Computability theory

The effective calculability of number-theoretic functions such as addition and multiplication has always been recognized, and for that judgment a rigorous notion of ‘computable function’ is not required. A sharp mathematical concept was defined only in the twentieth century, when issues including the decision problem for predicate logic required a precise delimitation of functions that can be viewed as effectively calculable. Predicate logic emerged from Frege’s fundamental ‘Begriffsschrift’ (1879) as an expressive formal language and was described with mathematical precision by Hilbert in lectures given during the winter of 1917–18. The logical calculus Frege had also developed allowed proofs to proceed as computations in accordance with a fixed set of rules; in principle, according to Gödel, the rules could be applied ‘by someone who knew nothing about mathematics, or by a machine’. Hilbert grasped the potential of this mechanical aspect and formulated the decision problem for predicate logic as follows: ‘The Entscheidungsproblem [decision problem] is solved if one knows a procedure that permits the decision concerning the validity, respectively, satisfiability of a given logical expression by a finite number of operations.’ Some, for example, von Neumann (1927), believed that the inherent freedom of mathematical thought provided a sufficient reason to expect a negative solution to the problem. But how could a proof of undecidability be given? The unsolvability results of other mathematical problems had always been established relative to a determinate class of admissible operations, for example, the impossibility of doubling the cube relative to ruler and compass constructions. A negative solution to the decision problem obviously required the characterization of ‘effectively calculable functions’. For two other important issues a characterization of that informal notion was also needed, namely, the general formulation of the incompleteness theorems and the effective unsolvability of mathematical problems (for example, of Hilbert’s tenth problem). The first task of computability theory was thus to answer the question ‘What is a precise notion of "effectively calculable function"?’. Many different answers invariably characterized the same class of number-theoretic functions: the partial recursive ones. Today recursiveness or, equivalently, Turing computability is considered to be the precise mathematical counterpart to ‘effective calculability’. Relative to these notions undecidability results have been established, in particular, the undecidability of the decision problem for predicate logic. The notions are idealized in the sense that no time or space limitations are imposed on the calculations; the concept of ‘feasibility’ is crucial in computer science when trying to capture the subclass of recursive functions whose values can actually be determined.