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12
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
A FirstOrder Isomorphism Theorem
 SIAM JOURNAL ON COMPUTING
, 1993
"... We show that for most complexity classes of interest, all sets complete under firstorder projections are isomorphic under firstorder isomorphisms. That is, a very restricted version of the BermanHartmanis Conjecture holds. ..."
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Cited by 24 (5 self)
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We show that for most complexity classes of interest, all sets complete under firstorder projections are isomorphic under firstorder isomorphisms. That is, a very restricted version of the BermanHartmanis Conjecture holds.
On the Structure of Complete Sets
 IN PROCEEDINGS 9TH STRUCTURE IN COMPLEXITY THEORY
, 1994
"... The many types of resource bounded reductions that are both object of study and research tool in structural complexity theory have given rise to a large variety of completeness notions. A complete set in a complexity class is a manageable object that represents the structure of the entire class. The ..."
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Cited by 21 (1 self)
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The many types of resource bounded reductions that are both object of study and research tool in structural complexity theory have given rise to a large variety of completeness notions. A complete set in a complexity class is a manageable object that represents the structure of the entire class. The study of its structure can reveal properties that are general in that complexity class, and the study of the structure of complete sets in different classes can reveal secrets about the relation between these classes. The research into all sorts of aspects and properties of complete sets has been and will be a major topic in structural complexity theory. In this expository paper we review the progress that has been made in recent years on selected topics of the study of complete sets.
Complete Sets and Structure in Subrecursive Classes
 In Proceedings of Logic Colloquium '96
, 1998
"... In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completene ..."
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Cited by 14 (1 self)
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In this expository paper, we investigate the structure of complexity classes and the structure of complete sets therein. We give an overview of recent results on both set structure and class structure induced by various notions of reductions. 1 Introduction After the demonstration of the completeness of several problems for NP by Cook [Coo71] and Levin [Lev73] and for many other problems by Karp [Kar72], the interest in completeness notions in complexity classes has tremendously increased. Virtually every form of reduction known in computability theory has found its way to complexity theory. This is usually done by imposing time and/or space bounds on the computational power of the device representing the reduction. Early on, Ladner et al. [LLS75] categorized the then known types of reductions and made a comparison between these by constructing sets that are reducible to each other via one type of reduction and not reducible via the other. They however were interested just in the rela...
On Growing ContextSensitive Languages
 Proc. 19th ICALP, Lecture Notes in Computer Science (W. Kuich,ed
, 1992
"... Growing contextsensitive grammars (GCSG) are investigated. The variable membership problem for GCSG is shown to be NPcomplete. This solves a problem posed in [DW86]. It is also shown that the languages generated by GCSG form an abstract family of languages and several implications are presented. ..."
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Cited by 12 (2 self)
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Growing contextsensitive grammars (GCSG) are investigated. The variable membership problem for GCSG is shown to be NPcomplete. This solves a problem posed in [DW86]. It is also shown that the languages generated by GCSG form an abstract family of languages and several implications are presented. Institut fur Informatik, Universitat Wurzburg, D8700 Wurzburg, Germany. y Instytut Informatyki, Uniwersytet Wroc/lawski, 51151 Wroc/law, Poland (permanent address). This research was supported by the Humboldt Foundation. 1 Introduction It is well known that the class of languages generated by contextsensitive grammars is equal to NSPACE(n) and that, even for fixed grammars, the membership problem can be PSPACEcomplete. On the other hand the contextfree grammars are known to have, for many applications, too weak derivative power. While many modifications extending contextfree grammars have been studied, only a few papers concern some restricted versions of contextsensitive gramm...
DSPACE(n) ? = NSPACE(n): A degree theoretic characterization
 in Proc. 10th Structure in Complexity Theory Conference
, 1995
"... It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1 ..."
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Cited by 3 (3 self)
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It is shown that the following are equivalent. 1. DSPACE(n) = NSPACE(n). 2. There is a nontrivial ≤1−NL mdegree that coincides with a ≤1−L mdegree. 3. For every class C closed under loglin reductions, the ≤1−NL m coincides with the ≤1−L mcomplete degree of C. 1
ON THE ISOMORPHISM CONJECTURE FOR 2DFA REDUCTIONS
 INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCEINTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
, 1999
"... The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under loglin reductions:  All complete sets for the class C under 2DFA reductions are also complete under oneone, lengthincreasing 2DFA reductions and are firstorder i ..."
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Cited by 2 (2 self)
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The degree structure of complete sets under 2DFA reductions is investigated. It is shown that, for any class C that is closed under loglin reductions:  All complete sets for the class C under 2DFA reductions are also complete under oneone, lengthincreasing 2DFA reductions and are firstorder isomorphic.  The 2DFAisomorphism conjecture is false, i.e., the complete sets under 2DFA reductions are not isomorphic to each other via 2DFA reductions.
For completeness, sublogarithmic space is no space
"... It is shown that for any class C closed under lineartime reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under firstorder reductions. ..."
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Cited by 2 (2 self)
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It is shown that for any class C closed under lineartime reductions, the complete sets for C under sublogarithmic reductions are also complete under 2DFA reductions, and thus are isomorphic under firstorder reductions.
NLprintable sets and Nondeterministic Kolmogorov Complexity
, 2003
"... This paper introduces nondeterministic spacebounded Kolmogorov complexity, and we show that it has some nice properties not shared by some other resourcebounded notions of Kcomplexity. ..."
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Cited by 2 (0 self)
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This paper introduces nondeterministic spacebounded Kolmogorov complexity, and we show that it has some nice properties not shared by some other resourcebounded notions of Kcomplexity.
NonUniform Reductions
, 2007
"... Reductions and completeness notions form the heart of computational complexity theory. Recently nonuniform reductions have been naturally introduced in a variety of settings concerning completeness notions for NP and other classes. We follow up on these results by strengthening some of them. In par ..."
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Cited by 1 (0 self)
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Reductions and completeness notions form the heart of computational complexity theory. Recently nonuniform reductions have been naturally introduced in a variety of settings concerning completeness notions for NP and other classes. We follow up on these results by strengthening some of them. In particular, we show that under certain well studied hypotheses: • manyone complete sets for NP are length increasing complete with a single bit of advice, strengthening a result of Hitchcock and Pavan [HP06]. • 1truthtable complete sets for NP are manyone complete with a single bit of advice. We tighten another result of [HP06] and give a tradeoff between the amount of advice that is needed for the reduction and its honesty, i.e., how length decreasing a reduction is for the manyone complete degree of NEXP. We also construct an oracle relative to which this tradeoff is optimal. For uniform reductions, the tradeoff only yields exponential honesty and the oracle witnesses an exponentially length decreasing manyone complete set for NEXP. Next, we start a more systematic study of nonuniform reductions and show, among other things, that in certain cases the nonuniformity can be removed at the cost of more queries. For example, for EXP we show that complete sets under manyone reductions that use O(logn) bits of advice are still Turing complete without advice. In line with Post’s program in complexity theory [BT05] we connect such ‘uniformization ’ properties to the separation of complexity classes. 1