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76
Nondeterministic Space is Closed Under Complementation
, 1988
"... this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the contextsensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt [4 ..."
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Cited by 236 (15 self)
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this paper we show that nondeterministic space s(n) is closed under complementation, for s(n) greater than or equal to log n. It immediately follows that the contextsensitive languages are closed under complementation, thus settling a question raised by Kuroda in 1964 [9]. See Hartmanis and Hunt [4] for a discussion of the history and importance of this problem, and Hopcroft and Ullman [5] for all relevant background material and definitions. The history behind the proof is as follows. In 1981 we showed that the set of firstorder inductive definitions over finite structures is closed under complementation [6]. This holds with or without an ordering relation on the structure. If an ordering is present the resulting class is P. Many people expected that the result was false in the absence of an ordering. In 1983 we studied firstorder logic, with ordering, with a transitive closure operator. We showed that NSPACE[log n] is equal to (FO + pos TC), i.e. firstorder logic with ordering, plus a transitive closure operation, in which the transitive closure operator does not appear within any negation symbols [7]. Now we have returned to the issue of complementation in the light of recent results on the collapse of the log space hierarchies [10, 2, 14]. We have shown that the class (FO + pos TC) is closed under complementation. Our
The quantitative structure of exponential time
 Complexity theory retrospective II
, 1997
"... ABSTRACT Recent results on the internal, measuretheoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with biimmunity, complexity cores, polynomialtime reductions, completeness, circuit ..."
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Cited by 90 (13 self)
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ABSTRACT Recent results on the internal, measuretheoretic structure of the exponential time complexity classes E and EXP are surveyed. The measure structure of these classes is seen to interact in informative ways with biimmunity, complexity cores, polynomialtime reductions, completeness, circuitsize complexity, Kolmogorov complexity, natural proofs, pseudorandom generators, the density of hard languages, randomized complexity, and lowness. Possible implications for the structure of NP are also discussed. 1
Two applications of inductive counting for complementation problems
 SIAM Journal of Computing
, 1989
"... nondeterministic spacebounded complexity classes are closed under complementation, two further applications of the inductive counting technique are developed. First, an errorless probabilistic algorithm for the undirected graph st connectivity problem that runs in O(log n) space and polynomial exp ..."
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Cited by 53 (4 self)
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nondeterministic spacebounded complexity classes are closed under complementation, two further applications of the inductive counting technique are developed. First, an errorless probabilistic algorithm for the undirected graph st connectivity problem that runs in O(log n) space and polynomial expected time is given. Then it is shown that the class LOGCFL is closed under complementation. The latter is a special case of a general result that shows closure under complementation of classes defined by semiunbounded fanin circuits (or, equivalently, nondeterministic auxiliary pushdown automata or treesize bounded alternating Turing machines). As one consequence, it is shown that small numbers of "role switches " in twoperson pebbling can be eliminated.
Counting Quantifiers, Successor Relations, and Logarithmic Space
 Journal of Computer and System Sciences
"... Given a successor relation S (i.e., a directed line graph), and given two distinguished points s and t, the problem ORD is to determine whether s precedes t in the unique ordering defined by S. We show that ORD is Lcomplete (via quantifierfree projections). We then show that firstorder logic with ..."
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Cited by 51 (2 self)
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Given a successor relation S (i.e., a directed line graph), and given two distinguished points s and t, the problem ORD is to determine whether s precedes t in the unique ordering defined by S. We show that ORD is Lcomplete (via quantifierfree projections). We then show that firstorder logic with counting quantifiers, a logic that captures TC 0 ([BIS90]) over structures with a builtin totalordering, can not express ORD. Our original proof of this in the conference version of this paper ([Ete95]) employed an EhrenfeuchtFraiss'e Game for firstorder logic with counting ([IL90]). Here we show how the result follows from a more general one obtained independently by Nurmonen, [Nur96]. We then show that an appropriately modified version of the EF game is "complete" for the logic with counting in the sense that it provides a necessary and sufficient condition for expressibility in the logic. We observe that the Lcomplete problem ORD is essentially sparse if we ignore reorderings of v...
On PolynomialTime Bounded TruthTable Reducibility of NP Sets to Sparse Sets
, 1991
"... We prove that if P ≠ NP, then there exists a set in NP that is not polynomial time bounded truthtable reducible (in short, p btt reducible) to any sparse set. In other words, we prove that no sparse p btt hard set exists for NP unless P = NP. By using the technique proving this result, we in ..."
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Cited by 44 (3 self)
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We prove that if P ≠ NP, then there exists a set in NP that is not polynomial time bounded truthtable reducible (in short, p btt reducible) to any sparse set. In other words, we prove that no sparse p btt hard set exists for NP unless P = NP. By using the technique proving this result, we investigate intractability of several number theoretic decision problems, i.e., decision problems defined naturally from number theoretic problems. We show that for these number theoretic decision problems, if it is not in P, then it is not p btt reducible to any sparse set.
Measure, Stochasticity, and the Density of Hard Languages
 SIAM Journal on Computing
, 1994
"... The main theorem of this paper is that, for every real number ff ! 1 (e.g., ff = 0:99), only a measure 0 subset of the languages decidable in exponential time are P n ff \Gammatt reducible to languages that are not exponentially dense. Thus every P n ff \Gammatt hard language for E is exp ..."
