Results 11  20
of
192
Polynomial size proofs of the propositional pigeonhole principle
 Journal of Symbolic Logic
, 1987
"... Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege syste ..."
Abstract

Cited by 72 (7 self)
 Add to MetaCart
(Show Context)
Abstract. Cook and Reckhow defined a propositional formulation of the pigeonhole principle. This paper shows that there are Frege proofs of this propositional pigeonhole principle of polynomial size. This together with a result of Haken gives another proof of Urquhart's theorem that Frege systems have an exponential speedup over resolution. We also discuss connections to provability in theories of bounded arithmetic. $1. Introduction. The motivation for this paper comes primarily from two sources. First, Cook and Reckhow [2] and Statman [7] discussed connections between lengths of proofs in propositional logic and open questions in computational complexity such as whether NP = coNP. Cook and Reckhow used the propositional pigeonhole principle as an example of a family of true formulae which
Some Connections between Bounded Query Classes and NonUniform Complexity
 In Proceedings of the 5th Structure in Complexity Theory Conference
, 1990
"... This paper is dedicated to the memory of Ronald V. Book, 19371997. ..."
Abstract

Cited by 71 (23 self)
 Add to MetaCart
This paper is dedicated to the memory of Ronald V. Book, 19371997.
The polynomial method in circuit complexity
 Structure in Comoexity Theory Conference
, 1993
"... ..."
(Show Context)
An Exponential Lower Bound to the Size of Bounded Depth Frege . . .
, 1994
"... We prove lower bounds of the form exp (n ffl d ) ; ffl d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for whic ..."
Abstract

Cited by 66 (9 self)
 Add to MetaCart
We prove lower bounds of the form exp (n ffl d ) ; ffl d ? 0; on the length of proofs of an explicit sequence of tautologies, based on the Pigeonhole Principle, in proof systems using formulas of depth d; for any constant d: This is the largest lower bound for the strongest proof system, for which any superpolynomial lower bounds are known.
Unprovability of Lower Bounds on the Circuit Size in Certain Fragments of Bounded Arithmetic
 IN IZVESTIYA OF THE RUSSIAN ACADEMY OF SCIENCE, MATHEMATICS
, 1995
"... We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators secure against the attack by smal ..."
Abstract

Cited by 54 (6 self)
 Add to MetaCart
(Show Context)
We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constantdepth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolationlike theorems for certain “split versions” of classical systems of Bounded Arithmetic introduced in this paper.
Lower Bounds to the Size of ConstantDepth Propositional Proofs
, 1994
"... 1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp ..."
Abstract

Cited by 53 (6 self)
 Add to MetaCart
1 LK is a natural modification of Gentzen sequent calculus for propositional logic with connectives : and V ; W (both of unbounded arity). Then for every d 0 and n 2, there is a set T d n of depth d sequents of total size O(n 3+d ) which are refutable in LK by depth d + 1 proof of size exp(O(log 2 n)) but such that every depth d refutation must have the size at least exp(n\Omega\Gamma21 ). The sets T d n express a weaker form of the pigeonhole principle. It is a fundamental problem of mathematical logic and complexity theory whether there exists a proof system for propositional logic in which every tautology has a short proof, where the length (equivalently the size) of a proof is measured essentially by the total number of symbols in it and short means polynomial in the length of the tautology. Equivalently one can ask whether for every theory T there is another theory S (both first order and reasonably axiomatized, e.g. by schemes) having the property that if a statement...
The History and Status of the P versus NP Question
, 1992
"... this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the re ..."
Abstract

Cited by 52 (1 self)
 Add to MetaCart
this article, I have attempted to organize and describe this literature, including an occasional opinion about the most fruitful directions, but no technical details. In the first half of this century, work on the power of formal systems led to the formalization of the notion of algorithm and the realization that certain problems are algorithmically unsolvable. At around this time, forerunners of the programmable computing machine were beginning to appear. As mathematicians contemplated the practical capabilities and limitations of such devices, computational complexity theory emerged from the theory of algorithmic unsolvability. Early on, a particular type of computational task became evident, where one is seeking an object which lies
Why does unsupervised pretraining help deep learning?
, 2010
"... Much recent research has been devoted to learning algorithms for deep architectures such as Deep Belief Networks and stacks of autoencoder variants with impressive results being obtained in several areas, mostly on vision and language datasets. The best results obtained on supervised learning tasks ..."
Abstract

Cited by 49 (11 self)
 Add to MetaCart
Much recent research has been devoted to learning algorithms for deep architectures such as Deep Belief Networks and stacks of autoencoder variants with impressive results being obtained in several areas, mostly on vision and language datasets. The best results obtained on supervised learning tasks often involve an unsupervised learning component, usually in an unsupervised pretraining phase. The main question investigated here is the following: why does unsupervised pretraining work so well? Through extensive experimentation, we explore several possible explanations discussed in the literature including its action as a regularizer (Erhan et al., 2009b) and as an aid to optimization (Bengio et al., 2007). Our results build on the work of Erhan et al. (2009b), showing that unsupervised pretraining appears to play predominantly a regularization role in subsequent supervised training. However our results in an online setting, with a virtually unlimited data stream, point to a somewhat more nuanced interpretation of the roles of optimization and regularization in the unsupervised pretraining effect.
An Optimal Approximation Algorithm For Bayesian Inference
 Artificial Intelligence
, 1997
"... Approximating the inference probability Pr[X = xjE = e] in any sense, even for a single evidence node E, is NPhard. This result holds for belief networks that are allowed to contain extreme conditional probabilitiesthat is, conditional probabilities arbitrarily close to 0. Nevertheless, all p ..."
Abstract

Cited by 48 (2 self)
 Add to MetaCart
(Show Context)
Approximating the inference probability Pr[X = xjE = e] in any sense, even for a single evidence node E, is NPhard. This result holds for belief networks that are allowed to contain extreme conditional probabilitiesthat is, conditional probabilities arbitrarily close to 0. Nevertheless, all previous approximation algorithms have failed to approximate efficiently many inferences, even for belief networks without extreme conditional probabilities. We prove that we can approximate efficiently probabilistic inference in belief networks without extreme conditional probabilities. We construct a randomized approximation algorithmthe boundedvariance algorithmthat is a variant of the known likelihoodweighting algorithm. The boundedvariance algorithm is the first algorithm with provably fast inference approximation on all belief networks without extreme conditional probabilities. From the boundedvariance algorithm, we construct a deterministic approximation algorithm u...
Optimal bounds for decision problems on the CRCW PRAM
 In Proceedings of the 19th ACM Symposium on Theory of Computing (New
"... Abstract. Optimal Q(logn/log logn) lower bounds on the time for CRCW PRAMS with polynomially bounded numbers of processors or memory cells to compute parity and a number of related problems are proven. A strict time hierarchy of explicit Boolean functions of n bits on such machines that holds up to ..."
Abstract

Cited by 46 (3 self)
 Add to MetaCart
(Show Context)
Abstract. Optimal Q(logn/log logn) lower bounds on the time for CRCW PRAMS with polynomially bounded numbers of processors or memory cells to compute parity and a number of related problems are proven. A strict time hierarchy of explicit Boolean functions of n bits on such machines that holds up to O(logn/loglogn) time is also exhibited. That is, for every time bound T within this range a function is exhibited that can be easily computed using polynomial resources in time T but requires more than polynomial resources to be computed in time T 1. Finally, it is shown that almost all Boolean functions of n bits require logn loglogn + fi ( 1) time when the number of processors is at most polynomial in n. The bounds do not place restrictions on the uniformity of the algorithms nor on the instruction sets of the machines.