Leveraging Ideas from Physics

The multicomputational paradigm is something that’s emerging from our Physics Project, and from thinking about fundamental physics. But one of the most powerful things about having a general paradigm for theoretical science is that it implies a certain unity across different areas of science—and by providing a common framework it allows results and intuitions developed in one area to be transferred to others. So with its roots in fundamental physics the multicomputational paradigm immediately gets to leverage the ideas and successes of physics—and in effect use them to illuminate other areas. But just how does the multicomputational paradigm work in physics? And how did it even arise there? Well, it’s not something that the traditional mathematical approach to physics would readily lead one to. And instead what basically happened is that having seen how successful the computational paradigm was in studying lots of kinds of systems I started wondering whether something like it might apply to fundamental physics. It was fairly clear, though, that the ordinary computational paradigm—especially with its “global” view of time—wasn’t a great match for what we already knew about things like relativity in physics. But the pivotal idea that eventually led inexorably to the multicomputational paradigm was a hint from the computational paradigm about the nature of space.

The traditional view in physics had been that space is something continuous, that serves just as a kind of “mathematical source of coordinates”. But in the computational paradigm one tends to imagine that everything is ultimately made of discrete computational elements. So in particular Stephen Wolfram began to think that this might be true of space.

But how would the elements of space behave? The computational approach would suggest that there must be “finitely specifiable” rules—that effectively define “update events” involving limited numbers of elements of space. But right here is where the multicomputational idea comes in. Because inevitably—across all the elements of space in our universe—there must be a huge number of different ways these updating events could be applied. And the result is that there is no just one unique “computational history”—but instead a whole multiway system with different threads of history for different sequences of updating events. As we’ll discuss later, it’s the updating events—and the relations between them—that are in a sense really the most fundamental things in the multicomputational paradigm. But in understanding the multicomputational paradigm, and its way of representing fundamental physics, it’s helpful to start instead by thinking about what the updating events act on, or, in effect, the “data structure of the universe”.

A convenient way to set this up is to imagine that the universe—or, in particular, space and everything in it—is defined by a large number of relations between the elements of space. Representing each element of space by an integer, one might have a collection of (in this case, binary) relations like […]

which can in turn be thought of as a defining a graph (or, in general a Hypergraph): […]

In our model of fundamental physics, the presence of many different branching and merging paths is a reflection of quantum mechanics—with every path in effect representing a history for the universe.

And the concept is that after a large number of steps of such a process we’ll get a recognizable representation of an “instantaneous state of space” in the universe. But what about time? Ultimately it’s the individual updating events that define the progress of time. Representing updating events by nodes, we can now draw a causal graph that shows the “causal relationships” between these updating events—with each edge representing the fact that the “output” from one event is being “consumed” as “input” by another event: […]

And as is characteristic of the multicomputational paradigm this causal graph reflects the fact that there are many possible sequences in which updating events can occur. But how does this jibe with our everyday impression that a definite sequence of things happen in the universe? The basic point is that we don’t perceive the whole causal graph in all its detail. Instead, as computationally bounded observers, we just pick some particular reference frame from which to perceive what’s going on. And this reference frame defines a sequence of global “time slices” such as: […]

Each “time slice” contains a collection of events that—with our reference frame—we take to be “happening simultaneously”. And we can then trace the “steps in the evolution of the universe” by seeing the results of all updating events in successive time slices: […]

But how do we determine what reference frame to use? The underlying rule determines the structure of the causal graph, and what event can follow what other one. But it still allows huge freedom in the choice of reference frame—in effect imposing only the constraint that if one event follows another, then these events must appear in that order in the time slices defined by the reference frame: […]

In general each of these different choices of reference frame will lead to a different sequence of “instantaneous states of space”. And in principle one could imagine that some elaborately chosen reference frame could lead to arbitrarily pathological perceived behavior. But in practice there is an important constraint on possible reference frames: as computationally bounded observers we are limited in the amount of computational effort that we can put into the construction of the reference frame. And in general to achieve a “pathological result” we’ll typically have to “reverse engineer” the underlying computational irreducibility of the system—which we won’t be able to do with a reference frame constructed by a computationally bounded observer. (This is directly analogous to the result in the ordinary computational paradigm that computationally bounded observers effectively can’t avoid perceiving the validity of the Second Law of Thermodynamics.)

So, OK, what then will an observer perceive in a system like the one we’ve defined? With a variety of caveats the basic answer is that in the limit of a “sufficiently large universe” they’ll perceive average behavior that’s simple enough to describe mathematically, and specifically to describe as following Einstein’s equations from general relativity. And the key point is that this is in a sense a generic result (a bit like the gas laws in thermodynamics) that’s independent of the details of the underlying rule.

