A Notion of Polynomial Time Computability for Set Functions

This talk discusses a new notion of polynomial time computability for general sets, based on $\epsilon$-recursion with a Cobham-style bounding using a new smash function tailored for sets. These are called the Cobham Recursive Set Functions (CRSF), and give a notion of polynomial time computability intrinsic to sets. The smash function accommodates polynomial growth rate of both the size of the transitive closure and the rank of sets. For suitable encodings of binary strings as hereditarily finite sets, the CRSF functions are precisely the usual polynomial time computable functions. We also discuss normal forms and closure properties for CRSF. This is joint work with A. Beckmann, S. Friedman, M. Mueller and N. Thapen.