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Cited by 43 (13 self)
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The main theorem of this paper is that, for every real number ff ! 1 (e.g., ff = 0:99), only a measure 0 subset of the languages decidable in exponential time are P n ff \Gammatt reducible to languages that are not exponentially dense. Thus every P n ff \Gammatt hard language for E is exponentially dense. This strengthens Watanabe's 1987 result, that every P O(log n)\Gammatt hard language for E is exponentially dense. The combinatorial technique used here, the sequentially most frequent query selection, also gives a new, simpler proof of Watanabe's result. The main theorem also has implications for the structure of NP under strong hypotheses. Ogiwara and Watanabe (1991) have shown that the hypothesis P 6= NP implies that every P btt hard language for NP is nonsparse (i.e., not polynomially sparse). Their technique does not appear to allow significant relaxation of either the query bound or the sparseness criterion. It is shown here that a stronger hypothesis na...
The Isomorphism Conjecture Fails Relative to a Random Oracle
 J. ACM
, 1996
"... Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kc ..."
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Cited by 40 (4 self)
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Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomialtime computable manyone (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NPcomplete setsthe kcreative setsand defined a class of sets (the K k f 's) that are necessarily kcreative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NPcomplete sets. Clearly, the BermanHartmanis and JosephYoung conjectures cannot both be correct. We introduce a family of strong oneway functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NPcomplete sets, as Joseph and Young conjectured, and the BermanHartmanis conjecture fails. Indeed, if scrambling functions exist, then the isomorphism also fails at higher complexity classes such as EXP and NEXP. As evidence for the existence of scramb...
PSelective Sets, and Reducing Search to Decision vs. SelfReducibility
, 1993
"... We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to de ..."
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Cited by 39 (9 self)
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We obtain several results that distinguish selfreducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6= E, then there exists a set L in NP \Gamma P such that search reduces to decision for L, search does not nonadaptively reduces to decision for L, and L is not selfreducible. Funding for this research was provided by the National Science Foundation under grant CCR9002292. y Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 z Department of Computer Science, State University of New York at Buffalo, 226 Bell Hall, Buffalo, NY 14260 x Research performed while visiting the Department of Computer Science, State University of New York at Buffalo, Jan. 1992Dec. 1992. Current address: Department of Computer Science, University of ElectroCommunications, Chofushi, Tokyo 182, Japan.  Department of Computer Science, State University of New York at Buffalo, 226...
Infeasibility of instance compression and succinct PCPs for NP
 Electronic Colloquium on Computational Complexity (ECCC
"... The ORSAT problem asks, given Boolean formulae φ1,..., φm each of size at most n, whether at least one of the φi’s is satisfiable. We show that there is no reduction from ORSAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the PolynomialTi ..."
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Cited by 34 (1 self)
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The ORSAT problem asks, given Boolean formulae φ1,..., φm each of size at most n, whether at least one of the φi’s is satisfiable. We show that there is no reduction from ORSAT to any set A where the length of the output is bounded by a polynomial in n, unless NP ⊆ coNP/poly, and the PolynomialTime Hierarchy collapses. This result settles an open problem proposed by Bodlaender et. al. [4] and Harnik and Naor [15] and has a number of implications. • A number of parametric NP problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless NP ⊆ coNP/poly. • Satisfiability does not have PCPs of size polynomial in the number of variables unless NP ⊆ coNP/poly. • An approach of Harnik and Naor to constructing collisionresistant hash functions from oneway functions is unlikely to be viable in its present form. • (BuhrmanHitchcock) There are no subexponentialsize hard sets for NP unless NP is in coNP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions. Categories and Subject Descriptors
Turing Machines With Few Accepting Computations And Low Sets For PP
, 1992
"... this paper we study two different ways to restrict the power of NP: We consider languages accepted by nondeterministic polynomial time machines with a small number of accepting paths in case of acceptance, and also investigate subclasses of NP that are low for complexity classes not known to be in t ..."
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Cited by 32 (5 self)
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this paper we study two different ways to restrict the power of NP: We consider languages accepted by nondeterministic polynomial time machines with a small number of accepting paths in case of acceptance, and also investigate subclasses of NP that are low for complexity classes not known to be in the polynomial time hierarchy. The first complexity class defined following the idea of bounding the number of accepting paths was Valiant's class UP (unique P) [Va76] of languages accepted by nondeterministic Turing machines that have exactly one accepting computation path for strings in the language, and none for strings not in the language. This class plays an important role in the areas of oneway functions and cryptography, for example in [GrSe84] it is shown that P6=UP if and only if oneway functions exist. The class UP can be generalized in a natural way by allowing a polynomial number of accepting paths. This gives rise to the class FewP defined by Allender [Al85] in connection with the notion of Pprintable sets. We study complexity classes defined by such pathrestricted nondeterministic polynomial time machines, and show results that exploit the fact that the machines for these classes have a bounded number of accepting computation paths. We will not only consider these subclasses of NP, namely UP and FewP, but also the class Few, an extension of FewP defined by Cai and Hemachandra [CaHe89], in which the accepting mechanism of the machine is more flexible. 1 The three classes UP, FewP and Few are all defined in terms of nondeterministic machines with a bounded number of accepting paths for every input string, but for the last two classes this number is not known beforehand, and can range over a space of polynomial size. We show in Section 3 that a polynomial numb...