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An Application of Category-Theoretic Semantics to the Characterisation of Complexity Classes Using Higher-Order Function Algebras

Martin Hofmann
The Bulletin of Symbolic Logic
Vol. 3, No. 4 (Dec., 1997), pp. 469-486
DOI: 10.2307/421100
Stable URL: http://www.jstor.org/stable/421100
Page Count: 18
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An Application of Category-Theoretic Semantics to the Characterisation of Complexity Classes Using Higher-Order Function Algebras
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Abstract

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PVω defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with ∑1b-induction over PVω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion.

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