An application of category-theoretic semantics to the characterisation of complexity classes using higher-order function algebras

Bulletin of Symbolic Logic 3 (4):469-486 (1997)
We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ω 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 ∑ 1 b -induction over PV ω 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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DOI 10.2307/421100
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Martin Hofmann (2000). Safe Recursion with Higher Types and BCK-Algebra. Annals of Pure and Applied Logic 104 (1-3):113-166.

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