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...

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 semi-membership algorithm for a set A is, informally, a program that when given any two strings determines which is logically more likely to be in A. A flurry of interest in this topic in the late seventies and early eighties was followed by a relatively quiescent half-decade. However, in the 1990s there has been a resurgence of interest in this topic. We survey recent work on the theory of semi-membership algorithms. 1 Introduction A membership algorithm M for a set A takes as its input any string x and decides whether x 2 A. Informally, a semi-membership algorithm M for a set A takes as its input any strings x and y and decides which is "no less likely" to belong to A in the sense that if exactly one of the strings is in A, then M outputs that one string. Semi-membership algorithms have been studied in a number of settings. Recursive semi-membership algorithms (and the associated semi-recursive sets---those sets having recursive semi-membership algorithms) were introduced in the 1...

In 1978, Hartmanis conjectured that there exist no sparse complete sets for P under logspace many-one reductions. In this paper, in support of the conjecture, it is shown that if P has sparse hard sets under logspace many-one reductions, then P ` DSPACE[log 2 n]. The result is derived from a more general statement that if P has 2 polylog sparse hard sets under poly-logarithmic space-computable many-one reductions, then P ` DSPACE[polylog]. 1 Introduction In 1978, Hartmanis conjectured that no P-complete sets under logspace many-one reductions can be polynomially sparse; i.e., for any P-complete set A, k fx 2 A j jxj ng k cannot be bounded by any polynomial in n [5]. The conjecture is interesting and fascinating. If the conjecture is true, then L 6= P, because L = P implies any nonempty finite set being P-complete. So, with expectation that L is different from P, one might believe the validity of the conjecture. Nevertheless, such a reasoning would be fallacious, for, proving thi...

A set is P-selective [Sel79] if there is a polynomial-time semi-decision algorithm for the set|an algorithm that given any two strings decides which is \more likely" to be in the set. This paper establishes a strict hierarchy among the various reductions and equivalences to P-selective sets. 1 Introduction Given the large number of important problems that do not appear to have easy solutions, researchers have explored more exible approaches to ecient set recognition (or near-recognition): almost polynomial time [MP79], average polynomial time (see [Gur91]), implicit membership testability [HH91], neartestability [GHJY91], P-closeness [Sch86,Yes83], P-selectivity [Sel79], and others. One such notion, that of the P-selective sets, has proven useful in many contexts, such as characterizing P [BvHT93] and understanding whether SAT may have unique solutions [HNOSb]. Intuitively, a set is P-selective if there is a 2-ary polynomial-time function that chooses which of its inputs is \more...

Building on a 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 .

this paper, as, e.g., UP, is not known to be closed under union, the distinction is nontrivial. 22 Chapter 3. UP: Boolean Hierarchies and Sparse Turing-Complete Sets Proof. The base case holds by definition. Suppose both statements of this fact to be true for n 1. Then, D 2n+1 (K) = K (coK D 2n-1 (K))

We study the relative computational power of logspace reduction models. In particular, we study the relationships between one-way and two-way oracle tapes, resetting of the oracle head, and blanking of the oracle tape. We show that oracle models letting information persist between queries can be quite powerful, even if the information is not readable by the querying machine. We show that logspace f(n)-Turing reductions are stronger than polynomial-time f(n)-Turing reductions when f(n) = !(log n), and that this is optimal if P = L. 1 Introduction Efficient reductions are a central object of study in computational complexity theory. Polynomial-time reductions have received wide attention, and logspace-bounded reductions have also long been studied as a potentially finer-grained reducibility than polynomial-time reducibility. But the extent to which logspace reducibilities provably provide a finer-grained stratification has remained open. We resolve this with respect to relativizable t...

This paper discusses advances, due to the work of Cai, Naik, and Sivakumar [CNS95] and Glaer [Gla00], in the complexity class collapses that follow if NP has sparse hard sets under reductions weaker than (full) truth-table reductions. 1 Quick Hits Most of this article will be devoted to presenting the work of Glaer [Gla00]. However, even before presenting the background and definitions for that, let us briefly note some improvements that follow from the work of Cai, Naik, and Sivakumar due to c fl Christian Glaer and Lane A. Hemaspaandra, 2000. Supported in part by grants NSFCCR -9322513 and NSF-INT-9815095/DAAD-315-PPP-gu-ab, and the Studienstiftung des Deutschen Volkes. Written in part while Lane A. Hemaspaandra was visiting Julius-Maximilians-Universitat Wurzburg. y E-mail: glasser@informatik.uni-wuerzburg.de. z E-mail: lane@cs.rochester.edu. 1 the results discussed in the first part of this article [GH00]. (See [GH00] for definitions of the terms and classes used here: U...

Sparse hard sets for complexity classes has been a central topic for two decades. The area is motivated by the desire to clarify relationships between completeness/hardness and density of languages and studies the existence of sparse complete/hard sets for various complexity classes under various reducibilities. Very recently, we have seen remarkable progress in this area for low-level complexity classes. In particular, the Hartmanis' sparseness conjectures for P and NL have been resolved. This article overviews the history of sparse hard set problems and exposes some of the recent results.