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Algebraic Proof Complexity: Progress, Frontiers and Challenges
[article]

2016
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arXiv
*
pre-print

, 2008b , Raz and

arXiv:1607.00443v1
fatcat:xdihpreuvfbyrnpkkxxv6znnra
*Tzameret*, 2008a ,*Tzameret*, 2011 in the context of the polynomial calculus proof system. ... Hrubeš-Tzameret's work [Hrubeš and*Tzameret*, 2009 ] investigated the PIT∈? ...##
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Complexity of Propositional Proofs under a Promise
[article]

2007
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arXiv
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pre-print

We study -- within the framework of propositional proof complexity -- the problem of certifying unsatisfiability of CNF formulas under the promise that any satisfiable formula has many satisfying assignments, where "many" stands for an explicitly specified function in the number of variables n. To this end, we develop propositional proof systems under different measures of promises (that is, different ) as extensions of resolution. This is done by augmenting resolution with axioms that,

arXiv:0707.4255v1
fatcat:5iigbvcyw5e5xp7g5z2leb23qa
## more »

... can eliminate sets of truth assignments defined by Boolean circuits. We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises: 1. Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is 2^n, for any constant 0<<1. 2. There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2^δ n (and the number of clauses is o(n^3/2)), for any constant 0<δ<1.##
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Algebraic Proofs over Noncommutative Formulas
[article]

2010
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arXiv
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pre-print

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege---yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in [BIKPRS96,GH03]. We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are written as ordered formulas (PC over

arXiv:1004.2159v2
fatcat:y4bunmfvcfevthvl64zhge3hue
## more »

... rdered formulas, for short): an ordered polynomial is a noncommutative polynomial in which the order of products in every monomial respects a fixed linear order on variables; an algebraic formula is ordered if the polynomial computed by each of its subformulas is ordered. We show that PC over ordered formulas is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR) and admits polynomial-size refutations for the pigeonhole principle and the Tseitin's formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas. The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in algebraic circuit complexity) for establishing lower bounds on propositional proofs.##
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Algebraic Proofs over Noncommutative Formulas
[chapter]

2010
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Lecture Notes in Computer Science
*

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in Buss et al. (1997) and Grigoriev and Hirsch (2003) . We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are

doi:10.1007/978-3-642-13562-0_7
fatcat:oxhqxiddd5fl5mcondl7ayhore
## more »

... tten as ordered formulas (PC over ordered formulas, for short). Given some fixed linear order on variables, an arithmetic formula is ordered if for each of its product gates the left subformula contains only variables that are less-than or equal, according to the linear order, than the variables in the right subformula of the gate. We show that PC over ordered formulas (when the base field is of zero characteristic) is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR), and admits polynomial-size refutations for the pigeonhole principle and Tseitin's formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas (Nisan, 1991) . The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs.##
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Uniform, Integral and Feasible Proofs for the Determinant Identities
[article]

2018
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arXiv
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pre-print

axiomatic assumptions in which one can develop basic linear algebra, we give a uniform and integral version of the short propositional proofs for the determinant identities demonstrated over GF(2) in Hrubes-

arXiv:1811.04313v1
fatcat:ppzlndegdbhsxlwwdjixir2wp4
*Tzameret*... Quite recently, Hrubeš and*Tzameret*[HT15] showed that at least in the propositional case, the determinant identities expressing the multiplicativity of the determinant over GF (2) can be proved with ... We achieved this by formalizing in the theory VNC 2 the construction of the PI-proof demonstrated in Hrubeš-*Tzameret*[HT15] , and using a reflection principle for PI-proofs in the theory. ...##
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The Strength of Multilinear Proofs

2008
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Computational Complexity
*

We introduce an algebraic proof system that manipulates multilinear arithmetic formulas. We show that this proof system is fairly strong, even when restricted to multilinear arithmetic formulas of a very small depth. Specifically, we show the following: 1. Algebraic proofs manipulating depth 2 multilinear arithmetic formulas polynomially simulate Resolution, Polynomial Calculus (PC) and Polynomial Calculus with Resolution (PCR) proofs; 2. Polynomial size proofs manipulating depth 3 multilinear

doi:10.1007/s00037-008-0246-0
fatcat:qgyc6hxogbc5rdejpnqdjbo57m
## more »

... rithmetic formulas for the functional pigeonhole principle; 3. Polynomial size proofs manipulating depth 3 multilinear arithmetic formulas for Tseitin's graph tautologies. By known lower bounds, this demonstrates that algebraic proof systems manipulating depth 3 multilinear arithmetic formulas are strictly stronger than Resolution, PC and PCR, and have an exponential gap over bounded-depth Frege for both the functional pigeonhole principle and Tseitin's graph tautologies. We also illustrate a connection between lower bounds on multilinear proofs and lower bounds on multilinear arithmetic circuits. In particular, we show that (an explicit) superpolynomial size separation between proofs manipulating general arithmetic circuits and proofs manipulating multilinear arithmetic circuits implies a super-polynomial size lower bound on multilinear arithmetic circuits for an explicit family of polynomials.##
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Simple Hard Instances for Low-Depth Algebraic Proofs
[article]

