Machine Models and Function Algebras

Abstract

A recurring theme in theory of computation is that of a function algebra ¹ i.e., a smallest class of functions containing certain initial functions and closed under certain operations (especially substitution and primitive recursion).² In 1904, G.H. Hardy [Har04] used related concepts to define sets of real numbers of cardinality ℵ₁. In 1923, Th. Skolem [Sko23] introduced the primitive recursive functions, and in 1925, as a technical tool in his claimed sketch proof of the continuum hypothesis, D. Hilbert [Hil25] defined classes of higher type functionals by recursion. In 1928, W. Ackermann [Ack28] furnished a proof that the diagonal function φ _a (a, a) of Hilbert [Hi125], a variant of the Ackermann function, is not primitive recursive. In 1931, K. Gödel [Göd31] defined the primitive recursive functions, there calling them “rekursive Funktionen” , and used them to arithmetize logical syntax via Gödel numbers for his incompleteness theorem. Generalizing Ackermann’s work, in 1936 R. Péter [Pét36] defined and studied the k-fold recursive functions. The same year saw the introduction of the fundamental concepts of Turing machine (A.M. Turing [Tur37]), λ-calculus (A. Church [Chu36]) and μ-recursive functions (S.C. Kleene [Kle36a]). By restricting the scheme of primitive recursion to allow only limited summations and limited products, the elementary functions were introduced in 1943 by L. Kalmár [Kal43].

Like so many other mathematical ideas, especially the more profoundly beautiful and fundamental ones, the idea of computability seems to have a kind of Platonic reality of its own. R. Penrose [Pen89]