Conservative subtheories of $${{R}^{1}_{2}}$$ and $${{S}^{1}_{2}}$$ are presented. For $${{S}^{1}_{2}}$$, a slight tightening of Jeřábek’s result (Math Logic Q 52(6):613–624, 2006) that $${T^{0}_{2} \preceq_{\forall \Sigma^{b}_{1}}S^{1}_{2}}$$ is presented: It is shown that $${T^{0}_{2}}$$ can be axiomatised as BASIC together with induction on sharply bounded formulas of one alternation. Within this $${\forall\Sigma^{b}_{1}}$$ -theory, we define a $${\forall\Sigma^{b}_{0}}$$ -theory, $${T^{-1}_{2}}$$, for the $${\forall\Sigma^{b}_{0}}$$ -consequences of $${S^{1}_{2}}$$. We show $${T^{-1}_{2}}$$ is weak by showing it cannot $${\Sigma^{b}_{0}}$$ -define division by 3. We then consider what (...) would be the analogous $${\forall\hat\Sigma^{b}_{1}}$$ -conservative subtheory of $${R^{1}_{2}}$$ based on Pollett (Ann Pure Appl Logic 100:189–245, 1999. It is shown that this theory, $${{T}^{0,\left\{2^{(||\dot{id}||)}\right\}}_{2}}$$, also cannot $${\Sigma^{b}_{0}}$$ -define division by 3. On the other hand, we show that $${{S}^{0}_{2}+open_{\{||id||\}}}$$ -COMP is a $${\forall\hat\Sigma^{b}_{1}}$$ -conservative subtheory of $${R^{1}_{2}}$$. Finally, we give a refinement of Johannsen and Pollett (Logic Colloquium’ 98, 262–279, 2000) and show that $${\hat{C}^{0}_{2}}$$ is $${\forall\hat\Sigma^{b}_{1}}$$ -conservative over a theory based on open cl-comprehension. (shrink)

A Ramsey statement denoted \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${n \longrightarrow (k)^2_2}$$\end{document} says that every undirected graph on n vertices contains either a clique or an independent set of size k. Any such valid statement can be encoded into a valid DNF formula RAM(n, k) of size O(nk) and with terms of size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\left(\begin{smallmatrix}k\\2\end{smallmatrix}\right)}$$\end{document}. Let rk be the minimal n for which the statement holds. We prove that (...) RAM(rk, k) requires exponential size constant depth Frege systems, answering a problem of Krishnamurthy and Moll [15]. As a consequence of Pudlák’s work in bounded arithmetic [19] it is known that there are quasi-polynomial size constant depth Frege proofs of RAM(4k, k), but the proof complexity of these formulas in resolution R or in its extension R(log) is unknown. We define two relativizations of the Ramsey statement that still have quasi-polynomial size constant depth Frege proofs but for which we establish exponential lower bound for R. (shrink)

We define a new NP search problem, the “local improvement” principle, about labellings of an acyclic, bounded-degree graph. We show that, provably in , it characterizes the consequences of and that natural restrictions of it characterize the consequences of and of the bounded arithmetic hierarchy. We also show that over V0 it characterizes the consequences of V1 and hence that, in some sense, a miniaturized version of the principle gives a new characterization of the consequences of . Throughout our search (...) problems are “type-2” NP search problems, which take second-order objects as parameters. (shrink)

Journal of Mathematical Logic, Volume 22, Issue 03, December 2022. We study a new class of NP search problems, those which can be proved total using standard combinatorial reasoning based on approximate counting. Our model for this kind of reasoning is the bounded arithmetic theory [math] of [E. Jeřábek, Approximate counting by hashing in bounded arithmetic, J. Symb. Log. 74(3) (2009) 829–860]. In particular, the Ramsey and weak pigeonhole search problems lie in the new class. We give a purely computational (...) characterization of this class and show that, relative to an oracle, it does not contain the problem CPLS, a strengthening of PLS. As CPLS is provably total in the theory [math], this shows that [math] does not prove every [math] sentence which is provable in bounded arithmetic. This answers the question posed in [S. Buss, L. A. Kołodziejczyk and N. Thapen, Fragments of approximate counting, J. Symb. Log. 79(2) (2014) 496–525] and represents some progress in the program of separating the levels of the bounded arithmetic hierarchy by low-complexity sentences. Our main technical tool is an extension of the “fixing lemma” from [P. Pudlák and N. Thapen, Random resolution refutations, Comput. Complexity, 28(2) (2019) 185–239], a form of switching lemma, which we use to show that a random partial oracle from a certain distribution will, with high probability, determine an entire computation of a [math] oracle machine. The introduction to the paper is intended to make the statements and context of the results accessible to someone unfamiliar with NP search problems or with bounded arithmetic. (shrink)

The purpose of this paper is to survey the correspondence between bounded arithmetic and propositional proof systems. In addition, it also contains some new results which have appeared as an extended abstract in the proceedings of the conference TAMC 2008 [11].Bounded arithmetic is closely related to propositional proof systems; this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Krajíček and (...) Pudlák [46]. Instead of focusing on the relation between particular proof systems and theories, we favour a general axiomatic approach to this correspondence. In the course of the development we particularly highlight the role played by logical closure properties of propositional proof systems, thereby obtaining a characterization of extensions of EF in terms of a simple combination of these closure properties (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim). (shrink)

The complexity class of [Formula: see text]-polynomial local search problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the [Formula: see text]-definable functions of [Formula: see text] are characterized in terms of [Formula: see text]-PLS problems. These [Formula: see text]-PLS problems can be defined in a weak base theory such as [Formula: see text], and proved to be total in [Formula: see text]. Furthermore, the [Formula: (...) see text]-PLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new [Formula: see text]-principle that is conjectured to separate [Formula: see text] and [Formula: see text]. (shrink)

In this note we will introduce a class of search problems, called nested Polynomial Local Search (nPLS) problems, and show that definable NP search problems, i.e., \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma^{b}_{1}}$$\end{document}-definable functions in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T^{2}_{2}}$$\end{document} are characterized in terms of the nested PLS.