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.

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

Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95], following progress made by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P = LOGSPACE. We extend the results to the case of sparse sets that are hard under more general reducibilities. Our main results are as follows. (1) If there exists a sparse set that is hard for P under bounded truth-table reductions, then P = NC 2 . (2) If there exists a sparse set that is hard for P under randomized logspace reductions with one-sided error, then P = Randomized LOGSPACE. (3) If there exists an NP-hard sparse set under randomized polynomial-time reductions with one-sided error, then NP = RP. (4) If there exists a 2 (log n) O(1) -sparse hard set for P under truth-table reductions, then P ` DSPACE[(logn) O(1) ]. As a by-product of (4), we obtain a uniform O(log 2 n log log n) time parallel algorithm for computing the rank of a 2 log 2 n \Theta n matrix o...

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...

this paper to give simple proofs, in a uniform format, of the major known (pre-1992) results relating how polynomial-time reductions of SAT to sparse sets collapse the polynomial-time hierarchy. To help the reader familiar with basic facts of complexity theory follow the main flow of ideas, while keeping the exposition self-contained, straight forward proofs from elementary complexity theory are relegated to footnotes. We treat polynomial-time Turing reductions (i.e., Cook reductions) in Section 2. Bounded truth-table reductions (and many-one reductions) are treated in Section 3. Sections 2 and 3 may be read independently of each other. Section 4 uses the definitions of Section 3 to give simple proofs of results on conjunctive and disjunctive reductions. A comprehensive discussion of early work on how reductions to sparse sets collapse the polynomial-time hierarchy may be found in [Ma-89]. Additional discussions of this topic, as well as extensive bibliographies, may be found in [JY-90] and in [Yo-90]. 2 Polynomial-Time Turing Reductions 2.1 Introduction In [Lo-82], Long explicitly used the result (KL-1) to prove:

Building on the recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace many-one reductions. We show that if there exists a sparse hard set for P under logspace many-one reductions, then P = LOGSPACE. We further prove that if P has a sparse hard set under many-one reductions computable in NC 1 , then P collapses to NC 1 . 1 Introduction A set S is called sparse if there are at most a polynomial number of strings in S up to length n. Sparse sets have been the subject of study in complexity theory for the past 20 years, as they reveal inherent structure and limitations of computation [BH77, HOW92, You92a, You92b]. For instance, it is well known that the class of languages polynomial time Turing reducible (i.e. by Cook reductions) to a sparse set is precisely the class of languages with polynomial size circuits. One major motivation for the study of sparse sets, and various reducib...

A basic question about NP is whether or not search reduces in polynomial time to decision. We indicate that the answer is negative: under a complexity assumption (that deterministic and nondeterministic doubleexponential time are unequal) we construct a language in NP for which search does not reduce to decision. These ideas extend in a natural way to interactive proofs and program checking. Under similar assumptions we present languages in NP for which it is harder to prove membership interactively than it is to decide this membership. Similarly we present languages where checking is harder than computing membership. Each of the following properties --- checkability, random-self-reducibility, reduction from search to decision, and interactive proofs in which the prover's power is limited to deciding membership in the language itself --- implies coherence, one of the weakest forms of self-reducibility. Under assumptions about triple-exponential time, we construct incoherent sets in NP....

In this paper, we study one word-decreasing self-reducible sets which are introduced by Lozano and Toran [21]. These are usual self-reducible sets with the peculiarity that the selfreducibility machine makes at most one query and this is lexicographically smaller than the input. We show rst that for all counting classes dened by a predicate on the number of accepting paths there exist complete sets which are one word-decreasing self-reducible. Using this fact we can prove that for any class K chosen from fPP; NP; C=P; MOD 2 P; MOD 3 P; g it holds that (1) if there is a sparse P btt -hard set for K then K P, and (2) if there is a sparse SN btt -hard set for K then K NP \ co-NP. This generalizes the result from [24] to the mentioned complexity classes. 1 Introduction One of the central roles in the study of structural complexity theory resides in nding structural differences or similarities among complexity classes. Since almost every complexity class is dened by us...

Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected topics in the area. Among the lowness properties we consider are polynomial-size circuit complexity, membership comparability, approximability, selectivity, and cheatability. Furthermore, we review some of the recent results concerning lowness for counting classes.