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A new “feasible” arithmetic

Published online by Cambridge University Press:  12 March 2014

Stephen Bellantoni  and
Martin Hofmann
Stephen Bellantoni
Affiliation:
Department of Computer Science, University of Toronto, Toronto, Canada, E-mail: [email protected]
Martin Hofmann
Affiliation:
Laboratory for Foundations of Computer Science, University of Edinburgh, Edinburgh, Scotland, UK, E-mail: [email protected]
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 67 , Issue 1 , March 2002 , pp. 104 - 116
Copyright
Copyright © Association for Symbolic Logic 2002

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