Ramified Recurrence and Computational Complexity III: Higher Type Recurrence and Elementary Complexity

Abstract We consider the functionals defined using an extension to higher types of ramified recurrence, which was introduced independently in [4,18,21] and [35]. Three styles of functional programs over free algebras are examined: equational recurrence, applicative programs with recurrence operators and purely applicative higher-type programs. We show that for every free algebra A and each one of these styles, the functions defined by ramified recurrence in finite types are precisely the functions over A computable in a number of steps elementary in the size of the input. This should be contrasted with unrestricted higher type recurrence which yields, for numeric computing, all provably recursive functions of first order arithmetic. This paper is revised and expanded from the proceedings paper [23]. The research project of which it is part is closely related to Rohit Parikh's longstanding interest in conceptual delineation of feasibility.

[1]  Daniel Leivant,et al.  Lambda Calculus Characterizations of Poly-Time , 1993, Fundam. Informaticae.

[2]  Yehoshua Bar-Hillel,et al.  The Intrinsic Computational Difficulty of Functions , 1969 .

[3]  Yiannis N. Moschovakis.  Logic from Computer Science , 1992 .

[4]  Daniel Leivant,et al.  Finitely Stratified Polymorphism , 1991, Inf. Comput..

[5]  Robert W. Ritchie,et al.  CLASSES OF PREDICTABLY COMPUTABLE FUNCTIONS , 1963 .

[6]  Harold Simmons,et al.  The realm of primitive recursion , 1988, Arch. Math. Log..

[7]  Daniel Leivant,et al.  A Foundational Delineation of Poly-time , 1994, Inf. Comput..

[8]  D. Leivant.  Ramified Recurrence and Computational Complexity I: Word Recurrence and Poly-time , 1995 .

[9]  Daniel Leivant,et al.  Intrinsic Theories and Computational Complexity , 1994, LCC.

[10]  Stephen Bellantoni.  Characterizing Parallel Time by Type 2 Recursions With Polynomial Output Length , 1994, LCC.

[11]  M. Nivat,et al.  Selected papers from the 6th international joint conference on Theory and practice of software development , 1996 .

[12]  Daniel Leivant,et al.  Contracting proofs to programs , 1989 .

[13]  Michael Detlefsen.  Proof, Logic and Formalization , 1992 .

[14]  Daniel Leivant.  Predicative Recurrence in Finite Types , 1994, LFCS.

[15]  Daniel Leivant,et al.  Subrecursion and lambda representation over free algebras , 1990 .

[16]  Piergiorgio Odifreddi,et al.  Logic and computer science , 1990 .

[17]  Daniel Leivant,et al.  Predicative Functional Recurrence and Poly-space , 1997, TAPSOFT.

[18]  Stephen Bellantoni.  Predicative Recursion and The Polytime Hierarchy , 1995 .

[19]  Daniel Leivant,et al.  Stratified functional programs and computational complexity , 1993, POPL '93.

[20]  Stephen A. Bloch.  Functional characterizations of uniform log-depth and polylog-depth circuit families , 1992, [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference.

[21]  J. Heijenoort.  From Frege To Gödel , 1967 .

[22]  Daniel Leivant,et al.  Semantic Characterizations of Number Theories , 1992 .

[23]  Richard Statman,et al.  The typed λ-calculus is not elementary recursive , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[24]  D. Hilbert.  Über das Unendliche , 1926 .

[25]  Daniel Leivant.  A characterization of NC by tree recurrence , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[26]  Daniel Leivant,et al.  Ramified Recurrence and Computational Complexity II: Substitution and Poly-Space , 1994, CSL.

[27]  Richard Statman,et al.  The Typed lambda-Calculus is not Elementary Recursive , 1979, Theor. Comput. Sci..

[28]  A. Grzegorczyk.  Some classes of recursive functions , 1964 .

[29]  Corrado Böhm,et al.  Automatic Synthesis of Typed Lambda-Programs on Term Algebras , 1985, Theor. Comput. Sci..

[30]  Von Kurt Gödel,et al.  ÜBER EINE BISHER NOCH NICHT BENÜTZTE ERWEITERUNG DES FINITEN STANDPUNKTES , 1958 .

[31]  David Isles.  What Evidence is There that 265536 is a Natural Number? , 1992, Notre Dame J. Formal Log..

[32]  Daniel Leivant,et al.  Ramified Recurrence and Computational Complexity IV : Predicative Functionals and Poly-Space , 2000 .

[33]  K. Gödel,et al.  Kurt Gödel : collected works , 1986 .