But inevitably functions like FixedPoint , ReplaceRepeated and FullSimplify can run into undecidability—so that ultimately they have to be limited by constructs such as $IterationLimit and TimeConstraint .

The Relationship of Space and Time…But such systems work not by being required to satisfy constraints, but instead by just repeatedly applying explicit rules. So is it in the end sensible to think of the universe as a single structure in spacetime whose form is determined by a set of constraints? … But while this view is closer to our everyday perception of time, it seems to

The Relationship of Space and Time…The laws of physics in effect provide a collection of constraints on the structure. … But now instead of defining constraints just in space, the laws of physics can be thought of as defining constraints on what can happen in both space and time. Just as for space, it is my strong belief that time is fundamentally discrete.

Time and Causal Networks…But the way this network is formed in effect treats space and time rather differently. … Any such rules can in principle be thought of as providing a set of constraints for the spacetime network. … Time and Causal Networks I argued in the last section that the progress of time should be viewed at a fundamental level much like the evolution of a system like a cellular automaton.

The Problem of Satisfying Constraints…But this process may still take a very long time. … In traditional science the notion of constraints is often introduced in an attempt to summarize the effects of evolution rules. Typically the idea is that after a sufficiently long time a system should be found only in states that are invariant under the application of its evolution rules.

Note (a) for Systems Based on Constraints…And in general what such equations do is to specify constraints that systems must satisfy. Sometimes these constraints just relate the state of a system at one time to its state at a previous time. … But if the constraints relate different features of a system at one particular time, then they cannot be converted into evolution rules.

Repetition in time is still easy to achieve—say just by different parts of a system behaving independently. … An example is rule 110, in which repetitive domains form with period 14 in space and 7 in time, but as the second picture below illustrates, the localized structures which separate these domains take a very long time to disappear. As we saw at the end of Chapter 5 , many systems based on constraints also in principle yield repetition—though from the discussion of the previous section it seems likely that this is rarely a good explanation for actual repetition that we see in nature.

Note (c) for The Problem of Satisfying Constraints…Usually, however, what is being optimized is some aspect of the form or behavior of an organism, which represents a very complicated constraint on the underlying genetic material. (It is as if one is defining constraints on the initial conditions for a cellular automaton by looking at the pattern generated by the cellular automaton after a long time.) But the strategies of biological evolution can also be used in trying to satisfy simpler constraints.

Constraints are yet another basis for uniformity. And as a trivial example, the constraint in a line of black or white cells that every cell should be the same color as both its neighbors immediately implies that the whole line must be either uniformly black or uniformly white. … Repetition in time corresponds just to a system repeatedly returning to a particular state.

Note (c) for Systems Based on Constraints…Explanations based on constraints In some areas of science it is common to give explanations in terms of constraints rather than mechanisms. … In fact, in finding out what configuration such molecules actually adopt, it is usually much more relevant to know how the molecule evolves in time as it is created than which of its configurations formally has minimum energy.