier 2023 à 23:22

Wolfram|Alpha as the Way to Bring Computational Knowledge Superpowers to ChatGPT

ChatGPT and Wolfram|Alpha

It’s always amazing when things suddenly “just work”. It happened to us with Wolfram|Alpha back in 2009. It happened with our Physics Project in 2020. And it’s happening now with OpenAI’s ChatGPT.

I’ve been tracking neural net technology for a long time (about 43 years, actually). And even having watched developments in the past few years I find the performance of ChatGPT thoroughly remarkable. Finally, and suddenly, here’s a system that can successfully generate text about almost anything—that’s very comparable to what humans might write. It’s impressive, and useful. And, as I discuss elsewhere, I think its success is probably telling us some very fundamental things about the nature of human thinking.

But while ChatGPT is a remarkable achievement in automating the doing of major human-like things, not everything that’s useful to do is quite so “human like”. Some of it is instead more formal and structured. And indeed one of the great achievements of our civilization over the past several centuries has been to build up the paradigms of mathematics, the exact sciences—and, most importantly, now computation—and to create a tower of capabilities quite different from what pure human-like thinking can achieve.

I myself have been deeply involved with the computational paradigm for many decades, in the singular pursuit of building a computational language to represent as many things in the world as possible in formal symbolic ways. And in doing this my goal has been to build a system that can “computationally assist”—and augment—what I and others want to do. I think about things as a human. But I can also immediately call on Wolfram Language and Wolfram|Alpha to tap into a kind of unique “computational superpower” that lets me do all sorts of beyond-human things.

It’s a tremendously powerful way of working. And the point is that it’s not just important for us humans. It’s equally, if not more, important for human-like AIs as well—immediately giving them what we can think of as computational knowledge superpowers, that leverage the non-human-like power of structured computation and structured knowledge.

We’ve just started exploring what this means for ChatGPT. But it’s pretty clear that wonderful things are possible. Wolfram|Alpha does something very different from ChatGPT, in a very different way. But they have a common interface: natural language. And this means that ChatGPT can “talk to” Wolfram|Alpha just like humans do—with Wolfram|Alpha turning the natural language it gets from ChatGPT into precise, symbolic computational language on which it can apply its computational knowledge power.

For decades there’s been a dichotomy in thinking about AI between “statistical approaches” of the kind ChatGPT uses, and “symbolic approaches” that are in effect the starting point for Wolfram|Alpha. But now—thanks to the success of ChatGPT—as well as all the work we’ve done in making Wolfram|Alpha understand natural language—there’s finally the opportunity to combine these to make something much stronger than either could ever achieve on their own.

A Basic Example

At its core, ChatGPT is a system for generating linguistic output that “follows the pattern” of what’s out there on the web and in books and other materials that have been used in its training. And what’s remarkable is how human-like the output is, not just at a small scale, but across whole essays. It has coherent things to say, that pull in concepts it’s learned, quite often in interesting and unexpected ways. What it produces is always “statistically plausible”, at least at a linguistic level. But—impressive as that ends up being—it certainly doesn’t mean that all the facts and computations it confidently trots out are necessarily correct.

Here’s an example I just noticed (and, yes, ChatGPT has intrinsic built-in randomness, so if you try this, you probably won’t get the same result):

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It sounds pretty convincing. But it turns out that it’s wrong, as Wolfram|Alpha can tell us:

How far is it from Chicago to Tokyo?

To be fair, of course, this is exactly the kind of thing that Wolfram|Alpha is good at: something that can be turned into a precise computation that can be done on the basis of its structured, curated knowledge.

But the neat thing is that one can think about Wolfram|Alpha automatically helping ChatGPT on this. One can programmatically ask Wolfram|Alpha the question (you can also use a web API, etc.):

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Now ask the question again to ChatGPT, appending this result:

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ChatGPT very politely takes the correction, and if you ask the question yet again it then gives the correct answer. Obviously there could be a more streamlined way to handle the back and forth with Wolfram|Alpha, but it’s nice to see that even this very straightforward pure-natural-language approach basically already works.

But why does ChatGPT get this particular thing wrong in the first place? If it had seen the specific distance between Chicago and Tokyo somewhere in its training (e.g. from the web), it could of course get it right. But this is a case where the kind of generalization a neural net can readily do—say from many examples of distances between cities—won’t be enough; there’s an actual computational algorithm that’s needed.

The way Wolfram|Alpha handles things is quite different. It takes natural language and then—assuming it’s possible—it converts this into precise computational language (i.e. Wolfram Language), in this case:

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The coordinates of cities and algorithms to compute distances between them are then part of the built-in computational knowledge in the Wolfram Language. And, yes, the Wolfram Language has a huge amount of built-in computational knowledge—the result of decades of work on our part, carefully curating what’s now a vast amount of continually updated data, implementing (and often inventing) methods and models and algorithms—and systematically building up a whole coherent computational language for everything.

