P-printable sets
01 Dec 1988-SIAM Journal on Computing (Society for Industrial and Applied Mathematics)-Vol. 17, Iss: 6, pp 1193-1202
TL;DR: It is shown that the class of sets of small generalized Kolmogorov complexity is exactly theclass of sets which are P-isomorphic to a tally language.
Abstract: P-printable sets arise naturally in the.studies of generalized Kolmogorov complexity and data compression, as well as in other areas. We present new characterizations of the P-printable sets and present necessary and sufficient conditions for the existence of sparse sets in P that are not P-printable. As a corollary to one of our results, we show that the class of sets of small generalized Kolmogorov complexity is exactly the class of sets which are P-isomorphic to a tally language.
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Book•
21 Dec 2011
TL;DR: This is the second volume of a systematic two-volume presentation of the various areas of research in the field of structural complexity, addressed to graduate students and researchers and assumes knowledge of the topics treated in the first volume but is otherwise nearly self-contained.
Abstract: This is the second volume of a systematic two-volume presentation of the various areas of research in the field of structural complexity. The mathematical theory of computation has developed into a broad and rich discipline within which the theory of algorithmic complexity can be approached from several points of view. This volume is addressed to graduate students and researchers and assumes knowledge of the topics treated in the first volume but is otherwise nearly self-contained. Topics covered include vector machines, parallel computation, alternation, uniform circuit complexity, isomorphism, biimmunity and complexity cores, relativization and positive relativization, the low and high hierarchies, Kolmogorov complexity and probability classes. Numerous exercises and references are given.
330 citations
02 Jan 1991
TL;DR: Kolmogorov complexity has its roots in probability theory, combinatorics, and philosophical notions of randomness, and came to fruition using the recent development of the theory of algorithms.
Abstract: Publisher Summary This chapter focuses on Kolmogorov complexity and its applications. The mathematical theory of Kolmogorov complexity contains deep and sophisticated mathematics. Yet, the amount of this mathematics that should be known to apply the notions fruitfully in widely divergent areas, from recursive function theory to chip technology, is very little. However, formal knowledge does not necessarily imply the wherewithal to apply it, especially so in the case of Kolmogorov complexity. Kolmogorov complexity has its roots in probability theory, combinatorics, and philosophical notions of randomness, and came to fruition using the recent development of the theory of algorithms. Shannon's classical information theory assigns a quantity of information to an ensemble of possible messages. All messages in the ensemble being equally probable, this quantity is the number of bits needed to count all possibilities. Each message in the ensemble can be communicated using this number of bits. However, it does not say anything about the number of bits needed to convey any individual message in the ensemble.
206 citations
TL;DR: A systematic comparison of several complexity classes of functions that are computed nondeterministically in polynomial time or with an oracle in NP shows that there exists a disjoint pair of NP-complete sets such that every separator is NP-hard.
180 citations
TL;DR: It is shown that the perfect matching problem is in the complexity class SPL (in the nonuniform setting), and if there are problems in DSPACE(n) requiring exponential-size circuits, then all of the results hold in the uniform setting.
107 citations
TL;DR: The potentially "off-by-one" nature of the definitions of counting (#P versus #NP), difference, and unambiguous (UP versus UNP; FewP versus FewNP) classes are explored, and suggestions as to logical approaches in each case are made.
Abstract: We explore the potentially "off-by-one" nature of the definitions of counting (#P versus #NP), difference (DP versus DNP), and unambiguous (UP versus UNP; FewP versus FewNP) classes, and make suggestions as to logical approaches in each case. We discuss the strangely differing representations that oracle and predicate models give for counting classes, and we survey the properties of counting classes beyond #P. We ask whether subtracting a #P function from a P function it is no greater than necessarily yields a #P function.
104 citations
References
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TL;DR: An application to the problem of defining a patternless sequence is proposed in terms of the concepts here developed to study the use of Turing machines for calculating finite binary sequences.
Abstract: The use of Turing machines for calculating finite binary sequences is studied from the point of view of information theory and the theory of recursive functions. Various results are obtained concerning the number of instructions in programs. A modified form of Turing machine is studied from the same point of view. An application to the problem of defining a patternless sequence is proposed in terms of the concepts here developed.
1,072 citations
01 Dec 1983
TL;DR: A time bounded variant of Kolmogorov complexity is studied, together with universal hashing, which can be used to show that problems solvable probabilistically in polynomial time are all within the second level of the polynometric time hierarchy.
Abstract: We study a time bounded variant of Kolmogorov complexity. This notion, together with universal hashing, can be used to show that problems solvable probabilistically in polynomial time are all within the second level of the polynomial time hierarchy. We also discuss applications to the theory of probabilistic constructions.
538 citations
TL;DR: A class of machines called auxiliary pushdown machines is introduced, characterized in terms of time-bounded Turing machines, and corollaries are derived which answer some open questions in the field.
Abstract: A class of machines called auxiliary pushdown machines is introduced. Several types of pushdown automata, including stack automata, are characterized in terms of these machines. The computing power of each class of machines in question is characterized in terms of time-bounded Turing machines, and corollaries are derived which answer some open questions in the field. ~
395 citations
312 citations
Proceedings Article•
01 Jan 1983
TL;DR: A generalized, two-parameter, Kolmogorov complexity of finite strings is defined which measures how much and how fast a string can be compressed and it is shown that this string complexity measure is an efficient tool for the study of computational complexity.
145 citations