A polynomially bounded recursive realizability, in which the recursive functions used in Kleene’s realizability are restricted to polynomially bounded functions, is introduced. It is used to show that provably total functions of Ruitenburg’s Basic Arithmetic are polynomially bounded (primitive) recursive functions. This sharpens our earlier result where those functions were proved to be primiti...

Matrix interpretations can be used to bound the derivational complexity of rewrite systems. We present a criterion that completely characterizes matrix interpretations that are polynomially bounded. It includes the method of upper triangular interpretations as a special case, and we prove that the inclusion is strict. The criterion can be expressed as a finite domain constraint system. It trans...

We prove a theorem which provides a method for constructing points on varieties defined by certain smooth functions. We require that the functions are definable in a definably complete expansion of a real closed field and are locally definable in a fixed o-minimal and polynomially bounded reduct. As an application we show that in certain o-minimal structures definable functions are piecewise im...

Forgetting is one of the most important concepts in logic based problem solving, both from a theoretical and a practical point of view. However, the size of the forgetting result is exponential in worst case. To address this issue, we consider the problem of polynomially bounded forgetting, i.e., when the size of the forgetting result can be expressed polynomially. We coin the notion of polynom...