We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson [44], and show that this analysis can be formalized in the bounded arithmetic system VNC^1 (corresponding to the “NC^1 reasoning”). As a corollary, we prove the assumption made by Jeřábek [28] that a construction of certain bipartite expander graphs can be formalized in VNC^1 . This in turn implies that every proof in Gentzen's sequent calculus LK of a (...) monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlák [9]. (shrink)

LetL0be a language on ℕ consisting of 0, 1, +, ∸, ·, ⌊½a⌋, ∣a∣, #, ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b), andμx≤ ∣s∣t(x). Hereμx≤ ∣s∣t(x) is the smallest numberx≤ ∣s∣ satisfyingt(x) > 0 and 0 if there exist no suchxand we stipulate that ifsandt(a) are terms inLo, thenμx≤ ∣s∣t(x) is also a term inLo. The defining axioms of functions ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b) are as follows:LetLa language on ℕ with only predicate constant = andL0⊆L. Letf(b,a1,…,am) be a function for ℕm+1into (...) ℕ. We say “fis weakly expressed by termst1(b,a1,…,am),…,tr(b,a1,…,am) inL” if for everyb,a1,…,am∈ ℕ,f(b,a1,…,am) is equal to one ofti(b,a1, …,am). The critical number ofbinfwith respect toLis the minimum numbernsuch that wheneverf(b,a1,…,an) is weakly expressed by termst1(b,a1, …,an),…, the number of occurrences ofbin someti(b,a1,…,an) is at leastn.As is defined in [1], a functionfis defined by a limited iteration fromgandhwith respect toLiff the following holds: Letτbe defined aswith the condition; andis defined bywhereandare terms inL. We say “fis defined by a short limited iteration fromgandh” ifis defined bywhereτ,s, tare the same as above satisfying the condition. (shrink)

We present two constructions of models of bounded arithmetic, both in the form of a generalization of the ultrapower construction, that yield nonelementary extensions but do not introduce new lengths. As an application we show, assuming the existence of a one-way permutation g hard against polynomial-size circuits, that strictR21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{strict}R^1_2$$\end{document} is weaker than R21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^1_2$$\end{document}. In particular, if such a permutation can be defined by (...) a term in the language of R21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^1_2$$\end{document}, then strictR21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{strict}R^1_2$$\end{document} is weaker than R21\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^1_2$$\end{document}. (shrink)

We develop an arithmetical theory and its variant , corresponding to “slightly nonuniform” . Our theories sit between and , and allow evaluation of log-depth bounded fan-in circuits under limited conditions. Propositional translations of -formulas provable in admit L-uniform polynomial-size Frege proofs.

We prove some independence results for the bounded arithmetic theoryR 2 0 , and we define a class of functions that is shown to be an upper bound for the class of functions definable by a certain restricted class of ∑ 1 b in extensions ofR 2 0.

A number of theories have been developed to characterize ALogTime (or uniform NC 1, or just NC 1), the class of languages accepted by alternating logtime Turing machines, in the same way that Buss’s theory ${{\bf S}^{1}_{2}}$ characterizes polytime functions. Among these, ALV′ (by Clote) is particularly interesting because it is developed based on Barrington’s theorem that the word problem for the permutation group S 5 is complete for ALogTime. On the other hand, ALV (by Clote), T 0 NC 0 (...) (by Clote and Takeuti) as well as Arai’s theory ${{\bf AID}+\Sigma_0^B{-}{\bf CA}}$ and its two-sorted version VNC 1 (by Cook and Morioka) are based on the circuit characterization of ALogTime. While the last three theories have been known to be equivalent, their relationship to ALV′ has been an open problem. Here we show that ALV′ is indeed equivalent to the other theories. (shrink)

We investigate the second order bounded arithmetical systems which is isomorphic to TAC i , TNC i or TLS.

In this paper we introduce a system AID of bounded arithmetic. The main feature of AID is to allow a form of inductive definitions, which was extracted from Buss’ propositional consistency proof of Frege systems F in Buss 3–29). We show that AID proves the soundness of F , and conversely any Σ 0 b -theorem in AID yields boolean sentences of which F has polysize proofs. Further we define Σ 1 b -faithful interpretations between AID+Σ 0 b -CA and (...) a quantified theory QALV of an equational system ALV in Clote 57–106). Hence ALV also proves the soundness of F. (shrink)

This paper defines natural hierarchies of function and relation classes □i,kc and Δi,kc, constructed from parallel complexity classes in a manner analogous to the polynomial-time hierarchy. It is easily shown that □i−1,kp □c,kc □i,kp and similarly for the Δ classes. The class □i,3c coincides with the single-valued functions in Buss et al.'s class , and analogously for other growth rates. Furthermore, the class □i,kc comprises exactly the functions Σi,kb-definable in Ski−1, and if Tki−1 is Σi,kb-conservative over Ski−1, then □i,kp is (...) completely parallelizable. All functions in □i,kc are Σi,kb-definable in Rki; this suffices to show that if the known Σi,kb conservativity between R3i and S3i−1 extends to R2i and S2i−1, then NC = NC1 relative to an oracle in PH. We prove a KPT-style witnessing theorem for Ski using constantly many rounds of □i,kc interactive computation, and thus show that if Ski ≡ Rki+1 then the bounded arithmetic hierarchy collapses, provably in Ski. (shrink)