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We show that an LK proof of size m of a monotone sequent (a sequent that contains only formulas in the basis ^; _) can be turned into a proof containing only monotone formulas of size m O(logm) and with the number of proof lines polynomial in m. Also we show that some interesting special cases, namely the functional and the onto versions of PHP and a… (More)

An exponential lower bound for the size of tree-like cutting planes refutations of a certain family of conjunctive normal form (CNF) formulas with polynomial size resolution refu-tations is proved. This implies an exponential separation between the tree-like versions and the dag-like versions of resolution and cutting planes. In both cases only… (More)

Proofs of computational effort were devised to control denial of service attacks. Dwork and Naor (CRYPTO '92), for example, proposed to use such proofs to discourage spam. The idea is to couple each email message with a proof of work that demonstrates the sender performed some computational task. A proof of work can be either CPU-bound or memory-bound. In a… (More)

We study the complexity of proving the Pigeon Hole Principle (PHP) in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We show that the standard encoding of the PHP as a monotone sequent admits quasipolynomial-size proofs in this system. This result is a consequence of deriving the basic properties of certain quasipolynomial-size… (More)

We devise a new combinatorial framework for proving space lower bounds in algebraic proof systems like Polynomial Calculus (Pc) and Polynomial Calculus with Resolution (Pcr). Our method can be thought as a Spoiler-Duplicator game, which is capturing boolean reasoning on polynomials instead that clauses as in the case of Resolution. Hence, for the first… (More)

There are methods to turn short refutations in <i>polynomial calculus</i> (Pc) and <i>polynomial calculus with resolution</i> (Pcr) into refutations of low degree. Bonet and Galesi [1999, 2003] asked if such size-degree tradeoffs for Pc [Clegg et al. 1996; Impagliazzo et al. 1999] and Pcr [Alekhnovich et al. 2004] are optimal.
We answer this question by… (More)

Parameterized Resolution and, moreover, a general framework for parameterized proof complexity was introduced by Dantchev, Martin, and Szeider [16] (FOCS'07). In that paper, Dantchev et al. show a complexity gap in tree-like Parameterized Resolution for propositional formulas arising from translations of first-order principles. We broadly investigate… (More)

We study the space complexity of the cutting planes proof system, in which the lines in a proof are integral linear inequalities. We measure the space used by a refutation as the number of linear inequalities that need to be kept on a blackboard while verifying it. We show that any unsatisfiable set of linear inequalities has a cutting planes refutation in… (More)

We present a new method for proving rank lower bounds for the cutting planes procedures of Gomory and Chvátal (GC) and Lovász and Schrijver (LS), when viewed as proof systems for unsatisfiability. We apply this method to obtain the following new results: First, we prove near-optimal rank bounds for GC and LS proofs for several prominent unsatisfiable CNF… (More)