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

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ! 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 \Sigma b 1 -induction over PV ! and use this to reestablish that the provably total functions in this system are in 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. 1 Introduction Cook and Urquhart's system PV ! [3] is a simply-typed lambda calculus providing constants to denote natural numbers and an operator for bounded recursion on notation like in Cobham's characterisation of polynomial-time computability. 1 Although functionals of arbitrary type can be defined in this system one can show that thei..