Safe Recursive Set Functions

We introduce the safe recursive set functions based on a Bellantoni-Cook style subclass of the primitive recursive set functions. We show that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets, are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. We also show that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets, are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations. We characterize the safe recursive set functions on arbitrary sets in definability-theoretic terms. In its strongest form, we show that a function on arbitrary sets is safe recursive if and only if it is uniformly definable in some polynomial level of a refinement of Jensen’s J-hierarchy, relativized to the transitive closure of the function’s arguments. We observe that safe recursive set functions on infinite binary strings are equivalent to functions computed by infinite-time Turing machines in time less than ωω. We also give a machine model for safe recursive set functions which is based on set-indexed parallel processors and the natural bound on running times. All three authors thank the John Templeton Foundation, Project #13152, for supporting their participation in the CRM Infinity Project at the Centre de Recerca Matemàtica, Barcelona, Catalonia, Spain during which this project was instigated. This research was partially done while the author was a visiting fellow at the Isaac Newton Institute for the Mathematical Sciences in the programme “Semantics & Syntax”. Supported in part by NSF grants DMS-0700533, DMS-1101228 and CCR-1213151, and by a grant from the Simons Foundation (#208717 to Sam Buss). Supported in part by the FWF (Austrian Science Fund) through FWF project number P 22430-N13.