Speedup for natural problems and noncomputability

Monroe, Hunter

doi:10.1016/j.tcs.2010.09.029

Cited by 9 publications

(11 citation statements)

References 13 publications

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“…While the problems exhibiting speedup considered by Blum, Schnorr and Stumpf were all artificially constructed for the purpose, Coppersmith and Winograd showed that for a particular class of algorithms for matrix multiplication in polynomial time, any of these algorithms could always be replaced by another with slightly smaller exponent. It is conjectured that a similar result holds true for any algorithm for matrix multiplication [25,26].…”

Section: Related Workmentioning

confidence: 81%

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Ben-Amram¹,

Christensen²,

Simonsen³

2012

Chicago J. of Theoretical Comp. Sci.

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The linear speedup theorem states, informally, that constants do not matter: It is essentially always possible to find a program solving any decision problem a factor of 2 faster. This result is a classical theorem in computing, but also one of the most debated. The main ingredient of the typical proof of the linear speedup theorem is tape compression, where a fast machine is constructed with tape alphabet or number of tapes far greater than that of the original machine. In this paper, we prove that limiting Turing machines to a fixed alphabet and a fixed number of tapes rules out linear speedup. Specifically, we describe a language that can be recognized in linear time (e. g., 1.51n), and provide a proof, based on Kolmogorov complexity, that the computation cannot be sped up (e. g., below 1.49n). Without the tape and alphabet limitation, the linear speedup theorem does hold and yields machines of time complexity of the form (1 + ε)n for arbitrarily small ε > 0. Earlier results negating linear speedup in alternative models of computation have often been based on the existence of very efficient universal machines. In the vernacular of programming language theory: These models have very efficient self-interpreters. As the second contribution of this paper, we define a class, PICSTI, of computation models that exactly captures this property, and we disprove the Linear Speedup Theorem for every model in this class, thus generalizing all similar, model-specific proofs.

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Section: Related Workmentioning

confidence: 81%

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Ben-Amram¹,

Christensen²,

Simonsen³

2012

Chicago J. of Theoretical Comp. Sci.

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“…For many interesting languages including the language of Boolean tautologies TAUT, the existence of an algorithm that is optimal on the positive instances only (such algorithm is called an optimal acceptor ) is equivalent to the existence of a p-optimal proof system (that is, a proof system that has the shortest possible proofs, and these proofs can be constructed by a polynomial-time algorithm given proofs in any other proof system) [KP89,Sad99,Mes99] (see [Hir10] for survey). Monroe [Mon11] recently gave a conjecture implying that optimal acceptors for TAUT do not exist.…”

Section: Optimal Algorithmsmentioning

confidence: 99%

Optimal heuristic algorithms for the image of an injective function

et al. 2012

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The existence of optimal algorithms is not known for any decision problem in NP \ P. We consider the problem of testing the membership in the image of an injective function. We construct optimal heuristic algorithms for this problem in both randomized and deterministic settings (a heuristic algorithm can err on a small fraction 1 d of the inputs; the parameter d is given to it as an additional input). Thus for this problem we improve an earlier construction of an optimal acceptor (that is optimal on the negative instances only) and also give a deterministic version.

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“…Concerning Theorem 6 (1) we should mention that Monroe [11] has shown that if the complement of (the classical problem underlying) p-Acc ≤ has an almost optimal algorithm (which by [9] holds if it has a p-optimal proof system), then p-Acc ≤ ∈ XP uni .…”

Section: Linking Slicewise Monotone Problems and Optimal Proof Systemsmentioning

confidence: 99%

On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT

Chen

Flum

2010

Computer Science Logic

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Abstract. For a reasonable sound and complete proof calculus for first-order logic consider the problem to decide, given a sentence ϕ of first-order logic and a natural number n, whether ϕ has no proof of length ≤ n. We show that there is a nondeterministic algorithm accepting this problem which, for fixed ϕ, has running time bounded by a polynomial in n if and only if there is an optimal proof system for the set TAUT of tautologies of propositional logic. This equivalence is an instance of a general result linking the complexity of so-called slicewise monotone parameterized problems with the existence of an optimal proof system for TAUT.

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Speedup for natural problems and noncomputability

Cited by 9 publications

References 13 publications

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Optimal heuristic algorithms for the image of an injective function

On Slicewise Monotone Parameterized Problems and Optimal Proof Systems for TAUT

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