Related papers: A Note on Induction Schemas in Bounded Arithmetic

We study variants of Buss's theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on $\hat\Pi^b_i$ induction schemes,…

We introduce system S^2_0E, a bounded arithmetic corresponding to Buss's S^2_0 with the predicate E which signifies the existence of the value. Then, we show that we can \Sigma^b_2-define truthness of S^2_0 E and therefore we can prove…

This paper proves Buss's hierarchy of bounded arithmetics $S^1_2 \subseteq S^2_2 \subseteq \cdots \subseteq S^i_2 \subseteq \cdots$ does not entirely collapse. More precisely, we prove that, for a certain $D$, $S^1_2 \subsetneq S^{2D+5}_2$…

One of the central open questions in bounded arithmetic is whether Buss' hierarchy of theories of bounded arithmetic collapses or not. In this paper, we reformulate Buss' theories using free logic and conjecture that such theories are…

We consider pure equational theories that allow substitution but disallow induction, which we denote as PETS, based on recursive definition of their function symbols. We show that the Bounded Arithmetic theory $S^1_2$ proves the consistency…

In this paper we introduce a system AID (Alogtime Inductive Definitions) 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…

This paper presents proof that Buss's $S^2_2$ can prove the consistency of a fragment of Cook and Urquhart's $\mathrm{PV}$ from which induction has been removed but substitution has been retained. This result improves Beckmann's result,…

We consider a minimal extension of the language of arithmetic, such that the bounded formulas provably total in a suitably-defined theory \`a la Buss (expressed in this new language) precisely capture polytime random functions. Then, we…

A set of integers is $S$-recognizable in an abstract numeration system $S$ if the language made up of the representations of its elements is accepted by a finite automaton. For abstract numeration systems built over bounded languages with…

We present a self-contained account of Woodin's extender algebra and its use in proving absoluteness results, including a proof of the $\Sigma^2_1$-absoluteness theorem. We also include a proof that the existence of an inner model with…

We introduce a new bounded theory RS^1_2 and show that the functions which are Sigma^b_1-representable in it are precisely random functions which can be computed in polynomial time. Concretely, we pass through a class of oracle functions…

A Turing degree d bounds a principle P of reverse mathematics if every computable instance of P has a d-computable solution. P admits a universal instance if there exists a computable instance such that every solution bounds P. We prove…

We prove the correctness of the AKS algorithm \cite{AKS} within the bounded arithmetic theory $T^{count}_2$ or, equivalently, the first-order consequences of the theory $VTC^0$ expanded by the smash function, which we denote by $VTC^0_2$.…

The elementary arithmetic operations $+,\cdot,\le$ on integers are well-known to be computable in the weak complexity class $\mathrm{TC}^0$, and it is a basic question what properties of these operations can be proved using only…

Let K be a number field and let S be a finite set of places of K which contains all the Archimedean places. For any f(z) in K(z) of degree d at least 2 which is not a d-th power in \bar{K}(z), Siegel's theorem implies that the image set…

We show that many principles of first-order arithmetic, previously only known to lie strictly between $\Sigma_1$-induction and $\Sigma_2$-induction, are equivalent to the well-foundedness of $\omega^\omega$. Among these principles are the…

We show that there is a constant $k$ such that Buss's intuitionistic theory $\mathsf{IS}^1_2$ does not prove that SAT requires co-nondeterministic circuits of size at least $n^k$. To our knowledge, this is the first unconditional…

G\"odel's second incompleteness theorem is proved for Herbrand consistency of some arithmetical theories with bounded induction, by using a technique of logarithmic shrinking the witnesses of bounded formulas, due to Z. Adamowicz [Herbrand…

We prove that the bounded arithmetic theory $S^1_2$ is consistent with EXP $\not\subseteq$ P/poly. More generally, we show that certain separations of $V^1_2$ from a theory $T$ imply the consistency of $T$ with EXP $\not\subseteq$ P/poly.…

It is proved that the continuous bounded cohomology of SL_2(k) vanishes in all positive degrees whenever k is a non-Archimedean local field. This holds more generally for boundary-transitive groups of tree automorphisms and implies low…