A Few More Examples

ChatGPT and Wolfram|Alpha work in very different ways, and have very different strengths. But in the interests of understanding where ChatGPT can take advantage of Wolfram|Alpha’s strengths, let’s discuss some cases where on its own ChatGPT doesn’t do quite the right thing. And one area where ChatGPT—like humans—often tends to struggle is math.

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It’s an interesting, essay-style response. But the actual result is wrong:

What is 3 to the power 73?

But if ChatGPT “consulted” Wolfram|Alpha it’d of course be able to get it right.

Let’s try something slightly more complex:

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At first glance, this result looks great, and I’d be inclined to believe it. It turns out, though, that it’s wrong, as Wolfram|Alpha can tell us:

Circumference of an ellipse with half axes 3 and 12

And, yes, doing math homework with ChatGPT (without it being able to consult Wolfram|Alpha) is probably a bad idea. It can give you a very plausible answer:

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But without “really understanding the math” it’s basically impossible for ChatGPT to reliably get the right answer. And in this case, the answer is again wrong:

What is the integral of x^2 cos(2x)

Still, ChatGPT can even make up a very plausible-looking explanation of “how it got its answer” (not that it’s in any way how it really “did it”). And, rather charmingly (and interestingly), the explanation it gives has mistakes very similar to what a human who didn’t understand the math might also make:

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There are all sorts of situations where “not really understanding what things mean” can cause trouble:

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That sounds convincing. But it’s not correct:

What are the largest countries in Central America?

ChatGPT seemed to have correctly learned this underlying data somewhere—but it doesn’t “understand what it means” enough to be able to correctly rank the numbers:

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And, yes, one can imagine finding a way to “fix this particular bug”. But the point is that the fundamental idea of a generative-language-based AI system like ChatGPT just isn’t a good fit in situations where there are structured computational things to do. Put another way, it’d take “fixing” an almost infinite number of “bugs” to patch up what even an almost-infinitesimal corner of Wolfram|Alpha can achieve in its structured way.

And the more complex the “computational chain” gets, the more likely you’ll have to call on Wolfram|Alpha to get it right. Here ChatGPT produces a rather confused answer:

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And, as Wolfram|Alpha tells us, its conclusion isn’t correct (as it already in a sense “knew”):

What planetary moons are larger than Mercury

Whenever it comes to specific (e.g. quantitative) data—even in fairly raw form—things very often tend to have to be more of a “Wolfram|Alpha story”. Here’s an example, inspired by a longtime favorite Wolfram|Alpha test query “How many turkeys are there in Turkey?”:

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Again, this seems (at first) totally plausible, and it’s even quoting a relevant source. Turns out, though, that this data is basically just “made up”:

Livestock populations in Turkey

Still, what’s very nice is that ChatGPT can easily be made to “ask for facts to check”:

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Now feed these through the Wolfram|Alpha API:

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Now we can ask ChatGPT to fix its original response, injecting this data (and even showing in bold where it did it):

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The ability to “inject facts” is particularly nice when it comes to things involving real-time (or location etc. dependent) data or computation. ChatGPT won’t immediately answer this:

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But here’s some relevant Wolfram|Alpha API output:

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And if we feed this to ChatGPT, it’ll generate a nice “essay-style” result:

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Sometimes there’s an interesting interplay between the computational and the human like. Here’s a rather whimsical question asked of Wolfram|Alpha (and it even checks if you want “soft-serve” instead):

How many calories are there in a cubic light year of ice cream?

ChatGPT at first gets a bit confused about the concept of volume:

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But then it seems to “realize” that that much ice cream is fairly silly:

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The Path Forward

Machine learning is a powerful method, and particularly over the past decade, it’s had some remarkable successes—of which ChatGPT is the latest. Image recognition. Speech to text. Language translation. In each of these cases, and many more, a threshold was passed—usually quite suddenly. And some task went from “basically impossible” to “basically doable”.

But the results are essentially never “perfect”. Maybe something works well 95% of the time. But try as one might, the other 5% remains elusive. For some purposes one might consider this a failure. But the key point is that there are often all sorts of important use cases for which 95% is “good enough”. Maybe it’s because the output is something where there isn’t really a “right answer” anyway. Maybe it’s because one’s just trying to surface possibilities that a human—or a systematic algorithm—will then pick from or refine.

It’s completely remarkable that a few-hundred-billion-parameter neural net that generates text a token at a time can do the kinds of things ChatGPT can. And given this dramatic—and unexpected—success, one might think that if one could just go on and “train a big enough network” one would be able to do absolutely anything with it. But it won’t work that way. Fundamental facts about computation—and notably the concept of computational irreducibility—make it clear it ultimately can’t. But what’s more relevant is what we’ve seen in the actual history of machine learning. There’ll be a big breakthrough (like ChatGPT). And improvement won’t stop. But what’s much more important is that there’ll be use cases found that are successful with what can be done, and that aren’t blocked by what can’t.

