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A new “feasible” arithmetic

  • Stephen Bellantoni (a1) and Martin Hofmann (a2)
Abstract
Abstract

A classical quantified modal logic is used to define a “feasible” arithmetic whose provably total functions are exactly the polynomial-time computable functions. Informally, one understands ⃞∝ as “∝ is feasibly demonstrable”.

differs from a system that is as powerful as Peano Arithmetic only by the restriction of induction to ontic (i.e., ⃞-free) formulas. Thus, is defined without any reference to bounding terms, and admitting induction over formulas having arbitrarily many alternations of unbounded quantifiers. The system also uses only a very small set of initial functions.

To obtain the characterization, one extends the Curry-Howard isomorphism to include modal operations. This leads to a realizability translation based on recent results in higher-type ramified recursion. The fact that induction formulas are not restricted in their logical complexity, allows one to use the Friedman A translation directly.

The development also leads us to propose a new Frege rule, the “Modal Extension” rule: if ⊢ ∝ a then ⊢ A ↔ ∝ for new symbol A.

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[14]M. Hofmann , Safe recursion with higher types and BCK-algebras, Annals of Pure and Applied Logic, vol. 104 (2000), pp. 113166.

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[23]S. Shapiro , Epistemic and intuitionistic arithmetic, Intensional mathematics (S. Shapiro , editor), Studies in Logic and The Foundations of Mathematics, vol. 113, North-Holland, 1985.

[24]H. Simmons , The realm of primitive recursion, Archive for Mathematical Logic, vol. 27 (1988), pp. 177+.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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