R. Raz. Resolution lower bounds for the weak pigeonhole principle. Technical Report 21, Electronic Colloquium on Computational Complexity, 2001. Avaliable at http://www.eccc.uni-trier.de/eccc/.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....to compute with depth d circuits of bottom fan in k. Because resolution may be viewed as Res(1) our results for Res(k) generalize known results for resolution. The weak pigeonhole principle (for any number of pigeons) is known to require an exponential number of steps to refute in resolution [31, 20, 32, 12, 6, 16, 26, 28, 29], and we generalize these lower bounds for the case of the cn to n pigeonhole principle. Resolution refutations of randomly chosen sets of clauses are also known to require exponential size [14, 6, 5, 9] We extend these results to general Res(k) systems, although as k increases, so does the width ....

....with small restrictions still suffers from the limitations of random restriction method. In particular, it seems ineffective against random 3 CNFs and very weak pigeonhole principles. The only techniques for understanding the refutation complexity of such CNFs seem specific to resolution [9, 8, 28, 29]. Understanding the refutation complexity of these principles in Res(k) is a necessary step before understanding them in more powerful systems, and the Res(k) systems might be simple enough for the development of new techniques. With this in mind, we suggest the following open problems as ....

R. Raz. Resolution lower bounds for the weak pigeonhole principle. In Proceedings of the ThirtyFourth Annual ACM Symposium on Theory of Computing (STOC), 2002.

....Haken [18] proved an exponential lower bound for the smallest resolution proofs of the Pigeonhole Principle, its strength has been studied in depth. The focus has been put in two related directions: 1) proving strong lower bounds for interesting tautologies arising from combinatorial principles [28, 11, 7, 9, 4, 24, 25], and (2) the study of the complexity of finding resolution proofs [7, 9, 3, 6] This research is still ongoing, and it seems that further study in both directions is necessary in order to completely understand the power of resolution. An important step towards the understanding of the strength ....

....to prove size lower bounds for some interesting cases such as the Weak Pigeonhole Principle. In fact, Bonet and Galesi [10] proved that the size width trade off is tight and therefore the technique cannot be applied to it. The problem about the Weak Pigeonhole Principle was finally solved by Raz [24] using a completely different technique. Our goal in this paper is to establish a tight connection between the resolution width of Ben Sasson and Wigderson, and the existential k pebble game, first introduced by Kolaitis and Vardi [19, 20] in the context of finite model theory. Research in this ....

R. Raz. Resolution lower bounds for the weak pigeonhole principle. In 34th Annual ACM Symposium on the Theory of Computing, 2002.

.... system PCR (that is a natural common extension of Polynomial Calculus and Resolution) but only when m o(n ) This poor input output ratio hindered their potential application to proving that NP P poly is hard even for Resolution, and this was established by somewhat di#erent methods in [RanRaz02, Raz02b] A prominent general way for enhancing the I O performance of pseudorandom generators in proof complexity has been recently proposed in [Kra02] Like the classical constructions in computational complexity [Yao82, GGM86] it is very natural to try to achieve this goal by composing the ....

....proof system and, moreover, Definition 2.2 severely restricts the choice of the encoding for the circuit C A,# . Thus, we should be careful in checking that the natural reduction can be indeed carried over with the limited tools at our disposal (cf. the previous arguments of this sort in [Raz98, RanRaz02, Raz02b, Kra02] Let f n be a Boolean function in n variables and n . We begin with reproducing the formal definition of the CNF Circuit t (f n ) from [Raz98, Raz02b] First, we list all variables of Circuit t (f n ) some of them have peculiar long names like InputType # # (v) along ....

R. Raz. Resolution lower bounds for the weak pigeonhole principle. In Proceedings of the 34th ACM Symposium on the Theory of Computing, pages 553--562, 2002.

....[Hak85] proved an exponential lower bound for the smallest resolution proofs of the Pigeonhole Principle, its strength has been studied in depth. The focus has been put in two related directions: 1) proving strong lower bounds for interesting tautologies arising from combinatorial principles [Urq87, CS88, BP96, BSW01, ABSRW00, Raz02a, Raz02b], and (2) the study of the complexity of finding resolution proofs [BP96, BSW01, AR01, AB02] This research is still ongoing, and it seems that further study in both directions is necessary in order to completely understand the power of resolution. An important step towards the understanding of ....

....to prove size lower bounds for some interesting cases such as the Weak Pigeonhole Principle. In fact, Bonet and Galesi [BG01] proved that the size width trade off is tight and therefore the technique cannot be applied to it. The problem about the Weak Pigeonhole Principle was finally solved by Raz [Raz02a] using a completely different technique. Our goal in this paper is to establish a tight connection between the resolution width of Ben Sasson and Wigderson, and the existential k pebble game, first introduced by Kolaitis and Vardi [KV95, KV00a] in the context of finite model theory. Research in ....

R. Raz. Resolution lower bounds for the weak pigeonhole principle. In 34th Annual ACM Symposium on the Theory of Computing, 2002. To appear.

.... years, several size lower bounds have been obtained for various families of CNF formulas, starting with the lower bound of Tseitin for regular resolution presented in 1968 [34] and followed by the lower bound of Haken in 1985 [24] that eventually let the way to many other such bounds (e.g. [35, 20, 6, 14, 30, 31] and many others) A second complexity measure, closely related to the size is the minimal width, measured as the maximal size of a clause in the proof. This measure, introduced by [14] is actually a space measure, as it counts the maximal space a single clause will occupy in a proof. ....

R. Raz. Resolution Lower Bounds for the Weak Pigeonhole Principle. Found at Electronic Colloqium on Computational Complexity, Reports Series 2001, Available at http://www.eccc.unitrier. de/eccc/. Technical Report TR97-021.

