Chapter 12: The Principle of Computational Equivalence

Section 6: Computational Irreducibility

History [of computational irreducibility]

The notion that there could be fundamental limits to knowledge or predictability has been discussed repeatedly since antiquity. But most often it has been assumed that the origin of this must be inadequacy in models, not difficulty in working out their consequences. And indeed already in the 1500s with the introduction of symbolic algebra and the discovery of formulas for solving cubic and quartic equations the expectation began to develop that with sufficient cleverness it should be possible to derive a formula for the solution to any purely mathematical problem. Infinitesimals were sometimes thought to get in the way of finite understanding—but this was believed to be overcome by calculus. And when mathematical models for natural systems became widespread in the late 1600s it was generally assumed that their basic consequences could always be found in terms of formulas or geometrical theorems, perhaps with fairly straightforward numerical calculations required for connection to practical situations. In discussing gravitational interactions between many planets Isaac Newton did however comment in 1684 that "to define these motions by exact laws admitting of easy calculation exceeds, if I am not mistaken, the force of any human mind". But in the course of the 1700s and 1800s formulas were successfully found for solutions to a great many problems in mathematical physics (see note below)—at least when suitable special functions (see page 1091) were introduced. The three-body problem (see page 972) nevertheless continued to resist efforts at general solution. In the 1820s it was shown that quintic equations cannot in general be solved in terms of radicals (see page 1137), and by the 1890s it was known that degree 7 equations cannot in general be solved even if elliptic functions are allowed. Around 1890 it was then shown that the three-body problem could not be solved in general in terms of ordinary algebraic functions and integrals (see page 972). However, perhaps in part because of a shift towards probabilistic theories such as quantum and statistical mechanics there remained the conviction that for relevant aspects of behavior formulas should still exist. The difficulty for example of finding more than a few exact solutions to the equations of general relativity was noted—but a steady stream of results (see note below) maintained the belief that with sufficient cleverness a formula could be found for behavior according to any model.

In the 1950s computers began to be used to work out numerical solutions to equations—but this was seen mostly as a convenience for applications, not as a reflection of any basic necessity. A few computer experiments were done on systems with simple underlying rules, but partly because Monte Carlo methods were sometimes used, it was typically assumed that their results were just approximations to what could in principle be represented by exact formulas. And this view was strengthened in the 1960s when solitons given by simple formulas were found in some of these systems.

The difficulty of solving equations for numerical weather prediction was noted even in the 1920s. And by the 1950s and 1960s the question of whether computer calculations would be able to outrun actual weather was often discussed. But it was normally assumed that the issue was just getting a better approximation to the underlying equations—or better initial measurements—not something more fundamental.

Particularly in the context of game theory and cybernetics the idea had developed in the 1940s that it should be possible to make mathematical predictions even about complex human situations. And for example starting in the early 1950s government control of economies based on predictions from linear models became common. By the early 1970s, however, such approaches were generally seen as unsuccessful, but it was usually assumed that the reason was not fundamental, but was just that there were too many disparate elements to handle in practice.

The notions of universality and undecidability that underlie computational irreducibility emerged in the 1930s, but they were not seen as relevant to questions arising in natural science. Starting in the 1940s they were presumably the basis for a few arguments made about free will and fundamental unpredictability of human behavior (see page 1135), particularly in the context of economics. And in the late 1950s there was brief interest among philosophers in connecting results like Gödel's Theorem to questions of determinism—though mostly there was just confusion centered around the difficulty of finding countable proofs for statements about the continuous processes assumed to occur in physics.

The development of algorithmic information theory in the 1960s led to discussion of objects whose information content cannot be compressed or derived from anything shorter. But as indicated on page 1067 this is rather different from what I call computational irreducibility. In the 1970s computational complexity theory began to address questions about overall resources needed to perform computations, but concentrated on computations that perform fairly specific known practical tasks. At the beginning of the 1980s, however, it was noted that certain problems about models of spin glasses were NP-complete. But there was no immediate realization that this was connected to any underlying general phenomenon.

Starting in the late 1970s there was increasing interest in issues of predictability in models of physical systems. And it was emphasized that when the equations in such models are nonlinear it often becomes difficult to find their solutions. But usually this was at some level assumed to be associated with sensitive dependence on initial conditions and the chaos phenomenon—even though as we saw on page 1098 this alone does not even prevent there from being formulas.

By the early 1980s it had become popular to use computers to study various models of natural systems. Sometimes the idea was to simulate a large collection of disparate elements, say as involved in a nuclear explosion. Sometimes instead the idea was to get a numerical approximation to some fairly simple partial differential equation, say for fluid flow. Sometimes the idea was to use randomized methods to get a statistical approximation to properties say of spin systems or lattice gauge theories. And sometimes the idea was to work out terms in a symbolic perturbation series approximation, say in quantum field theory or celestial mechanics. With any of these approaches huge amounts of computer time were often used. But it was almost always implicitly assumed that this was necessary in order to overcome the approximations being used, and not for some more fundamental reason.

Particularly in physics, there has been some awareness of examples such as quark confinement in QCD where it seems especially difficult to deduce the consequences of a theory—but no general significance has been attached to this.

When I started studying cellular automata in the early 1980s I was quickly struck by the difficulty of finding formulas for their behavior. In traditional models based for example on continuous numbers or approximations to them there was usually no obvious correspondence between a model and computations that might be done about it. But the evolution of a cellular automaton was immediately reminiscent of other computational processes—leading me by 1984 to formulate explicitly the concept of computational irreducibility.

No doubt an important reason computational irreducibility was not identified before is that for more than two centuries students had been led to think that basic theoretical science could somehow always be done with convenient formulas. For almost all textbooks tend to discuss only those cases that happen to come out this way. Starting in earnest in the 1990s, however, the influence of Mathematica has gradually led to broader ranges of examples. But there still remains a very widespread belief that if a theoretical result about the behavior of a system is truly fundamental then it must be possible to state it in terms of a simple mathematical formula.

Image Source Notebooks:

From Stephen Wolfram: A New Kind of Science [citation]