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129
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
On Uniformity within NC¹
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by ..."
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Cited by 127 (19 self)
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In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by firstorder formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic logtime reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for lowlevel circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].
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
Cook versus KarpLevin: Separating Completeness Notions If NP Is Not Small
 Theoretical Computer Science
, 1992
"... Under the hypothesis that NP does not have pmeasure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is show n that there is a language that is P T complete ("Cook complete "), but not P m complete ("KarpLevin complete"), for NP. This conclusion, widely be ..."
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Cited by 56 (12 self)
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Under the hypothesis that NP does not have pmeasure 0 (roughly, that NP contains more than a negligible subset of exponential time), it is show n that there is a language that is P T complete ("Cook complete "), but not P m complete ("KarpLevin complete"), for NP. This conclusion, widely believed to be true, is not known to follow from P 6= NP or other traditional complexitytheoretic hypotheses. Evidence is presented that "NP does not have pmeasure 0" is a reasonable hypothesis with many credible consequences. Additional such consequences proven here include the separation of many truthtable reducibilities in NP (e.g., k queries versus k+1 queries), the class separation E 6= NE, and the existence of NP search problems that are not reducible to the corresponding decision problems. This research was supported in part by National Science Foundation Grant CCR9157382, with matching funds from Rockwell International. 1 Introduction The NPcompleteness of decision problems has...
Almost Every Set in Exponential Time is PBiImmune
 Theoretical Computer Science
, 1994
"... . A set A is Pbiimmune if neither A nor its complement has an infinite subset in P. We investigate here the abundance of Pbiimmune languages in linearexponential time (E). We prove that the class of Pbiimmune sets has measure 1 in E. This implies that `almost' every language in E is Pbiimmun ..."
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Cited by 51 (5 self)
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. A set A is Pbiimmune if neither A nor its complement has an infinite subset in P. We investigate here the abundance of Pbiimmune languages in linearexponential time (E). We prove that the class of Pbiimmune sets has measure 1 in E. This implies that `almost' every language in E is Pbiimmune, that is to say, almost every set recognizable in linear exponential time has no algorithm that recognizes it and works in polynomial time on an infinite number of instances. A bit further, we show that every prandom (pseudorandom) language is Ebiimmune. Regarding the existence of Pbiimmune sets in NP, we show that if NP does not have measure 0 in E, then NP contains a Pbiimmune set. Another consequence is that the class of p m complete languages for E has measure 0 in E. In contrast, it is shown that in E, and even in REC, the class of Pbiimmune languages lacks the property of Baire (the Baire category analogue of Lebesgue measurability). * This work was supported by a Spani...
An oracle builder’s toolkit
, 2002
"... We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and ..."
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Cited by 47 (10 self)
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We show how to use various notions of genericity as tools in oracle creation. In particular, 1. we give an abstract definition of genericity that encompasses a large collection of different generic notions; 2. we consider a new complexity class AWPP, which contains BQP (quantum polynomial time), and infer several strong collapses relative to SPgenerics; 3. we show that under additional assumptions these collapses also occur relative to Cohen generics; 4. we show that relative to SPgenerics, ULIN ∩ coULIN ̸ ⊆ DTIME(n k) for any k, where ULIN is unambiguous linear time, despite the fact that UP ∪ (NP ∩ coNP) ⊆ P relative to these generics; 5. we show that there is an oracle relative to which NP/1∩coNP/1 ̸ ⊆ (NP∩coNP)/poly; and 6. we use a specialized notion of genericity to create an oracle relative to which NP BPP ̸ ⊇ MA.
The Complexity and Distribution of Hard Problems
 SIAM JOURNAL ON COMPUTING
, 1993
"... Measuretheoretic aspects of the P m reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in ..."
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Cited by 45 (16 self)
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Measuretheoretic aspects of the P m reducibility structure of the exponential time complexity classes E=DTIME(2 linear ) and E 2 = DTIME(2 polynomial ) are investigated. Particular attention is given to the complexity (measured by the size of complexity cores) and distribution (abundance in the sense of measure) of languages that are P m  hard for E and other complexity classes. Tight upper and lower bounds on the size of complexity cores of hard languages are derived. The upper bound says that the P m hard languages for E are unusually simple, in the sense that they have smaller complexity cores than most languages in E. It follows that the P m complete languages for E form a measure 0 subset of E (and similarly in E 2 ). This latter fact is seen to be a special case of a more general theorem, namely, that every P m degree (e.g., the degree of all P m complete languages for NP) has measure 0 in E and in E 2 .
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...
QueryLimited Reducibilities
, 1995
"... We study classes of sets and functions computable by algorithms that make a limited number of queries to an oracle. We distinguish between queries made in parallel (each question being independent of the answers to the others, as in a truthtable reduction) and queries made in serial (each question ..."
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Cited by 41 (14 self)
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We study classes of sets and functions computable by algorithms that make a limited number of queries to an oracle. We distinguish between queries made in parallel (each question being independent of the answers to the others, as in a truthtable reduction) and queries made in serial (each question being permitted to depend on the answers to the previous questions, as in a Turing reduction). We define computability by a set of functions, and we show that it captures the informationtheoretic aspects of computability by a fixed number of queries to an oracle. Using that concept, we prove a very powerful result, the Nonspeedup Theorem, which states that 2 n parallel queries to any fixed nonrecursive oracle cannot be answered by an algorithm that makes only n queries to any oracle whatsoever. This is the tightest general result possible. A corollary is the intuitively obvious, but nontrivial result that additional parallel queries to an oracle allow us to compute additional functions; t...
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...