The Basic Concepts of Computability Theory

Abstract

The fundamental notion of computability theory is that of the computable functions f : ℕ^k→ ℕ, where ℕ ={0,1,2,...} is the set of natural numbers and the dotted arrow indicates a partial function. By definition, a function f : ℕ^k→ℕ (k∈ℕ) is computable iff there is a register machine which computes it (c f. Chapter 1.2, Def. 3). The domain of f consists of those inputs for which the register machine halts. There are several other equivalent definitions (see Part 1 of this book). By Church’s Thesis the computable functions are exactly those number functions which are intuitively computable. This thesis is not a theorem since the concept “intuitively computable” is not precisely defined. The theory of computability does not depend on the validity of Church’s Thesis which is only used for interpreting the results. By P^(k)we denote the set of k-ary partial recursive functions f : ℕ^k→ℕ, by R^(k) we denote the set of k-ary total recursive functions (c f. Def. 1.7.6). For proving that a function f : ℕ^k→ℕ is computable, usually it is shown that a register machine exists which computes f. In practice such a proof is not elaborated in all details. The first essential step consists of the definition of an abstract machine M ( = input encoding + output encoding + flowchart, c f. Chapter 1.1) and a proof of f = f_M (correctness). The flowchart of the machine should be sufficiently detailed such that transformation of the machine M into a register machine by refinement and simulation is a routine exercise which generally is not executed. Usually it suffices to specify the flowchart only informally and incompletely using e.g. ALGOL like notation such that a precise specification can be derived easily. In future we shall proceed in this manner.