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.

Structural Complexity II

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.

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Kolmogorov complexity and its applications

A taxonomy of complexity classes of functions

Isolation, Matching, and Counting Uniform and Nonuniform Upper Bounds

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.

The satanic notations: counting classes beyond #P and other definitional adventures

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.

P-printable sets

The class of self-P-printable sets, those sets that are enumerable in polynomial time relative to themselves, has been shown to be the same as the class of sets that have small generalized Kolmogorov complexity relative to themselves [J. Hartmanis and L. Hemachandra, Proc. 3rd Annual Symposium on Theoretical Aspects of Computer Science, Springer-Verlag, Berlin, New York, 1986, pp. 321–333] and [J. Balcazar and R. Book, Acta Informatics, 23 (1986), pp. 679–688]. This class properly includes the sets of small generalized Kolmogorov complexity, including the tally sets. The class of sets polynomial time Turing equivalent to a tally set includes the class of self-P-printable sets [J. Balcazar and R. Book, op. cit.], and the inclusion is clearly proper because every self-P-printable set must be sparse while there are many nonsparse sets (such as all nonsparse sets in P) that are polynomial time Turing equivalent to a tally set.The question of whether or not the class of self-P-printable sets is the same as the...

Roy S. Rubinstein

Papers

P-printable sets

Self-P-printability and polynomial time Turing equivalence to a tally set