Typical forcings, NP search problems and an extension of a theorem of Riis

TL;DR

This paper develops a general forcing framework to analyze the complexity and provability of NP search problems in bounded arithmetic, extending existing theorems and exploring independence results for combinatorial principles.

Contribution

It introduces typical forcings for bounded arithmetic, extends Riis' finitization theorem, and establishes independence results for combinatorial principles within relativized $T^1_2$.

Findings

General forcing conditions produce models of relativized $T^1_2$.

Extended Riis' finitization theorem with model-theoretic properties.

Showed certain combinatorial principles are independent of relativized $T^1_2$.

Abstract

We define typical forcings encompassing many informal forcing arguments in bounded arithmetic and give general conditions for such forcings to produce models of the universal variant of relativized $T^1_2$. We apply this result to study the relative complexity of total (type 2) NP search problems associated to finitary combinatorial principles. Complexity theory compares such problems with respect to polynomial time many-one or Turing reductions. From a logical perspective such problems are graded according to the bounded arithmetic theories that prove their totality. The logical analogue of a reduction is to prove the totality of one problem from the totality of another. The link between the two perspectives is tight for what we call universal variants of relativized bounded arithmetics. We strengthen a theorem of Buss and Johnson (2012) that infers relative bounded depth Frege proofs of totality from polynomial time Turing reducibility. As an application of our general forcing method we derive a strong form of Riis' finitization theorem (1993). We extend it by exhibiting a simple model-theoretic property that implies independence from the universal variant of relativized $T^1_2$ plus the weak pigeonhole principle. More generally, we show that the universal variant of relativized $T^1_2$ does not prove (the totality of the total NP search problem associated to) a strong finitary combinatorial principle from a weak one. Being weak or strong are simple model-theoretic properties based on the behaviour of the principles with respect to finite structures that are only partially defined.