The authors give a new characterization of NP: the class NP contains exactly those languages L for which membership proofs (a proof that an input x is in L) can be verified probabilistically in polynomial time using logarithmic number of random bits and sub-logarithmic number of queries to the proof. This is a non-relativizing characterization of NP. They discuss implications of this characterization; specifically, they show that approximating clique (or independent set) is NP-hard. >

Probabilistic checking of proofs; a new characterization of NP

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

Freeness of partial words

Immunity and pseudorandomness of context-free languages

The proofs of major results of Computability Theory like Rice, Rice-Shapiro or Kleene's fixed point theorem hidemore information of what is usually expressed in theirrespective statements. We make this information explicit, allowing to state stronger, complexity theoretic-versions of all these theorems. In particular, we replace the notion of extensional set of indices of programs, by a set of indices of programs having not only the same extensional behavior but also similar complexity (Complexity Clique). We prove, under very weak complexity assumptions, that any recursive Complexity Clique is trivial, and any r.e. Complexity Clique is an extensional set (and thus satisfies Rice-Shapiro conditions). This allows, for instance, to use Rice's argument to prove that the property of having polynomial complexity is not decidable, and to use Rice-Shapiro to conclude that it is not even semi-decidable. We conclude the paper with a discussion of "complexity-theoretic" versions of Kleene's Fixed Point Theorem.

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The intensional content of Rice's theorem

Starting from six kinds of periodicity of words we define six sets of words which are primitive in different senses and we investigate their relationships. We show that only three of the sets are external Marcus contextual languages with choice but none of them is an external contextual language without choice or an internal contextual language. For the time complexity of deciding any of our sets by one-tape Turing machines, n2 is a lower bound and this is optimal in two cases. The notions of roots and degrees of words and languages, which are strongly connected to periodicity and primitivity, are also considered, and we show that there can be an arbitrarily large gap between the complexity of a language and that of its roots. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Generalized periodicity and primitivity for words

For a set H of natural numbers, the H-power of a language L is the set of all words pk where p ∈ L and k ∈ H. The root of L is the set of all primitive words p such that pn belongs to L for some n ≥ 1. There is a strong connection between the root and the powers of a regular language L namely, the H-power of L for an arbitrary finite set H with 0, 1, 2 ∉ H is regular if and only if the root of L is finite. If the root is infinite then the H-power for most regular sets H is context-sensitive but not context-free. The stated property is decidable.

Roots and powers of regular languages

In the algebraic theory of codes and formal languages, the set Q of all primitive words over some alphabet has received special inter- est. With this survey article we give an overview about relevant research to this topic during the last twenty years including own investigations and some new results. In Section 1 after recalling the most important notions from formal language theory we illustrate the connection between coding theory and primitive words by some facts. We dene primitive words as words having only a trivial representation as the power of another word. Nonprimitive words (without the empty word) are exactly the periodic words. Every nonempty word is a power of an uniquely determined prim- itive word which is called the root of the former one. The set of all roots of nonempty words of a language is called the root of the language. The primitive words have interesting combinatorial properties which we con- sider in Section 2. In Section 3 we investigate the relationship between the set Q of all primitive words over some xed alphabet and the lan- guage classes of the Chomsky Hierarchy and the contextual languages over the same alphabet. The computational complexity of the set Q and of the roots of languages are considered in Section 4. The set of all pow- ers of the same degree of all words from a language is the power of this language. We examine the powers of languages for dierent sets of ex- ponents, and especially their regularity and context-freeness, in Section 5, and the decidability of appropriate questions in Section 6. Section 7 is dedicated to several generalizations of the notions of periodicity and primitivity of words.

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Primitive words and roots of words

Punctured languages are languages whose words are partial words in the sense that the letters at some positions are unknown We investigate to which extent restoration of punctured languages is possible if the number of unknown positions or the proportion of unknown positions per word, respectively, is bounded, and we study their relationships for different boundings The considered restoration classes coincide with similarity classes according to some kind of similarity for languages Thus all results we can also formulate in the language of similarity We show some hierarchies of similarity classes for each class from the Chomsky hierarchy and prove the existence of linear languages which are not δ -similar to any regular language for any δ < ½ For δ ≥ ½ this is unknown but it could only be possible in the case of non-slender linear languages (© 2006 WILEY-VCH Verlag GmbH & Co KGaA, Weinheim)

Gerhard Lischke

Papers

Generalized periodicity and primitivity for words

Roots and powers of regular languages

Primitive words and roots of words

Restorations of punctured languages and similarity of languages

Oracle‐Constructions to Prove All Possible Relationships Between Relativizations of P, NP, EL, NEL, EP and NEP