Syntactic characterization in Lisp of the polynomial complexity classes and hierarchy

Abstract

The de nition of a class C of functions is syntactic if membership to C can be decided from the construction of its elements. Syntactic characterizations of PTIMEF, of PSPACEF, of the polynomial hierarchy PH, and of its subclasses p n are presented. They are obtained by progressive restrictions of recursion in Lisp, and may be regarded as predicative according to a foundational point raised by L e i v ant.

A predicative approach to the classification problem

“…A PR definition like f(x, y + 1) = h(x, y + 1, f(x, y)) is asking to compute h for a number of times, depending on the entity f being introduced. After the resource-free characterization of PTIMEF by Bellantoni and Cook (1992) and Bellantoni (1992), other complexity classes have been captured by means of variants of Safe Recursion (SR) schemes (Leivant, 1994;Leivant and Marion, 1995;Bellantoni, 1995;Caporaso et al, 1997Caporaso et al, , 2000. Many of them reduce the circularity implicit in all recursions by denying the role of the principal variable to all variables already used as auxiliary in a previous recursion; they differ from one another in the choice of the initial functions and by the kind of SR scheme.…”