Feasible set functions have small circuits

Arnold Beckmann, Sam Buss, Sy-David Friedman, Moritz Müller, Neil Thapen
2018 Computability - The Journal of the Assosiation  
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 is our main result: 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
more » ... ransitive closure and subset-bounded recursion. , ordered lexicographically, and [a#b]. Definition 2.9. We let π 1,a,b : [a#b] → [a] and π 2,a,b : [a#b] → [b] be projection functions inverting σ a,b , so that σ a,b (π 1,a,b (z), π 2,a,b (z)) = z for z ∈ [a#b]. Lemma 2.10. For sets a and b, (1) rank(a ⊙ b) = rank(b) + rank(a) (2) |tc(a ⊙ b)| = |tc(a)| + |tc(b)| (3) rank(a#b) + 1 = (rank(b) + 1)(rank(a) + 1) (4) |tc(a#b)| + 1 = (|tc(b)| + 1)(|tc(a)| + 1), equivalently, |[a#b]| = |[a]| · |[b]|. Definition 2.11. A smash-term is a term built from variables, the constant 0 and the functions pairing, cartesian product, transitive closure, ⊙ and #. Smash-terms will play the role usually played by polynomials in computational complexity, providing bounds for various complexity measures. Notice that the rank, and respectively the size of the transitive closure, of a smashterm is at most polynomially larger than those of its arguments. (The corresponding definition of #-term in [4] is stricter, only allowing variables, the constant 1, ⊙ and #.) Subset-bounded recursion and CRSF This section is modelled on the similar development of CRSF in [4]. Definition 3.1. Let g( a, b, x) and h( a, b) be functions from sets to sets. The function f ( a, b) is obtained from g( a, b, x) by subset-bounded recursion with bound h( a, b) if f ( a, b) = g( a, b, { f ( a, c) : c ∈ b}) ∩ h( a, b).
doi:10.3233/com-180096 fatcat:exngvdxcyvfexko562thvw37ia