Some Subsystems of Constant-Depth Frege with Parity

Resolution over linear equations is a natural extension of the popular resolution refutation system, augmented with the ability to carry out basic counting. Denoted Res(linR), this refutation system operates with disjunctions of linear equations with boolean variables over a ring R, to refute unsatisfiable sets of such disjunctions. Beginning in the work of [26], through the work of [17] which focused on tree-like lower bounds, this refutation system was shown to be fairly strong. Subsequent work (cf. [18, 17, 19, 13]) made it evident that establishing lower bounds against general Res(linR) refutations is a challenging and interesting task since the system captures a "minimal" extension of resolution with counting gates for which no super-polynomial lower bounds are known to date.We provide the first super-polynomial size lower bounds on general (dag-like) resolution over linear equations refutations in the large characteristic regime. In particular we prove that the subset-sum principle 1 + x1 + • • • + 2 n xn = 0 requires refutations of exponential-size over Q. Our proof technique is nontrivial and novel: roughly speaking, we show that under certain conditions every refutation of a subset-sum instance f = 0, where f is a linear polynomial over Q, must pass through a fat clause containing an equation f = α for each α in the image of f under boolean assignments. We develop a somewhat different approach to prove exponential lower bounds against tree-like refutations of any subset-sum instance that depends on n variables, hence also separating tree-like from dag-like refutations over the rationals.We then turn to the finite fields regime, showing that the work of Itsykson and Sokolov [17] who obtained tree-like lower bounds over F2 can be carried over and extended to every finite field. We establish new lower bounds and separations as follows: (i) for every pair of distinct primes p, q, there exist CNF formulas with short tree-like refutations in Res(lin Fp ) that require exponential-size treelike Res(lin Fq ) refutations; (ii) random k-CNF formulas require exponential-size tree-like Res(lin Fp ) refutations, for every prime p and constant k; and (iii) exponential-size lower bounds for tree-like Res(lin F ) refutations of the pigeonhole principle, for every field F.

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“…denotes an open problem. The results marked with [17,13] were proved in the respective papers. All other results are from the current work.…”

Section: Resolution With Countingmentioning

confidence: 76%

Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Part¹,

Tzameret

2021

comput. complex.

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“…In this sense, resolution over linear equations over prime fields and over the integers is interesting as a first step towards AC 0 [p]-Frege lower and TC 0 -Frege lower bounds, respectively. Works by Krajíček [Kra17], Garlik-Ko lodziejczyk [GK17] and Krajíček-Oliveira [KO18] had suggested possible approaches to attack daglike Res(lin F 2 ) lower bounds.…”

Section: Introduction 1backgroundmentioning

confidence: 99%

Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Part¹,

Tzameret²

2018

Preprint

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Resolution over linear equations (introduced in [RT08]) emerged recently as an important object of study. This refutation system, denoted Res(lin R ), operates with disjunction of linear equations over a ring R. 1 On the one hand, the system captures a natural "minimal" extension of resolution in which efficient counting can be achieved; while on the other hand, as observed by, e.g., Krajíček [Kra17] (cf. [IS14, KO18, GK17]), when considered over prime fields, and specifically F 2 , super-polynomial lower bounds on (dag-like) Res(lin F 2 ) is a first step towards the long-standing open problem of establishing constant-depth Frege with counting gates (AC 0 [2]-Frege) lower bounds.In this work we develop new lower bound techniques for resolution over linear equations and extend existing ones to work over different rings. We obtain a host of new lower bounds, separations and upper bounds, while calibrating the relative strength of different sub-systems. We first establish, over fields of characteristic zero, exponential-size dag-like lower bounds against resolution over linear equations refutations of instances with large coefficients. Specifically, we demonstrate that the subset sum principle α 1 x 1 + . . . + α n x n = β, for β not in the image of the linear form, requires refutations proportional to the size of the image. Moreover, for instances with small coefficients, we separate the tree and dag-like versions of Res(lin F ), when F is of characteristic zero, by employing the notion of immunity from Alekhnovich-Razborov [AR01], among other techniques.We then study resolution over linear equations over different finite fields, extending the work of Itsykson and Sokolov [IS14] who developed tree-like Res(lin F 2 ) lower bounds techniques. We obtain new lower bounds and separations as follows: (i) exponential-size lower bounds for tree-like Res(lin Fp ) Tseitin mod q formulas, for every pair of distinct primes p, q. As a corollary we obtain an exponential-size separation between tree-like Res(lin Fp ) and tree-like Res(lin Fq ); (ii) exponential-size lower bounds for tree-like Res(lin Fp ) refutations of random k-CNF formulas, for every prime p and constant k; and (iii) exponential-size lower bounds for tree-like Res(lin F ) refutations of the pigeonhole principle, for every field F.

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Notes on Resolution over Linear Equations

Gryaznov

2019

Computer Science – Theory and Applications

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Some Subsystems of Constant-Depth Frege with Parity

Cited by 9 publications

References 28 publications

Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Resolution with Counting: Dag-Like Lower Bounds and Different Moduli

Notes on Resolution over Linear Equations

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