2022
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arXiv
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pre-print

relies on extending the recent breakthrough lower bounds against constant-depth algebraic circuits by Limaye, Srinivasan and Tavenas (FOCS'21) to the functional lower bound framework of Forbes, Shpilka,

arXiv:2205.07175v1
fatcat:5rpfawkfqrco7bkbbkwcao4mgu
*Tzameret*... Forbes, Shpilka,*Tzameret*and Wigderson [8] considered subsystems of IPS using read-once oblivious algebraic programs (roABP) and multilinear formulas over large fields. ... This idea has circulated in proof complexity starting from Pitassi [18, 19] , and subsequently in Grigoriev and Hirsch [10] , Raz and*Tzameret*[22, 21, 27] , and finally in the introduction of the Ideal ...##
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Short Proofs for the Determinant Identities
[article]

2013
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arXiv
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pre-print

We study arithmetic proof systems P_c(F) and P_f(F) operating with arithmetic circuits and arithmetic formulas, respectively, that prove polynomial identities over a field F. We establish a series of structural theorems about these proof systems, the main one stating that P_c(F) proofs can be balanced: if a polynomial identity of syntactic degree d and depth k has a P_c(F) proof of size s, then it also has a P_c(F) proof of size poly(s,d) and depth O(k+^2 d + d s). As a corollary, we obtain a

arXiv:1112.6265v2
fatcat:ymrua6bsyfg4hj3udd4ujqjmr4
## more »

... asipolynomial simulation of P_c(F) by P_f(F), for identities of a polynomial syntactic degree. Using these results we obtain the following: consider the identities det(XY) = det(X)det(Y) and det(Z)= z_11... z_nn, where X,Y and Z are nxn square matrices and Z is a triangular matrix with z_11,..., z_nn on the diagonal (and det is the determinant polynomial). Then we can construct a polynomial-size arithmetic circuit det such that the above identities have P_c(F) proofs of polynomial-size and O(^2 n) depth. Moreover, there exists an arithmetic formula det of size n^O( n) such that the above identities have P_f(F) proofs of size n^O( n). This yields a solution to a basic open problem in propositional proof complexity, namely, whether there are polynomial-size NC^2-Frege proofs for the determinant identities and the hard matrix identities, as considered, e.g., in Soltys and Cook (2004) (cf., Beame and Pitassi (1998)). We show that matrix identities like AB=I → BA=I (for matrices over the two element field) as well as basic properties of the determinant have polynomial-size NC^2-Frege proofs, and quasipolynomial-size Frege proofs.##
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Algebraic proofs over noncommutative formulas

2011
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Information and Computation
*

We study possible formulations of algebraic propositional proof systems operating with noncommutative formulas. We observe that a simple formulation gives rise to systems at least as strong as Frege, yielding a semantic way to define a Cook-Reckhow (i.e., polynomially verifiable) algebraic analog of Frege proofs, different from that given in Buss et al. (1997) and Grigoriev and Hirsch (2003) . We then turn to an apparently weaker system, namely, polynomial calculus (PC) where polynomials are

doi:10.1016/j.ic.2011.07.004
fatcat:ykgqwbjuxnd5xehouzbqs5aeqm
## more »

... tten as ordered formulas (PC over ordered formulas, for short). Given some fixed linear order on variables, an arithmetic formula is ordered if for each of its product gates the left subformula contains only variables that are less-than or equal, according to the linear order, than the variables in the right subformula of the gate. We show that PC over ordered formulas (when the base field is of zero characteristic) is strictly stronger than resolution, polynomial calculus and polynomial calculus with resolution (PCR), and admits polynomial-size refutations for the pigeonhole principle and Tseitin's formulas. We conclude by proposing an approach for establishing lower bounds on PC over ordered formulas proofs, and related systems, based on properties of lower bounds on noncommutative formulas (Nisan, 1991) . The motivation behind this work is developing techniques incorporating rank arguments (similar to those used in arithmetic circuit complexity) for establishing lower bounds on propositional proofs.##
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First-Order Reasoning and Efficient Semi-Algebraic Proofs
[article]

2021
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arXiv
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pre-print

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical

arXiv:2105.07531v2
fatcat:amomceoul5ex3ky3fymqhitsry
## more »

... gation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds. This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones. We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.##
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Complexity of propositional proofs under a promise

2010
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ACM Transactions on Computational Logic
*

We study -within the framework of propositional proof complexity -the problem of certifying unsatisfiability of CNF formulas under the promise that any satisfiable formula has many satisfying assignments, where "many" stands for an explicitly specified function Λ in the number of variables n. To this end, we develop propositional proof systems under different measures of promises (that is, different Λ) as extensions of resolution. This is done by augmenting resolution with axioms that, roughly,

doi:10.1145/1740582.1740586
fatcat:ehy3fl6bcze3ndcpckgl535dpq
## more »

