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 context-sensitive 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 first-order 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 first-order logic, with ordering, with a transitive closure operator. We showed that NSPACE[log n] is equal to (FO + pos TC), i.e. first-order 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

ABSTRACT Recent results on the internal, measure-theoretic 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 bi-immunity, complexity cores, polynomial-time reductions, completeness, circuit-size 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

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 L-complete (via quantifier-free projections). We then show that first-order logic with counting quantifiers, a logic that captures TC 0 ([BIS90]) over structures with a built-in total-ordering, can not express ORD. Our original proof of this in the conference version of this paper ([Ete95]) employed an Ehrenfeucht-Fraiss'e Game for first-order 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 L-complete problem ORD is essentially sparse if we ignore reorderings of v...

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 non-sparse (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...

We prove that if P ≠ NP, then there exists a set in NP that is not polynomial time bounded truth-table 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.

Berman and Hartmanis [BH77] conjectured that there is a polynomialtime computable isomorphism between any two languages complete for NP with respect to polynomial-time computable many-one (Karp) reductions. Joseph and Young [JY85] gave a structural definition of a class of NP-complete sets---the k-creative sets---and defined a class of sets (the K k f 's) that are necessarily k-creative. They went on to conjecture that certain of these K k f 's are not isomorphic to the standard NP-complete sets. Clearly, the Berman--Hartmanis and Joseph--Young conjectures cannot both be correct. We introduce a family of strong one-way functions, the scrambling functions. If f is a scrambling function, then K k f is not isomorphic to the standard NP-complete sets, as Joseph and Young conjectured, and the Berman-Hartmanis 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...

We obtain several results that distinguish self-reducibility 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 self-reducible. 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. 1992--Dec. 1992. Current address: Department of Computer Science, University of Electro-Communications, Chofu-shi, Tokyo 182, Japan. -- Department of Computer Science, State University of New York at Buffalo, 226...

This paper surveys investigations into how strong these commonalities are. More concretely, we are concerned with: What do NP-complete sets look like? To what extent are the properties of particular NP-complete sets, e.g., SAT, shared by all NP-complete sets? If there are are structural differences between NP-complete 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 NP-completeness. There are a number of competing definitions of NP-completeness. (See [Har78a, p. 7] for a discussion.) The most common, and the one we use, is based on the notion of m-reduction, also known as polynomial-time manyone reduction and Karp reduction. A set A is m-reducible to B if and only if there is a (total) polynomial-time computable function f such that for all x, x 2 A () f(x) 2 B: (1) 1

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 one-way functions and cryptography, for example in [GrSe84] it is shown that P6=UP if and only if one-way 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 P-printable sets. We study complexity classes defined by such path-restricted 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...

The Random Oracle Hypothesis, attributed to Bennett and Gill, essentially states that the relationships between complexity classes which hold for almost all relativized worlds must also hold in the unrelativized case. Although this paper is not the first to provide a counterexample to the Random Oracle Hypothesis, it does provide a most compelling counterexample by showing that for almost all oracles A, IP A 6= PSPACE A . If the Random Oracle Hypothesis were true, it would contradict Shamir's result that IP = PSPACE. In fact, it is shown that for almost all oracles A, co-NP A 6` IP A . These results extend to the multi-prover proof systems of Ben-Or, Goldwasser, Kilian and Wigderson. In addition, this paper shows that the Random Oracle Hypothesis is sensitive to small changes in the definition. A class IPP, similar to IP, is defined. Surprisingly, the IPP = PSPACE result holds for all oracle worlds. 1 Department of Computer Science, Cornell University, Ithaca, NY 14853, U.S.A...