Computability over an Arbitrary Structure. Sequential and Parallel Polynomial Time

Abstract

Abstract. We provide several machine-independent characterizations of deterministic complexity classes in the model of computation proposed by L. Blum, M. Shub and S. Smale. We provide a characterization of partial recursive functions over any arbitrary structure. We show that polynomial time computable functions over any arbitrary structure can be characterized in term of safe recursive functions. We show that polynomial parallel time decision problems over any arbitrary structure can be characterized in terms of safe recursive functions with substitutions.

Tailoring Recursion to Characterize Non-Deterministic Complexity Classes Over Arbitrary Structures

“…In a previous paper [5], we provided machine independent characterizations of the class of computable functions and of the class of functions computable in polynomial time. Since this work is based on the latter characterization, we next briefly recall our previous result.…”

“…In addition the two inclusions PAR R ⊂ PAT R and PAR R ⊂ EXP R are known to be strict. Concerning classical complexity, our characterizations of PAT and DPAT, combined with our previous one of PAR in [5], provide several new original alternative characterizations of PSPACE.…”

“…In a previous paper [5], based on classical characterizations in [3] and [19], we exhibited machine-independent characterizations of the classes of functions over an arbitrary structure computable in polynomial sequential or parallel time. Our aim here is to provide such machine-independent characterizations over an arbitrary structure for polynomial hierarchy and polynomial alternating time.…”

“…Based on previous characterizations of deterministic complexity classes [5], recalled in Section 2 and 3, we provide in Section 4 a characterization of the polynomial hierarchy. Minor changes allow us to characterize the digital polynomial hierarchy in Section 5.…”

There is no safe pairing function over an arbitrary structure

“…In [1], Bournez et al defined the class of safe recursive functions over an arbitrary structure as the smallest class of functions containing a set of basic functions and closed under safe composition and simultaneous safe recursion. These functions have two sorts of variables, normal and safe.…”