Computable and uncomputable sets

Recursion theory originated with work of Kurt Gödel, Alonzo Church, Alan Turing, Stephen Kleene and Emil Post in the 1930s.

The fundamental results the researchers obtained established Turing computability as the correct formalization of the informal idea of effective calculation. These results led Stephen Kleene (1952) to coin the two names "Church's thesis" (Kleene 1952:300) and "Turing's Thesis" (p. 376). Nowadays these are often considered as a single hypothesis, the Church–Turing thesis, which states that any function that is computable by an algorithm is a computable function. Although initially skeptical, by 1946 Gödel argued in favor of this thesis.

"Tarski has stressed in his lecture (and I think justly) the great importance of the concept of general recursiveness (or Turing's computability). It seems to me that this importance is largely due to the fact that with this concept one has for the first time succeeded in giving an absolute notion to an interesting epistomological notion, i.e., one not depending on the formalism chosen."(Gödel 1946 in Davis 1965: 84)

With a definition of effective calculation came the first proofs that there are problems in mathematics that cannot be effectively decided. Church (1936a, 1936b) and Turing (1936), inspired by techniques used in by Gödel (1931) to prove his incompleteness theorems, independently demonstrated that the Entscheidungsproblem is not effectively decidable. This result showed that there is no algorithmic procedure that can correctly decide whether arbitrary mathematical propositions are true or false.

Many problems of mathematics have been shown to be undecidable after these initial examples were established. In 1947, Markov and Post published independent papers showing that the word problem for semigroups cannot be effectively decided. Extending this result, Pyotr Novikov and William Boone showed independently in the 1950s that the word problem for groups is not effectively solvable: there is no effective procedure that, given a word in a finitely presented group, will decide whether the element represented by the word is the identity element of the group. In 1970, Yuri Matiyasevich proved Matiyasevich's theorem, which implies that Hilbert's tenth problem has no effective solution; this problem asked whether there is an effective procedure to decide whether a Diophantine equation over the integers has a solution in the integers. The list of undecidable problems gives additional examples of problems with no computable solution.

The study of which mathematical constructions can be effectively performed is sometimes called recursive mathematics; the Handbook of Recursive Mathematics (Ershov et al. 1998) covers many of the known results in this field.

Turing computability

The main form of computability studied in recursion theory was introduced by Turing (1936). A set of natural numbers is said to be a computable set (also called a decidable, recursive, or Turing computable set) if there is a Turing machine that, given a number n, halts with output 1 if n is in the set and halts with output 0 if n is not in the set. A function f from the natural numbers to themselves is a recursive or (Turing) computable function if there is a Turing machine that, on input n, halts and returns output f(n). The use of Turing machines here is not necessary; there are many other models of computation that have the same computing power as Turing machines; for example the μ-recursive functions obtained from primitive recursion and the μ operator.

The terminology for recursive functions and sets is not completely standardized. The definition in terms of μ-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for sets and functions computable by a Turing machine. The word decidable stems from the German word Entscheidungsproblem which was used in the original papers of Turing and others. In contemporary use, the term "computable function" has various definitions: according to Cutland (1980), it is a partial recursive function (which can be undefined for some inputs), while according to Soare (1987) it is a total recursive (equivalently, general recursive) function. This article follows the second of these conventions. Soare (1996) gives additional comments about the terminology.

Not every set of natural numbers is computable. The halting problem, which is the set of (descriptions of) Turing machines that halt on input 0, is a well known example of a noncomputable set. The existence of many noncomputable sets follows from the facts that there are only countably many Turing machines, and thus only countably many computable sets, but there are uncountably many sets of natural numbers.

Although the Halting problem is not computable, it is possible to simulate program execution and produce an infinite list of the programs that do halt. Thus the halting problem is an example of a recursively enumerable set, which is a set that can be enumerated by a Turing machine (other terms for recursively enumerable include computably enumerable and semidecidable). Equivalently, a set is recursively enumerable if and only if it is the range of some computable function. The recursively enumerable sets, although not decidable in general, have been studied in detail in recursion theory.