And yes, there’ll be plenty of cases where “raw ChatGPT” can help with people’s writing, make suggestions, or generate text that’s useful for various kinds of documents or interactions. But when it comes to setting up things that have to be perfect, machine learning just isn’t the way to do it—much as humans aren’t either.

And that’s exactly what we’re seeing in the examples above. ChatGPT does great at the “human-like parts”, where there isn’t a precise “right answer”. But when it’s “put on the spot” for something precise, it often falls down. But the whole point here is that there’s a great way to solve this problem—by connecting ChatGPT to Wolfram|Alpha and all its computational knowledge “superpowers”.

Inside Wolfram|Alpha, everything is being turned into computational language, and into precise Wolfram Language code, that at some level has to be “perfect” to be reliably useful. But the crucial point is that ChatGPT doesn’t have to generate this. It can produce its usual natural language, and then Wolfram|Alpha can use its natural language understanding capabilities to translate that natural language into precise Wolfram Language.

In many ways, one might say that ChatGPT never “truly understands” things; it just “knows how to produce stuff that’s useful”. But it’s a different story with Wolfram|Alpha. Because once Wolfram|Alpha has converted something to Wolfram Language, what it’s got is a complete, precise, formal representation, from which one can reliably compute things. Needless to say, there are plenty of things of “human interest” for which we don’t have formal computational representations—though we can still talk about them, albeit it perhaps imprecisely, in natural language. And for these, ChatGPT is on its own, with its very impressive capabilities.

But just like us humans, there are times when ChatGPT needs a more formal and precise “power assist”. But the point is that it doesn’t have to be “formal and precise” in saying what it wants. Because Wolfram|Alpha can communicate with it in what amounts to ChatGPT’s native language—natural language. And Wolfram|Alpha will take care of “adding the formality and precision” when it converts to its native language—Wolfram Language. It’s a very good situation, that I think has great practical potential.

And that potential is not only at the level of typical chatbot or text generation applications. It extends to things like doing data science or other forms of computational work (or programming). In a sense, it’s an immediate way to get the best of both worlds: the human-like world of ChatGPT, and the computationally precise world of Wolfram Language.

What about ChatGPT directly learning Wolfram Language? Well, yes, it could do that, and in fact it’s already started. And in the end I fully expect that something like ChatGPT will be able to operate directly in Wolfram Language, and be very powerful in doing so. It’s an interesting and unique situation, made possible by the character of the Wolfram Language as a full-scale computational language that can talk broadly about things in the world and elsewhere in computational terms.

The whole concept of the Wolfram Language is to take things we humans think about, and be able to represent and work with them computationally. Ordinary programming languages are intended to provide ways to tell computers specifically what to do. The Wolfram Language—in its role as a full-scale computational language—is about something much larger than that. In effect, it’s intended to be a language in which both humans and computers can “think computationally”.

Many centuries ago, when mathematical notation was invented, it provided for the first time a streamlined medium in which to “think mathematically” about things. And its invention soon led to algebra, and calculus, and ultimately all the various mathematical sciences. The goal of the Wolfram Language is to do something similar for computational thinking, though now not just for humans—and to enable all the “computational X” fields that can be opened up by the computational paradigm.

I myself have benefitted greatly from having Wolfram Language as a “language to think in”, and it’s been wonderful to see over the past few decades so many advances being made as a result of people “thinking in computational terms” through the medium of Wolfram Language. So what about ChatGPT? Well, it can get into this too. Quite how it will all work I am not yet sure. But it’s not about ChatGPT learning how to do the computation that the Wolfram Language already knows how to do. It’s about ChatGPT learning how to use the Wolfram Language more like people do. It’s about ChatGPT coming up with the analog of “creative essays”, but now written not in natural language but in computational language.

I’ve long discussed the concept of computational essays written by humans—that communicate in a mixture of natural language and computational language. Now it’s a question of ChatGPT being able to write those—and being able to use Wolfram Language as a way to deliver “meaningful communication”, not just to humans, but also to computers. And, yes, there’s a potentially interesting feedback loop involving actual execution of the Wolfram Language code. But the crucial point is that the richness and flow of “ideas” represented by the Wolfram Language code is—unlike in an ordinary programming language—something much closer to the kind of thing that ChatGPT has “magically” managed to work with in natural language.

Or, put another way, Wolfram Language—like natural language–is something expressive enough that one can imagine writing a meaningful “prompt” for ChatGPT in it. Yes, Wolfram Language can be directly executed on a computer. But as a ChatGPT prompt it can be used to “express an idea” whose “story” could be continued. It might describe some computational structure, leaving ChatGPT to “riff” on what one might computationally say about that structure that would—according to what it’s learned by reading so many things written by humans—be “interesting to humans”.

There are all sorts of exciting possibilities, suddenly opened up by the unexpected success of ChatGPT. But for now there’s the immediate opportunity of giving ChatGPT computational knowledge superpowers through Wolfram|Alpha. So it can not just produce “plausible human-like output”, but output that leverages the whole tower of computation and knowledge that’s encapsulated in Wolfram|Alpha and the Wolfram Language.

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