P. Beame and T. Pitassi. Propositional proof complexity: past, present and future. Technical Report 67, ECCC, 1998.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....explored. Among the most studied are Frege and extended Frege Proof systems [Urq87] and [KP89] refutation systems, most notably resolution [Rob65] and circuit based proof systems [Ajt83] and [Bus87] We brie y discuss the complexity of resolution systems here, but see Beame and Pitassi [BP98] for a nice overview of results concerning these other proof systems. Resolution proof systems are the most well studied model. Resolution is a very restricted proof system and so has provided the setting for the rst lower bound proofs. Resolution proof systems are refutation systems where a ....

P. Beame and T. Pitassi. Propositional proof complexity: Past, present and future. Bull. of the EATCS, 65:66-89, 1998.

....proof systems has close connections to # Supported in part by NSF grant DMS 0100589 and CCR0098197. Supported in part by NSF grant DMS 0100589. # Supported in part by NSF grant CCR 0098197 and USA Israel BSF Grant 97 00188. open problems in computational and circuit complexity (see [15, 23, 27, 7]) as well as implications for the run times of satisfiability algorithms and automated theorem provers. Resolution is one of the most studied proof systems, and is used as the basis for many satisfiability algorithms. Back tracking algorithms such as DPLL that branch on a single variable provide ....

P. Beame and T. Pitassi. Propositional proof complexity: Past, present, and future. Bulletin of the EATCS, 65:66--89, 1998.

....measures related to size, which is the most interesting measure in the circuit complexity framework. In other words, the main e ort in proof complexity was invested in investigating the amount of time (or at least time like resources) taken by proofs; we recommend the excellent recent survey [BP98] for further reading on this subject. During the workshop Complexity Lower Bounds held at the Fields Institute in Toronto in 1998, A. Haken raised the question of whether something intelligent can be said about the amount of space taken by propositional proofs. Quite surprisingly, it turned out ....

P. Beame and T. Pitassi, Propositional proof complexity: Past, present and future, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 65 (1998), pp. 66-89.

.... approach to this problem and use it to prove Theorem 1 For all r 3:145, F 3 (n; rn) is satis able with probability 1 o(1) For the connections of random formulas to proof complexity and computational hardness we refer the interested reader to the excellent surveys by Beame and Pitassi [1] and Cook and Mitchell [8] respectively. The rest of the paper is organized as follows. In Section 2 we summarize most known results regarding the conjecture. In Section 3 we give a more detailed account of our contribution and its relationship to past work. In Section 4 we give the preliminaries ....

Paul Beame and Toniann Pitassi, Propositional proof complexity: past, present, and future, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS (1998), no. 65, 66-89.

....p such that all tautologies F (i.e. all F 2 TAUT ) have proofs in S of size p(jF j) The Cook Reckhow paper led to an extensive and fruitful study of lengths of proofs for standard proof systems of increasing complexity and strength. A recent survey of the area by Beame and Pitassi is contained in [2]. A restatement of the Cook Reckhow result is that if NP 6= coNP then for every sound propositional proof system P and polynomial bound p there is an in nite collection of hard tautologies: i.e. the shortest proofs in P of each such tautology F is of length more than p(jF j) A stronger notion ....

.... p 2 (cf. 6] also described as P NP [O(log n) consists of all sets decidable by a polynomial time oracle machine that can make O(log n) queries to an NP oracle. We start with a de nition of proof systems that is equivalent to the original Cook Reckhow formulation [4] De nition 1. [2] A (sound and complete) propositional proof system S is de ned to be a polynomial time predicate S such that for all F , F 2 TAUT , 9p : S(F; p) In other words, a proof system can be identi ed with an ecient procedure for checking correctness of proofs. Thus, in complexity theoretic terms we ....

P. Beame and T. Pitassi, Propositional proof complexity: past, present, and future, ECCC Report TR98-067, 1998.

.... work, has attracted a lot of attention in computer science, mathematics and, more recently, in mathematical physics [29, 30, 32, 31] For the connections of random fomulae to proof complexity and computational hardness we refer the interested reader to the excellent surveys by Beame and Pitassi [5] and Cook and Mitchell [13] respectively. Allowing replacement simpli es calculations greatly. Moreover, in the interesting range m = n) the expected number of repeated clauses is O(1) and thus it is virtually inconsequential. In particular, all the results discussed in this paper hold also ....

Paul Beame and Toniann Pitassi, Propositional proof complexity: past, present, and future, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS (1998), no. 65, 66-89.

....p such that all tautologies F (i.e. all F 2 TAUT ) have proofs in S of size p(jF j) The Cook Reckhow paper led to an extensive and fruitful study of lengths of proofs for standard proof systems of increasing complexity and strength. A recent survey of the area by Beame and Pitassi is contained in [2]. A restatement of the Cook Reckhow result is that if NP 6= coNP then for every sound propositional proof system P and polynomial bound p there is an infinite collection of hard tautologies: i.e. the shortest proofs in P of each such tautology F is of length more than p(jF j) A stronger notion ....

....the definitions of standard complexity classes like P, NP, coNP, Theta p 2 , NEXP, the polynomial hierarchy PH etc. we refer the reader to a standard textbook, e.g. 1] We start with a definition of proof systems that is equivalent to the original CookReckhow formulation [4] Definition 1 [2] A (sound and complete) propositional proof system S is defined to be a polynomial time predicate S such that for all F , F 2 TAUT , 9p : S(F; p) 2 In other words, a proof system can be identified with an efficient procedure for checking correctness of proofs. Thus, in complexity theoretic ....

P. Beame and T. Pitassi, Propositional proof complexity: past, present, and future, ECCC Report TR98-067, 1998.

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P. Beame and T. Pitassi. Propositional proof complexity: past, present and future. Technical Report 67, ECCC, 1998.

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Paul Beame and Toniann Pitassi, Propositional proof complexity: Past, present, and future, Bulletin of the EATACS, TR98-067, 1998.

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Paul Beame and Toniann Pitassi. Propositional proof complexity: Past, present, and future. Bulletin of the EATACS, TR98-067, 1998.

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P. Beame and T. Pitassi. Propositional proof complexity: Past, present, and future. Bulletin of the EATCS, 65:66--89, 1998.

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P. Beame and T. Pitassi. Propositional proof complexity: past, present, and future, Bulletin of the EATCS 65:66--89, June 1998. external. nj.nec.com/homepages/fortnow/beatcs/column65.ps.

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P. Beame and T. Pitassi, Propositional proof complexity: Past, present and future, Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, 65 (1998), pp. 66--89.

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P. Beame and T. Pitassi, Propositional Proof Complexity: Past, Present and Future, Tech. Rep. TR98067, ECCC, 1998.

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P. Beame and T. Pitassi. Propositional proof complexity: Past, present, and future. Bulletin of the EATCS, 65:66--89, 1998.

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P. Beame and T. Pitassi. Propositional proof complexity: past, present, and future. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS, (65):66-89, 1998.

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