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  • Cited by 5
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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Japaridze, Giorgi 2011. Introduction to clarithmetic I. Information and Computation, Vol. 209, Issue. 10, p. 1312.


    Marion, Mathieu 2009. Radical anti-realism, Wittgenstein and the length of proofs. Synthese, Vol. 171, Issue. 3, p. 419.


    Schwichtenberg, Helmut 2006. An arithmetic for polynomial-time computation. Theoretical Computer Science, Vol. 357, Issue. 1-3, p. 202.


    Simmons, Harold 2005. Tiering as a Recursion Technique. Bulletin of Symbolic Logic, Vol. 11, Issue. 03, p. 321.


    Leivant, D. 2002. Proceedings 17th Annual IEEE Symposium on Logic in Computer Science. p. 345.

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A new “feasible” arithmetic

  • Stephen Bellantoni (a1) and Martin Hofmann (a2)
  • DOI: http://dx.doi.org/10.2178/jsl/1190150032
  • Published online: 01 March 2014
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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[1]S. Abramsky , Computational interpretations of linear logic, Theoretical Computer Science, vol. 111 (1993), pp. 357.

[3]S. Bellantoni and S. Cook , A new recursion-theoretic characterization of the polytime functions, Computational Complexity, vol. 2 (1992), pp. 97110.

[5]S. Bellantoni , K. H. Niggl , and H. Schwichtenberg , Ramification, modality, and linearity in higher type recursion, Annals of Pure and Applied Logic, (2000).

[8]T. Coquand , Computational content of classical proofs, Semantics and logics of computation (A. Pitts and P. Dybjer , editors), Cambridge University Press, 1997.

[10]J. Y. Girard , A. Scedrov , and P. J. Scott , Bounded linear logic: a modular approach to polynomial-time computability, Theoretical Computer Science, vol. 97 (1992), pp. 166.

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

[16]J. Krajicek , Bounded arithmetic, prepositional logic, and complexity theory, Cambridge University Press, 1995.

[20]E. Nelson , Predicative arithmetic, Princeton University Press, Princeton, NJ, 1986.

[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
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