....Proofs in every one of the two subsystems of Resolution considered in [24] are in fact regular. The following result by Pitassi and Raz made a major improvement on [24] Theorem 14 ( 25] Every regular resolution proof of FPHP n 2 n must have size exp(n= log n) O(1) Shortly after Raz [26] came up with a complete solution for the basic version PHP m n . By Theorem 11, this immediately extends to the onto version. Theorem 15 ( 26] SR (onto PHP n 2 n ) exp(n= log n) O(1) Razborov [27] gave a simpler proof of the same result. In the next paper [28] the lower bound was ....

....major improvement on [24] Theorem 14 ( 25] Every regular resolution proof of FPHP n 2 n must have size exp(n= log n) O(1) Shortly after Raz [26] came up with a complete solution for the basic version PHP m n . By Theorem 11, this immediately extends to the onto version. Theorem 15 ([26]) SR (onto PHP n 2 n ) exp(n= log n) O(1) Razborov [27] gave a simpler proof of the same result. In the next paper [28] the lower bound was extended to the functional case, and, nally, in [29] the weakest functional onto version was also analyzed. Theorem 16 ( 27 29] SR (onto FPHP ....

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Raz, R.: Resolution lower bounds for the weak pigeonhole principle. Technical Report TR01-021, Electronic Colloquium on Computational Complexity (2001)

....Resolution Lower Bounds for the Weak Pigeonhole Principle Alexander A. Razborov July 11, 2001 Abstract Recently, Raz [Raz01] established exponential lower bounds on the size of resolution proofs of the weak pigeonhole principle. We give another proof of this result which leads to better numerical bounds. Specifically, we show that every resolution proof of PHP m n must have size exp Gamma Omega Gamma n= log m) ....

.... as the resolution size is concerned, the case of generator tautologies is still completely open) RWY97] proved exponential 2 lower bounds for a subsystem of regular resolution (so called rectangular calculus) PR00] proved such bounds for unrestricted regular resolution, and recently Raz [Raz01] completely solved the case of general resolution proofs for the version of the weak pigeonhole principle in which the axioms forbidding pigeons to split between several holes are missing. The main goal of this paper is to present another (and, probably, simpler) proof of the latter result; we ....

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. Manuscript, 2001.

....for m n , no non trivial lower bound was known until very recently. In the last two years, these problems were completely solved. An exponential lower bound for any Regular Resolution proof was proved in [8] and an exponential lower bound for any Resolution proof was nally proved in [9]. More precisely, it was proved in [9] that for any m, any Resolution proof for the weak pigeonhole principle WPHP n is of length ) where 0 is some global constant ( 1=8) The lower bound was further improved in several results by Razborov. The rst result [13] presents a proof for an ....

....lower bound was known until very recently. In the last two years, these problems were completely solved. An exponential lower bound for any Regular Resolution proof was proved in [8] and an exponential lower bound for any Resolution proof was nally proved in [9] More precisely, it was proved in [9] that for any m, any Resolution proof for the weak pigeonhole principle WPHP n is of length ) where 0 is some global constant ( 1=8) The lower bound was further improved in several results by Razborov. The rst result [13] presents a proof for an improved lower bound of ) for ....

[Article contains additional citation context not shown here]

Raz, R., \Resolution lower bounds for the weak pigeonhole principle," Symposium on Theory of Computing, 2002.

....the first super polynomial lower bounds for unrestricted Resolution proofs of PHP n , for m = n 1 [10] This lower bound was generalized by Buss and Turan [8] for m n . For the next 10 years, the resolution complexity of PHP n for m n was completely open. A recent result due to Raz [17] gives exponential Resolution lower bounds for the weak pigeonhole principle, and subsequently Razborov has resolved the problem for most interesting variants of the PHP [20] Substantially less is known about the complexity of the pigeonhole principle in bounded depth Frege systems, although ....

R. Raz. Resolution lower bounds for the weak pigeonhole principle. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pages 553--562, Montreal, Quebec, Canada, May 2002.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. Technical Report 21, Electronic Colloquium on Computational Complexity, 2001. Avaliable at http://www.eccc.uni-trier.de/eccc/.

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R. Raz, Resolution Lower Bounds for the Weak Pigeonhole Principle, in: Proc. of the 34th STOC, (2002), pp.553-562.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. Manuscript, 2001.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. Manuscript, 2001.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. 34th ACM Symposium on Theory of computing, STOC 2002, pp. 553--562.

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R. Raz, Resolution lower bounds for the weak pigeonhole principle, in: Proceedings of the 34th ACM Symposium on the Theory of Computing, 2002, pp. 553--562.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. Technical Report TR01-021, Electronic Colloquium on Computational Complexity, 2001.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. Technical Report TR01-021, Electronic Colloquium in Computation Complexity, http://www.eccc.uni-trier.de/eccc/, 2001.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. Journal of the ACM, 51(2):115--138, 2004.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. In Proceedings of the ThirtyFourth Annual ACM Symposium on Theory of Computing (STOC), 2002.

No context found.

R. Raz. Resolution lower bounds for the weak pigeonhole principle. In Proceedings of the 34th ACM Symposium on the Theory of Computing, pages 553-562, 2002.

No context found.

R. Raz. Resolution lower bounds for the weak pigeonhole principle, in Proceedings of ACM STOC'2002.

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R. Raz. Resolution lower bounds for the weak pigeonhole principle. Manuscript, 2001.

No context found.

R. Raz. Resolution lower bounds for the weak pigeonhole principle. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pages 553--562, 2002.

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