The theory , axiomatized by the induction scheme for sharply bounded formulae in Buss’ original language of bounded arithmetic , has recently been unconditionally separated from full bounded arithmetic S2. The method used to prove the separation is reminiscent of those known from the study of open induction.We make the connection to open induction explicit, showing that models of can be built using a “nonstandard variant” of Wilkie’s well-known technique for building models of IOpen. This makes it possible to transfer (...) many results and methods from open to sharply bounded induction with relative ease.We provide two applications: the Shepherdson model of IOpen can be embedded into a model of , which immediately implies some independence results for ; extended by an axiom which roughly states that every number has a least 1 bit in its binary notation, while significantly stronger than plain , does not prove the infinity of primes. (shrink)

We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(PV)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in PV₁ + dWPHP(PV).

We show how to formalize approximate counting via hash functions in subsystems of bounded arithmetic, using variants of the weak pigeonhole principle. We discuss several applications, including a proof of the tournament principle, and an improvement on the known relationship of the collapse of the bounded arithmetic hierarchy to the collapse of the polynomial-time hierarchy.

We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of Π1 + ¬Ω1has a proper end-extension to a model of Π1, and so Π1 + ¬Ω ⊢ BΣ1. Under an even stronger complexity-theoretic assumption which nevertheless seems hard to disprove using present-day methods, Π1 + ¬Exp ⊢ BΣ1. Both assumptions can be modified to versions which make it possible to replace Π1 by IΔ0as the base theory.We also show that any proof that IΔ0+ ¬Exp does not (...) prove a given finite fragment of BΣ1has to be “nonrelativizing”, in the sense that it will not work in the presence of an arbitrary oracle. (shrink)

We investigate the provability of some properties of abelian groups and quadratic residues in variants of bounded arithmetic. Specifically, we show that the structure theorem for finite abelian groups is provable in S22 + iWPHP, and use it to derive Fermat's little theorem and Euler's criterion for the Legendre symbol in S22 + iWPHP extended by the pigeonhole principle PHP. We prove the quadratic reciprocity theorem in the arithmetic theories T20 + Count2 and I Δ0 + Count2 with modulo-2 counting (...) principles. (shrink)

The elementary arithmetic operations +,·,≤\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${+,\cdot,\le}$$\end{document} on integers are well-known to be computable in the weak complexity class TC0, and it is a basic question what properties of these operations can be proved using only TC0-computable objects, i.e., in a theory of bounded arithmetic corresponding to TC0. We will show that the theory VTC0 extended with an axiom postulating the totality of iterated multiplication proves induction for quantifier-free formulas in the language ⟨+,·,≤⟩\documentclass[12pt]{minimal} (...) \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle{+,\cdot,\le}\rangle}$$\end{document}, and more generally, minimization for Σ0b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Sigma_{0}^{b}}$$\end{document} formulas in the language of Buss’s S2. (shrink)

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)

The question of whether the bounded arithmetic theories and are equal is closely connected to the complexity question of whether is equal to. In this paper, we examine the still open question of whether the prenex version of,, is equal to. We give new dependent choice‐based axiomatizations of the ‐consequences of and. Our dependent choice axiomatizations give new normal forms for the ‐consequences of and. We use these axiomatizations to give an alternative proof of the finite axiomatizability of and to (...) show new results such as is finitely axiomatized and that there is a finitely axiomatized theory,, containing and contained in. On the other hand, we show that our theory for splits into a natural infinite hierarchy of theories. We give a diagonalization result that stems from our attempts to separate the hierarchy for. (shrink)

We study the long-standing open problem of giving$\forall {\rm{\Sigma }}_1^b$separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the$\forall {\rm{\Sigma }}_1^b$Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole (...) principle for FPNPfunctions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of$T_2^1$augmented with the surjective weak pigeonhole principle for polynomial time functions. (shrink)