J. Kraj  cek, P. Pudl ak, and A. Woods, Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, 7(1), (1995), pp.15-39.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....x 2 c and (in a very nontrivial way, see e.g. 10, Chapter 5.3] jxj. Unfortunately, the standard proof of this lemma involves conditioning on second order objects (see e.g. the quantifier on F in [5, Lemma 3:3 ] and hence it is not clear a priori how to place it even into V 1 . Woods (see [14], 13, Chapter 15] gave a variant of this proof which avoids such conditioning. Apparently, this is already formalizable in V 1 . For our purposes, however, we need the following more constructive version of Hastad s proof which probably is interesting in its own right. We denote by R the ....

J. Kraj ' icek, P. Pudl'ak, and A. R. Woods. Exponential lower bounds to the size of bounded depth frege proofs of the pigeonhole principle. Submitted to Random Structures and Algorithms, 1993.

.... proof [Bus87] involves a complicated construction of addition circuits while our proof is very simple and intuitive (actually, we employ the ability of F PC to count) Our proof also has depth bounded by a constant, while constant depth Frege systems do not have polynomial size proofs of PHP [KPW95,PBI93]. In fact, our proof can be conducted in constant depth F NS. The results of Section 6, however, do not suce to prove an exponential gap between the lengths of proofs in PC and F PC as propositional proof systems (see the discussion in the end of Section 6) In Section 7 we demonstrate this gap ....

J. Krajcek, P. Pudlak, and A. Woods. Exponential lower bound to the size of bounded depth frege proofs of the pigeonhole principle. Random Structures and Algorithms, 7:15-39, 1995.

....IST 99 14186. Buss proved that PHP n 1 n has polynomial size proofs in Frege systems [3] Ajtai proved a superpolynomial lower bound for bounded depth Frege systems [1] and that was improved to an exponential lower bound by Pitassi, Beame and Impagliazzo [19] and Kraj icek, Pudl ak and Woods [13] independently. For the Weak Pigeonhole Principle PHP 2n n , the situation is quite different. While it is known that PHP 2n n requires exponential size proofs in Resolution [4] Paris, Wilkie and Woods [18, 12] proved that it has quasipolynomial size (n O(log(n) proofs in bounded depth ....

J. Kraj'icek, P. Pudl'ak, and A. Woods. Exponential lower bound to the size of bounded depth frege proofs of the pigeon hole principle. Random Structures and Algorithms, 7(1):15--39, 1995.

....at the same time extracting from it the following explicit bound: Theorem 4 ( 13, 14] For every xed d 0, SFd (onto FPHP n 1 n ) n d log n , where d is a constant depending only on d. Finally, Pitassi, Beame, Impagliazzio [15] and, independently, Kraj cek, Pudlak and Woods [16] improved this lower bound to truly exponential. Theorem 5 ( 15, 16] For every xed d 0, SFd (onto FPHP n 1 n ) exp(n d ) where d is a constant depending only on d. Another proof system for which PHP was the rst tautology to be shown to be hard is Polynomial Calculus. 5 The ....

....bound: Theorem 4 ( 13, 14] For every xed d 0, SFd (onto FPHP n 1 n ) n d log n , where d is a constant depending only on d. Finally, Pitassi, Beame, Impagliazzio [15] and, independently, Kraj cek, Pudlak and Woods [16] improved this lower bound to truly exponential. Theorem 5 ([15, 16]) For every xed d 0, SFd (onto FPHP n 1 n ) exp(n d ) where d is a constant depending only on d. Another proof system for which PHP was the rst tautology to be shown to be hard is Polynomial Calculus. 5 The following lower bound proved by Razborov [17] is applicable to an ....

Krajcek, J., Pudlak, P., Woods, A.R.: Exponential lower bounds to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms 7 (1995) 15-39

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J. Kraj  cek, P. Pudl ak, and A. Woods, Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, 7(1), (1995), pp.15-39.

....sit in n pigeonholes, there must be a pigeonhole occupied by more than one pigeon. One of the major achievements of proof complexity has been to show that this simple claim is hard to prove in various proof systems such as resolution [9] the polynomial calculus [16] and bounded depth Frege [2, 11, 13]. In this paper we give an alternative proof of the hardness of the pigeonhole principle for bounded depth Frege proofs. 1.1 Previous Results The rst super polynomial lower bounds for the pigeonhole principle in bounded depth Frege were presented by Ajtai [2] This proof was simpli ed and ....

.... The rst super polynomial lower bounds for the pigeonhole principle in bounded depth Frege were presented by Ajtai [2] This proof was simpli ed and improved by Bellantoni et al. 4] The rst exponential lower bounds were given by Pitassi et al. 13] and independently by Kraj cek et al. [11]. Several extensions of this result have appeared over the years (see e.g. 6] and the recent [5] Our paper gives an alternative proof of the following exponential lower bound of [13] and [11] Theorem 1 ( 13, 11] For any Frege system F , and any integer d, there exists a constant 0 such ....

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Jan Krajcek, Pavel Pudlak, and Alan Woods. Exponential lower bounds to the size of bounded depth frege proofs of the pigeonhole principle. Random Structure and Algorithms, 7(1):15-39, 1995. 18

.... general we can take fragments of the quanti ed propositional calculus obtained by an appropriate restriction on the quanti er complexity of formulas [18] A large part of the activity is concentrated on proving lower bounds on the lengths of propositional proofs and we can report steady progress [1, 12, 13, 21, 31, 32]. Unfortunately the proof systems for which one can prove that they are not polynomially bounded are still much weaker than Extended Frege; even proving a superpolynomial lower bound for Frege systems would be a breakthrough. Thus we do not expect that concrete independent 1 sentences will be ....

J. Kraj  cek, P. Pudl ak, and A. Woods, An exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random structures and Algorithms 7/1 (1995), pp. 15-39.

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J. Kraj cek, P. Pudl ak, and A. Woods, Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, 7 (1995), pp. 15--39.

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J. Kraj cek, P. Pudl ak, and A. Woods, Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, 7 (1995), pp. 15--39.

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J. Kraj cek, P. Pudl ak, and A. Woods, Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, 7 (1995), pp. 15--39.

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Jan Krajcek, Pavel Pudlak, Alan Woods.: Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms. 7, 15--39 (1995)

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J. Kraj cek, P. Pudl ak, and A. Woods, Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, 7 (1995), pp. 15--39.

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J. Kraj cek, P. Pudl ak, and A. Woods, Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, submitted, (1991).

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Jan Krajcek, Pavel Pudlak, and Alan Woods. Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle. Random Structures and Algorithms, 7:15--39, 1995.

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J. Kraj cek, P. Pudl ak, and A. Woods, Exponential lower bound to the size of bounded depth Frege proofs of the pigeonhole principle, Random Structures and Algorithms, 7 (1995), pp. 15--39.

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