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Machine Models and Function Algebras

  • Peter Clote
  • Evangelos Kranakis
Part of the Texts in Theoretical Computer Science. An EATCS Series book series (TTCS)

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].

Keywords

Turing Machine Initial Function Machine Model Function Algebra Recursion Scheme 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Peter Clote
    • 1
  • Evangelos Kranakis
    • 2
  1. 1.Department of Computer Science and Department of BiologyBoston CollegeChestnut HillUSA
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada

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