... can eliminate sets of truth assignments defined by Boolean circuits. We then investigate the complexity of such systems, obtaining an exponential separation in the average-case between resolution under different size promises: (i) Resolution has polynomial-size refutations for all unsatisfiable 3CNF formulas when the promise is ε·2 n , for any constant 0 < ε < 1. (ii) There are no sub-exponential size resolution refutations for random 3CNF formulas, when the promise is 2 δn (and the number of clauses is o(n 3/2 )), for any constant 0 < δ < 1.##
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The Proof Complexity of Polynomial Identities

2009
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2009 24th Annual IEEE Conference on Computational Complexity
*

Devising an efficient deterministic -or even a nondeterministic sub-exponential time -algorithm for testing polynomial identities is a fundamental problem in algebraic complexity and complexity at large. Motivated by this problem, as well as by results from proof complexity, we investigate the complexity of proving polynomial identities. To this end, we study a class of equational proof systems, of varying strength, operating with polynomial identities written as arithmetic formulas over a

doi:10.1109/ccc.2009.9
dblp:conf/coco/HrubesT09
fatcat:pjxxycpsqfgj5hazpxmhq5gw5a
## more »

... ring. A proof in these systems establishes that two arithmetic formulas compute the same polynomial, and consists of a sequence of equations between polynomials, written as arithmetic formulas, where each equation in the sequence is derived from previous equations by means of the polynomial-ring axioms. We establish the first non-trivial upper and lower bounds on the size of equational proofs of polynomial identities, as follows: 1) Polynomial-size upper bounds on equational proofs of identities involving symmetric polynomials and interpolation-based identities. In particular, we show that basic properties of the elementary symmetric polynomials are efficiently provable already in equational proofs operating with depth-4 formulas, over infinite fields. This also yields polynomial-size depth-4 proofs of the Newton identities, providing a positive answer to a question posed by Grigoriev and Hirsch [5] . 2) Exponential-size lower bounds on (full, unrestricted) equational proofs of identities over certain specific rings. 3) Exponential-size lower bounds on analytic proofs operating with depth-3 formulas, under a certain regularity condition. The "analytic" requirement is, roughly, a condition that forbids introducing arbitrary formulas in a proof and the regularity condition is an additional structural restriction. 4) Exponential-size lower bounds on one-way proofs (of unrestricted depth) over infinite fields. Here, one-way proofs are analytic proofs, in which one is also not allowed to introduce arbitrary constants. Furthermore, we determine basic structural characteriza-tions of equational proofs, and consider relations with polynomial identity testing procedures. Specifically, we show that equational proofs efficiently simulate the polynomial identity testing algorithm provided by Dvir and Shpilka [3] .##
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Short Propositional Refutations for Dense Random 3CNF Formulas
[article]

2011
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arXiv
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pre-print

Random 3CNF formulas constitute an important distribution for measuring the average-case behavior of propositional proof systems. Lower bounds for random 3CNF refutations in many propositional proof systems are known. Most notably are the exponential-size resolution refutation lower bounds for random 3CNF formulas with Ω(n^1.5-ϵ) clauses [Chvatal and Szemeredi (1988), Ben-Sasson and Wigderson (2001)]. On the other hand, the only known non-trivial upper bound on the size of random 3CNF

arXiv:1101.3970v2
fatcat:ljar3wx6vjbmbkqdgn57uwrkqu
## more »

... s in a non-abstract propositional proof system is for resolution with Ω(n^2/ n) clauses, shown by Beame et al. (2002). In this paper we show that already standard propositional proof systems, within the hierarchy of Frege proofs, admit short refutations for random 3CNF formulas, for sufficiently large clause-to-variable ratio. Specifically, we demonstrate polynomial-size propositional refutations whose lines are TC^0 formulas (i.e., TC^0-Frege proofs) for random 3CNF formulas with n variables and Ω(n^1.4) clauses. The idea is based on demonstrating efficient propositional correctness proofs of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek (2006). Since the soundness of these witnesses is verified using spectral techniques, we develop an appropriate way to reason about eigenvectors in propositional systems. To carry out the full argument we work inside weak formal systems of arithmetic and use a general translation scheme to propositional proofs.##
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Sparser Random 3-SAT Refutation Algorithms and the Interpolation Problem
[chapter]

2014
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Lecture Notes in Computer Science
*

Comparison with Müller and

doi:10.1007/978-3-662-43948-7_84
fatcat:jcna7uxo3bgpplvyhbmoabzlem
*Tzameret*[25] . In [25] a polynomial-size TC 0 -Frege proof of the correctness of the Feige et al. witnesses was shown. ...##
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Witnessing matrix identities and proof complexity

2018
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International journal of algebra and computation
*

Hrubeš and

doi:10.1142/s021819671850011x
fatcat:aevrlmha6bd5djqugsialn55fu
*Tzameret*[14] obtained polynomial-size (algebraic and propositional) proofs for certain (suitably encoded) identities concerning matrices. ... Hrubeš-*Tzameret*[13] raised the question whether, assuming that the PIT problem does posses short witnesses, a proof system using only symbolic manipulation (resembling a logical proof system) is enough ...
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