### arnold beckmann's pages

## Cobham Recursive Set Functions

**File:** PDF-File

**Author:** Arnold Beckmann, Samuel R. Buss, Sy-David Friedman, Moritz Müller and Neil Thapen

**Title:** Cobham Recursive Set Functions

**Journal:** APAL 2016, **167**(3): 335-369

**Pages:** 35

**DOI:** 10.1016/j.apal.2015.12.005

**Abstract:**
This paper introduces the
*Cobham Recursive Set Functions* (CRSF)
as a version of polynomial time computable
functions on general sets,
based on a limited (bounded) form of epsilon-recursion. This is inspired
by Cobham's classic definition of polynomial time functions
based on limited recursion on notation. We introduce a
new set composition function, and a new smash
function for sets which allows polynomial increases in the
ranks and in the cardinalities of transitive closures.
We bootstrap CRSF, prove closure under (unbounded) replacement,
and prove that
any CRSF function is embeddable into a smash term.
When restricted to natural encodings of binary strings as hereditarily finite
sets, the CRSF functions define precisely the polynomial time
computable functions on binary strings. Prior work
of Beckmann, Buss and Friedman and of Arai introduced
set functions based on safe-normal recursion in the
sense of Bellantoni-Cook.
We prove an equivalence between our class CRSF and
a variant of Arai's predicatively computable set functions.