No abstract

We introduce a class of safe Turing machines which execute structured while, if-then- else programs and operate on stacks and on a read-only input tape. A hierarchy is obtained by taking as Si the class of all functions computed by programs of loop-depth i. The main result is that S1 equals Lintime and S2 equals Polytime while, for i≥3, we have that Si equals the i-th Grzegorczyk class. By adding to the language a non-deterministic construct choose we take S2 into a class equivalent to NP. This gives a syntactical characterization in a pure-machine model of the mentioned classes.

No abstract

We harmonize many time-complexity classes DTIMEF(f(n)) (f(n) [ges ] n) with the PR functions (at and above the elementary level) in a transfinite hierarchy of classes of functions [Tscr ]α. Class [Tscr ]α is obtained by means of unlimited operators, namely: a variant Π of the predicative or safe recursion scheme, introduced by Leivant, and by Bellantoni and Cook, if α is a successor; and constructive diagonalization if α is a limit. Substitution (SBST) is discarded because the time complexity classes are not closed under this scheme. [Tscr ]α is a structure for the PR functions finer than [Escr ]α, to the point that we have [Tscr ]ε0 = [Escr ]3 (elementary functions). Although no explicit use is made of hierarchy functions, it is proved that f(n) ∈ [Tscr ]α implies f(n) [les ] nGα(n), where Gα belongs to the slow growing hierarchy (of functions) studied, in particular, by Girard and Wainer.

Characterizations of PTIME, PSPACE, the polynomial hierarchy and its elements are given, which are decidable (membership can be decided by syntactic inspection to the constructions), predicative (according to points of view by Leivant and others), and are obtained by means of increasing restrictions to course-of-values recursion on trees (represented in a dialect of Lisp). (C) 2001 Elsevier Science B.V. All rights reserved