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A New "Feasible" Arithmetic

Stephen Bellantoni and Martin Hofmann
The Journal of Symbolic Logic
Vol. 67, No. 1 (Mar., 2002), pp. 104-116
Stable URL: http://www.jstor.org/stable/2694998
Page Count: 13
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Abstract

A classical quantified modal logic is used to define a "feasible" arithmetic A12 whose provably total functions are exactly the polynomial-time computable functions. Informally, one understands $\Box\alpha$ as "α is feasibly demonstrable". A12 differs from a system A2 that is as powerful as Peano Arithmetic only by the restriction of induction to ontic (i.e., $\Box$-free) formulas. Thus, A12 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 $\vdash \alpha$ then $\vdash A \leftrightarrow \alpha$ a for new symbol A.

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