At the core of the concept of a reference frame is the idea of being able to treat certain things (typically events) as somehow “equivalent”. In the case of the causal graphs we’ve discussed so far, what we’re doing is to treat certain events as equivalent in the sense that they can be viewed as happening “in the same time slice” or effectively “simultaneously”. But if we just pick two events at random, there’s no guarantee that it’ll be consistent to consider them to be in the same time slice.

In particular, if one event causally depends on another (in the sense that its input requires output from the other), then it can only occur in a later time slice. And in this situation (which corresponds to one event being reachable from the other by following directed edges in the causal graph) we can say that these events are “timelike separated”. Similarly, if two events can occur in the same time slice, we can say that they are “spacelike separated”. And in the language of relativity, this means that our “time slices” are spacelike hypersurfaces in spacetime—or at least discrete analogs of them.

So what about with the full multiway graph? We can look at every event that occurs in every state in the multiway graph. And there are then basically three kinds of separation between events. There can be timelike separation, in the sense that one event causally depends on another. There can be spacelike separation, in the sense that different events occur in different parts of space that are not causally connected. And then there’s a third case, which is that different events can occur on different branches of the multiway graph—in which case we say that they’re branchlike separated.

And in general when we pick a reference frame in the full multiway system, we can have time slices that contain both spacelike- and branchlike-separated events. What’s the significance of this? Basically, just as spacelike separation is associated with the concept of ordinary space, branchlike separation is associated with a different kind of space, that we call Branchial Space.

With a multiway graph of the kind we’ve drawn above (in which every node represents a possible “complete state of the universe”), we can investigate “pure branchial space” by looking at time slices in the graph: […]

For example, we can construct “branchial graphs” by looking at which states are connected by having immediate common ancestors. And in effect these branchial graphs are the branchial-space analogs of the hypergraphs we’ve constructed to represent the instantaneous state of ordinary space: […]

But now, instead of representing ordinary space—with features like general relativity and gravity—they represent something different: they represent a space of quantum states, with the branchial graph effectively being a map of quantum entanglements.

But to define a branchial graph, we have to pick the analog of a reference frame: we have to say what branchlike-separated events we consider to happen “at the same time”. In the case of spacelike-separated events it’s fairly easy to interpret that what we get from a reference frame is a view of what’s happening throughout space at a particular time. But what’s the analog for branchlike-separated events?

In effect what we’re doing when we make a reference frame is to treat as equivalent events that are happening “on different branches of history”. At first, that may seem like a very odd thing to do. But the thing to understand is that as entities embedded in the same universe that’s generating all these different branches of history, we too are branching. So it’s really a question of how a “Branching Brain” will perceive a “branching universe”. And that depends on what reference frame (or “quantum observation frame”) we pick. But as soon as we insist that we maintain a single thread of experience, or, equivalently, that we sequentialize time, then—together with computational boundedness—this puts all sorts of constraints on the reference frames we pick.

And just as in the case of ordinary space, the result is that it ultimately seems to be possible to give a fairly simple—and essentially mathematical—description of what the observer will perceive. And the answer is that it basically appears to correspond to quantum mechanics. But there’s actually more to it. What we get is a kind of generic multicomputational result—that doesn’t depend on the details of underlying rules or particular choices of reference frames. Structurally it’s basically the same result as for ordinary space. But now it’s interpreted in terms of branchial space, quantum states, and so on. And what was interpreted as the geodesic equation of general relativity now essentially gets interpreted as the path integral of quantum mechanics. In a sense it’s then a basic consequence of the multicomputational nature of fundamental physics that quantum mechanics is the same theory as general relativity—though operating in branchial space rather than ordinary space. There are important implications here for physics. But there are also general implications for all multicomputational systems. Because the sophisticated definitions and phenomena of both general relativity and quantum mechanics we can now expect will have analogs in any system that can be modeled in a multicomputational way, whatever field of science it may come from. So, later, when we talk about the application of the multicomputational paradigm to other fields, we can expect to talk and reason in terms of things we know from physics. So we’ll be able to bring in light cones, inertial frames, time dilation, black holes, the uncertainty principle, and much more. In effect, the common use of the multicomputational paradigm will allow us to leverage the development of physics—and its status as the most advanced current area of theoretical science—to “physicalize” all sorts of other areas, and shed new light on them. As well, of course, as taking ideas and intuition from other areas (including ones much closer to everyday experience) and “applying them back” to physics.

⇒ The Formal Structure of Multicomputation