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Average-case complexity
About: Average-case complexity is a research topic. Over the lifetime, 1749 publications have been published within this topic receiving 44972 citations.
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TL;DR: A logical complexity measure — Horn complexity — for Boolean functions which measures the minimal length of quasi-Horn definitions of such functions by propositional formulae is proposed.
Abstract: We propose in this paper a logical complexity measure -- Horn complexity -- for Boolean functions which measures the minimal length of quasi-Horn definitions of such functions by propositional formulae. The interest for this complexity measure comes on the one hand from the observation that the satisfiability problem for Horn formulae is in P, on the other hand from a strong connection to Cook's problem. We show the proposed Horn complexity to be polynomially equivalent to network complexity and therefore to Turing complexity for Boolean functions.
15 citations
16 Jul 1979
TL;DR: The authors showed that parsing strings of length n is harder than recognizing such strings by a factor of only 0(log n), at most, for linear and/or unambiguous context-free languages.
Abstract: Several results on the computational complexity of general context-free language parsing and recognition are given. In particular we show that parsing strings of length n is harder than recognizing such strings by a factor of only 0(log n), at most. The same is true for linear and/or unambiguous context-free languages. We also show that the time to multiply \(\sqrt n \times \sqrt n\) Boolean Matrices is a lower bound on the time to recognize all prefixes of a string (or do on-line recognition), which in turn is a lower bound on the time to generate a particular convenient representation of all parses of a string (in an ambiguous grammar). Thus these problems are solvable in linear time only if n×n Boolean matrix multiplication can be done in 0(n2).
15 citations
Journal Article•
TL;DR: In this article, a lower bound of (p n) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem was given. But this lower bound was later broken by Gavinsky, Kempe, de Wolf, and de Wolf.
Abstract: We give a tight lower bound of ( p n) for the randomized one-way communication complexity of the Boolean Hidden Matching Problem [BJK04]. Since there is a quantum one-way communication complexity protocol of O(logn) qubits for this problem, we obtain an exponential separation of quantum and classical one-way communication complexity for partial functions. A similar result was independently obtained by Gavinsky, Kempe, de Wolf [GKdW06]. Our lower bound is obtained by Fourier analysis, using the Fourier coecients inequality of Kahn Kalai and Linial [KKL88].
15 citations
07 Feb 2006
TL;DR: The idea is that information is an extension of the concept 'algorithmic complexity' from a class of desirable and concrete processes to a class more general that can only in pragmatic terms be regarded as existing in the conception.
Abstract: We study complexity and information and introduce the idea that while complexity is relative to a given class of processes, information is process independent: Information is complexity relative to the class of all conceivable processes. In essence, the idea is that information is an extension of the concept 'algorithmic complexity' from a class of desirable and concrete processes, such as those represented by binary decision trees, to a class more general that can only in pragmatic terms be regarded as existing in the conception. It is then precisely the fact that information is defined relative to such a large class of processes that it becomes an effective tool for analyzing phenomena in a wide range of disciplines.We test these ideas on the complexity of classical states. A domain is used to specify the class of processes, and both qualitative and quantitative notions of complexity for classical states emerge. The resulting theory is used to give new proofs of fundamental results from classical information theory, to give a new characterization of entropy in quantum mechanics, to establish a rigorous connection between entanglement transformation and computation, and to derive lower bounds on algorithmic complexity. All of this is a consequence of the setting which gives rise to the fixed point theorem: The least fixed point of the copying operator above complexity is information.
15 citations
13 Feb 1992
TL;DR: It will be shown that for Communieation Complexity MOD p -P and MOD q -P are uncomparable via inclusion for all pairs of distinct primes p, q and it is proved that the same is true for PP and MOD -P for any prime number p.
Abstract: We develope new lower bound arguments on communication complexity and establish a number of separation results for Counting Communication Classes. In particular, it will be shown that for Communieation Complexity MOD p -P and MOD q -P are uncomparable via inclusion for all pairs of distinct primes p, q. Further we prove that the same is true for PP and MOD p -P for any prime number p. Our results are due to mathematical characterization of modular and probabilistic communication complexity by the minimum rank of matrices belonging to certain equivalence classes. We use arguments from algebra and analytic geometry.
15 citations