Using the Groebner basis algorithm to find proofs of unsatisfiability

Clegg, Matthew; Edmonds, Jeffery; Impagliazzo, Russell

doi:10.1145/237814.237860

Cited by 261 publications

(289 citation statements)

References 8 publications

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“…For instance, Davis et al (1962), and Beame & Pitassi (1996) gave algorithms for resolution, and Clegg et al (1996) for polynomial calculus. The algorithms of and Beame & Pitassi (1996) are both weakly exponential for resolution, and that of Clegg et al (1996) is polynomial for the system of polynomial calculus with bounded polynomial degree. Therefore this bounded-degree polynomial calculus is automatizable.…”

Section: Introductionmentioning

confidence: 99%

Optimality of size-width tradeoffs for resolution

Bonet¹,

Galesi²

2001

Computational Complexity

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Abstract. This paper is concerned with the complexity of proofs and of searching for proofs in resolution. We show that the recently proposed algorithm of Ben-Sasson & Wigderson for searching for proofs in resolution cannot give better than weakly exponential performance. This is a consequence of our main result: we show the optimality of the general relationship called size-width tradeoff in Ben-Sasson & Wigderson. Moreover we obtain the optimality of the size-width tradeoff for the widely used restrictions of resolution: regular, Davis-Putnam, negative, positive.

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Section: Introductionmentioning

confidence: 99%

Optimality of size-width tradeoffs for resolution

Bonet¹,

Galesi²

2001

Computational Complexity

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“…The main theorems of this paper remain valid under a larger class of notions of uniformity. In [15], we use these notions of uniformity to show gaps and lower bounds on the complexity of algebraic proofs of ideal membership [1], [7], [4], [6], for S n -closed, uniformly generated polynomial systems. Another interesting use of the results in this paper is based on the following observation.…”

Section: Discussionmentioning

confidence: 99%

Uniformly Generated Submodules of Permutation Modules

Riis¹,

Sitharam²

1998

BRICS

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This paper is motivated by a link between algebraic proof complexity and the representation theory of the finite symmetric groups. Our perspective leads to a series of non-traditional problems in the representation theory of Sn. Most of our technical results concern the structure of "uniformly" generated submodules of permutation modules. We consider (for example) sequences Wn of submodules of the permutation modules M(n−k;1k) and prove that if the modules Wn are given in a uniform way - which we make precise - the dimension p(n) of Wn (as a vector space) is a single polynomial with rational coefficients, for all but finitely many "singular" values of n. Furthermore, we show that dim(Wn) < p(n) for each singular value of n >= 4k. The results have a non-traditional flavor arising from the study of the irreducible structure of the submodules Wn beyond isomorphism types. We sketch the link between our structure theorems and proof complexity questions, which can be viewed as special cases of the famous NP vs. co-NP problem in complexity theory. In particular, we focus on the efficiency of proof systems for showing membership in polynomial ideals, for example, based on Hilbert's Nullstellensatz.

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“…4) Our motivation for the lower bound came from [23]. A similar lower bound is mentioned in [28,29], based on the manuscript of Stålmarck.…”

Section: Asymmetric Widthmentioning

confidence: 99%

Unified Characterisations of Resolution Hardness Measures

Beyersdorff

Kullmann

2014

Lecture Notes in Computer Science

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Abstract. Various "hardness" measures have been studied for resolution, providing theoretical insight into the proof complexity of resolution and its fragments, as well as explanations for the hardness of instances in SAT solving. In this paper we aim at a unified view of a number of hardness measures, including different measures of width, space and size of resolution proofs. Our main contribution is a unified game-theoretic characterisation of these measures. As consequences we obtain new relations between the different hardness measures. In particular, we prove a generalised version of Atserias and Dalmau's result on the relation between resolution width and space from [5].

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Using the Groebner basis algorithm to find proofs of unsatisfiability

Cited by 261 publications

References 8 publications

Optimality of size-width tradeoffs for resolution

Optimality of size-width tradeoffs for resolution

Uniformly Generated Submodules of Permutation Modules

Unified Characterisations of Resolution Hardness Measures

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