Chapter 12: The Principle of Computational Equivalence

Section 8: Undecidability and Intractability

Examples of undecidability

Once universality exists in a system it is known from Gordon Rice's 1953 theorem and its generalizations that most questions about ultimate behavior will be undecidable unless their answers are always trivially the same. Undecidability has been demonstrated in various seemingly rather different types of systems, most often by reduction to halting (termination) problems for multiway systems.

In formal language theory, questions about regular languages are always decidable, but ones about context-free languages (see page 1103) are already often not. It is decidable whether such a language is finite, but not whether it contains every possible string, is regular, is unambiguous, or is equivalent to a language with a different grammar.

In mathematical logic, it can be undecidable whether statements are provable from a given axiom system—say predicate logic or Peano arithmetic (see page 782). It is also undecidable whether one axiom system is equivalent to another—even for basic logic (see page 1170).

In algebra and related areas of mathematics problems of equivalence between objects built up from elements that satisfy relations are often in general undecidable. Examples are word problems for groups and semigroups (see page 1141), and equivalence of finitely specified 4D manifolds (see page 1051). (Equivalence for 3D manifolds is thought to be decidable.) A related undecidable problem is whether two integer matrices can be multiplied together in some sequence to yield the zero matrix. It is also undecidable whether two sets of relations specify the same group or semigroup.

In combinatorics it is known in general to be undecidable whether a given set of tiles can cover the plane (see page 1139). And from this follows the undecidability of various problems about 2D cellular automata (see note below) and spin systems. Also undecidable are many questions about whether strings exist that satisfy particular constraints (see below).

In number theory it is known to be undecidable whether Diophantine equations have solutions (i.e. whether algebraic equations have integer solutions) (see page 786). And this means for example that it is in general undecidable whether expressions that involve both algebraic and trigonometric functions can be zero for real values of variables, or what the values of integrals are in which such expressions appear as denominators (compare page 916).

In computer science, general problems about verifying the possible behavior of programs tend to be undecidable, usually being directly related to halting problems. It is also for example undecidable whether a given program is the shortest one that produces particular output (see page 1067).

It is in general undecidable whether a given system exhibits universality—or undecidability.

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