Subset-Bounded Recursion and a Circuit Model for Cobham Recursive Set Functions

The Cobham Recursive Set Functions (CRSF) provide an analogue of polynomial time computation which applies to arbitrary sets. We give three new equivalent characterizations of CRSF. The first is algebraic, using subset-bounded recursion and a form of Mostowski collapse. The second uses a circuit model of computation; the CRSF functions are shown to be precisely the functions computed by a class of uniform, infinitary, Boolean circuits. The third is in terms of a simple extension of the rudimentary functions by transitive closure and subset-bounded recursion. ∗Supported in part by NSF grants DMS-1101228 and CCR-1213151, by the Simons Foundation, award 306202, and by the Skolkovo Institute for Science and Technology. †Supported by the Austrian Science Fund (FWF) under project number P24654. ‡Supported by the Austrian Science Fund (FWF) under project number P28699. §Partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 339691. The Institute of Mathematics of the Czech Academy of Sciences is supported by RVO:67985840.