Related papers: Polylogarithmic Cuts in Models of V^0
Proof-theoretic methods are developed for subsystems of Johansson's logic obtained by extending the positive fragment of intuitionistic logic with weak negations. These methods are exploited to establish properties of the logical systems.…
We investigate the proof complexity of extended Frege (EF) systems for basic transitive modal logics (K4, S4, GL, ...) augmented with the bounded branching axioms $\mathbf{BB}_k$. First, we study feasibility of the disjunction property and…
We prove that (additive) ordered group reducts of nonstandard models of the bounded arithmetical theory $\mathsf{VTC^0}$ are recursively saturated in a rich language with predicates expressing the integers, rationals, and logarithmically…
In this note we show that unsatisfiable systems of linear equations with a constant number of variables per equation over prime finite fields have polynomial-size constant-degree semi-algebraic proofs of unsatisfiability. These are proofs…
In this note we show that any $k$-CNF which can be refuted by a quasi-polynomial $\mathsf{Res}^*(\mathsf{polylog})$ refutation has a "narrow" refutation in $\mathsf{Res}$ (i.e., of poly-logarithmic width). We also show the converse…
We investigate non-wellfounded proof systems based on parsimonious logic, a weaker variant of linear logic where the exponential modality ! is interpreted as a constructor for streams over finite data. Logical consistency is maintained at a…
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 $VTC^0_2$.…
Motivated by the necessity to include so-called logarithmic operators in conformal field theories (Gurarie, 1993) at values of the central charge belonging to the logarithmic series c_{1,p}=1-6(p-1)^2/p, reducible but indecomposable…
We analyze Kumar's recent quadratic algebraic branching program size lower bound proof method (CCC 2017) for the power sum polynomial. We present a refinement of this method that gives better bounds in some cases. The lower bound relies on…
Cyclic and non-wellfounded proofs are now increasingly employed to establish metalogical results in a variety of settings, in particular for type systems with forms of (co)induction. Under the Curry-Howard correspondence, a cyclic proof can…
Frege's theorem says that second-order Peano arithmetic is interpretable in Hume's Principle and full impredicative comprehension. Hume's Principle is one example of an abstraction principle, while another paradigmatic example is Basic Law…
This paper presents new constructions of models of Hume's Principle and Basic Law V with restricted amounts of comprehension. The techniques used in these constructions are drawn from hyperarithmetic theory and the model theory of fields,…
The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas --…
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…
The \emph{sum-of-squares (SoS) complexity} of a $d$-multiquadratic polynomial $f$ (quadratic in each of $d$ blocks of $n$ variables) is the minimum $s$ such that $f = \sum_{i=1}^s g_i^2$ with each $g_i$ $d$-multilinear. In the case $d=2$,…
We study a mathematical model of a compressible viscous fluid driven by stochastic forces under slip boundary conditions of friction type. We introduce a notion of a weak solution that is analytically and probabilistically consistent with…
We prove super-polynomial lower bounds for low-depth arithmetic circuits using the shifted partials measure [Gupta-Kamath-Kayal-Saptharishi, CCC 2013], [Kayal, ECCC 2012] and the affine projections of partials measure [Garg-Kayal-Saha, FOCS…
We introduce a natural notion of depth that applies to individual cutting planes as well as entire families. This depth has nice properties that make it easy to work with theoretically, and we argue that it is a good proxy for the practical…
Cut-elimination is the bedrock of proof theory with a multitude of applications from computational interpretations to proof analysis. It is also the starting point for important meta-theoretical investigations including decidability,…
We extend the classical Feferman-Vaught theorem to logic for metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent, and therefore they are isomorphic under the Continuum…