Homogenization and the Polynomial Calculus

Buresh-Oppenheim, Joshua; Clegg, Matthew; Impagliazzo, Russell; Pitassi, Toniann

doi:10.1007/3-540-45022-x_78

Cited by 12 publications

(17 citation statements)

References 12 publications

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“…polynomial calculus exponential separation [48] cutting planes exponential separation [44] Lovász-Schrijver unknown OBDD refutations unknown Figure 5. Comparisons between the DAG-like and treelike forms of some proof systems.…”

Section: Constant-depth Frege With Counting Gates: a Natural Extensiomentioning

confidence: 99%

The Complexity of Propositional Proofs

Segerlind¹

2007

Bull. symb. log.

173

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Propositional proof complexity is the study of the sizes of propositional proofs, and more generally, the resources necessary to certify propositional tautologies. Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms. This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.

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Section: Constant-depth Frege With Counting Gates: a Natural Extensiomentioning

confidence: 99%

The Complexity of Propositional Proofs

Segerlind¹

2007

Bull. symb. log.

173

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“…There exist several approaches and applications of this traslation, which allow the use of algebraic tools (as Gröbner Basis) for solving logical problems (see e.g. [5,13,10] and the application given in [19]). This section is devoted to review the main features.…”

Section: Propositional Logic and The Ring F [X]mentioning

confidence: 99%

Conservative Retractions of Propositional Logic Theories by Means of Boolean Derivatives: Theoretical Foundations

Aranda-Corral

Borrego-Díaz

Fernández-Lebrón

2009

Lecture Notes in Computer Science

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Abstract. We present a specialised (polynomial-based) rule for the propositional logic called the Independence Rule, which is useful to compute the conservative retractions of propositional logic theories. In this paper we show the soundness and completeness of the logical calculus based on this rule, as well as other applications. The rule is defined by means of a new kind of operator on propositional formulae. It is based on the boolean derivatives on the polynomial ring F2 [x].

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“…N is indivisible by m), the Nullstellensatz system [5,14,8,15,12], which captures static polynomial reasoning, and the polynomial calculus [16,26,18,10,13], which captures iterative polynomial reasoning.…”

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confidence: 99%

Constant-depth Frege systems with counting axioms polynomially simulate Nullstellensatz refutations

Impagliazzo

Segerlind

2006

ACM Trans. Comput. Logic

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Abstract. We show that constant-depth Frege systems with counting axioms modulo m polynomially simulate Nullstellensatz refutations modulo m. Central to this is a new definition of reducibility from formulas to systems of polynomials with the property that, for most previously studied translations of formulas to systems of polynomials, a formula reduces to its translation. When combined with a previous result of the authors, this establishes the first size separation between Nullstellensatz and polynomial calculus refutations. We also obtain new, small refutations for certain CNFs by constant-depth Frege systems with counting axioms.

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Homogenization and the Polynomial Calculus

Cited by 12 publications

References 12 publications

The Complexity of Propositional Proofs

The Complexity of Propositional Proofs

Conservative Retractions of Propositional Logic Theories by Means of Boolean Derivatives: Theoretical Foundations

Constant-depth Frege systems with counting axioms polynomially simulate Nullstellensatz refutations

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