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A new “feasible” arithmetic

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

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
  • URL: /core/journals/journal-of-symbolic-logic
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