The provably total NP search problems of weak second order bounded arithmetic

The provably total NP search problems of weak second order bounded arithmetic

We define a new NP search problem, the “local improvement” principle, about labellings of an acyclic, bounded-degree graph. We show that, provably in PV, it characterizes the ∀Σb1 consequences of V 1 2 and that natural restrictions of it characterize the ∀Σb1 consequences of U 1 2 and of the bounded arithmetic hierarchy. We also show that over V 0 it characterizes the ∀ΣB0 consequences of V 1 a...

The Provably Total Search Problems of Bounded Arithmetic

We give combinatorial principles GIk, based on k-turn games, which are complete for the class of NP search problems provably total at the kth level T k 2 of the bounded arithmetic hierarchy and hence characterize the ∀Σ̂1 consequences of T k 2 , generalizing the results of [22]. Our argument uses a translation of first order proofs into large, uniform propositional proofs in a system in which th...

NP search problems in low fragments of bounded arithmetic

We give combinatorial and computational characterizations of the NP search problems definable in the bounded arithmetic theories T 2 2 and T 3 2 . By a search problem in full generality we mean simply a binary relation R(x, y) such that ∀x∃yR(x, y) holds. The search task is: given an instance x find a solution y such that R(x, y) holds. The most important class of search problems are NP search ...

Provably Total Functions in Bounded Arithmetic Theories . . .

Bounded Second-Order Unification Is NP-Complete

Bounded Second-Order Unification is the problem of deciding, for a given second-order equation t ? = u and a positive integer m, whether there exists a unifier σ such that, for every second-order variable F , the terms instantiated for F have at most m occurrences of every bound variable. It is already known that Bounded Second-Order Unification is decidable and NP-hard, whereas general Second-...