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The Structure of Complete Degrees
, 1990
"... This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences ..."
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Cited by 29 (3 self)
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This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NPcomplete sets look like? To what extent are the properties of particular NPcomplete sets, e.g., SAT, shared by all NPcomplete sets? If there are are structural differences between NPcomplete sets, what are they and what explains the differences? We make these questions, and the analogous questions for other complexity classes, more precise below. We need first to formalize NPcompleteness. There are a number of competing definitions of NPcompleteness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of mreduction, also known as polynomialtime manyone reduction and Karp reduction. A set A is mreducible to B if and only if there is a (total) polynomialtime computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1
Completeness for Nondeterministic Complexity Classes
, 1991
"... We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomialtime bounded (even logarithmic space bounded) reducibilities turn out to be different for any class ..."
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Cited by 9 (0 self)
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We demonstrate differences between reducibilities and corresponding completeness notions for nondeterministic time and space classes. For time classes the studied completeness notions under polynomialtime bounded (even logarithmic space bounded) reducibilities turn out to be different for any class containing NE . For space classes the completeness notions under logspace reducibilities can be separated for any class properly containing LOGSPACE . Key observation in obtaining the separations is the honesty property of reductions, which was recently observed to hold for the time classes and can be shown to hold for space classes. 1 Introduction Efficient reducibilities and completeness are two of the central concepts of complexity theory. Since the first use of polynomial time bounded Turing reductions by Cook [4] and the introduction of polynomial time bounded manyone reductions by Karp[9], considerable effort has been put in the investigation of properties and the relative strengt...
Immunity and Pseudorandomness of ContextFree Languages
, 902
"... Abstract. We examine the computational complexity of contextfree languages, mainly concentrating on two wellknown structural properties—immunity and pseudorandomness. An infinite language is REGimmune (resp., CFLimmune) if it contains no infinite subset that is a regular (resp., contextfree) la ..."
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Cited by 1 (1 self)
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Abstract. We examine the computational complexity of contextfree languages, mainly concentrating on two wellknown structural properties—immunity and pseudorandomness. An infinite language is REGimmune (resp., CFLimmune) if it contains no infinite subset that is a regular (resp., contextfree) language. We prove that (i) there is a contextfree REGimmune language outside REG/n and (ii) there is a REGbiimmune language that can be computed deterministically using logarithmic space. We also show that (iii) there is a CFLsimple set, where a CFLsimple language is an infinite contextfree language whose complement is CFLimmune. Similar to the REGimmunity, a REGprimeimmune language has no polynomially dense subsets that are also regular. We further prove that (iv) there is a contextfree language that is REG/nbiprimeimmune but not even REGimmune. Concerning pseudorandomness of contextfree languages, we show that (v) CFL contains REG/npseudorandom languages. Finally, we prove that (vi) against REG/n, there exists an almost 11 pseudorandom generator computable in nondeterministic pushdown automata equipped with a writeonly output tape and (vii) against REG, there is no almost 11 weak pseudorandom generator computable deterministically in linear time by a singletape Turing machine.
Resource Bounded Immunity and Simplicity ∗ Tomoyuki Yamakami a Toshio Suzuki b
, 2005
"... Abstract: Revisiting the thirty yearsold notions of resourcebounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immun ..."
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Abstract: Revisiting the thirty yearsold notions of resourcebounded immunity and simplicity, we investigate the structural characteristics of various immunity notions: strong immunity, almost immunity, and hyperimmunity as well as their corresponding simplicity notions. We also study limited immunity and simplicity, called kimmunity and feasible kimmunity, and their simplicity notions. Finally, we propose the kimmune hypothesis as a working hypothesis that guarantees the existence of simple sets in NP.