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...

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...

A set A is k(n) membership comparable if there is a polynomial time computable function that, given k(n) instances of A of length at most n, excludes one of the 2 k(n) possibilities for the memberships of the given strings in A. We show that if SAT is O(log n) membership comparable, then UniqueSAT 2 P. This extends the work of Ogihara; Beigel, Kummer, and Stephan; and Agrawal and Arvind [Ogi94, BKS94, AA94], and answers in the affirmative an open question suggested by Buhrman, Fortnow, and Torenvliet [BFT97]. Our proof also shows that if SAT is o(n) membership comparable, then UniqueSAT can be solved in deterministic time 2 o(n) . Our main technical tool is an algorithm of Ar et al. [ALRS92] to reconstruct polynomials from noisy data through the use of bivariate polynomial factorization.

We describe three orthogonal complexity measures: parallel time, amount of hardware, and degree of nonuniformity, which together parametrize most complexity classes. We show that the descriptive complexity framework neatly captures these measures using the parameters: quantifier depth, number of variable bits, and type of numeric predicates respectively. A fairly simple picture arises in which the basic questions in complexity theory -- solved and unsolved -- can be understood as questions about tradeoffs among these three dimensions. 1 Introduction An initial presentation of complexity theory usually makes the implicit assumption that problems, and hence complexity classes, are linearly ordered by "difficulty ". In the Chomsky Hierarchy each new type of automaton can decide more languages, and the Time Hierarchy Theorem tells us adding more time allows a Turing machine to decide more languages. Indeed the word "complexity" is often used (e.g., in the study of algorithms) to mean "wo...

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...

en a graph G and a pair of vertices s; t, this reduction produces a polynomial number of graphs G 1 ; : : : ; G k of polynomial size, together with distinguished vertex-pairs (s 1 ; t 1 ); : : : ; (s k ; t k ), that satisfy the following conditions. If there is no path from s to t in G, then no G i has a path from s i to t i ; if there is a path from s to t in G, then with high probability, at least one of the G i 's has a unique path from s i to t i . This reduction is due to Avi Wigderson [Wig94], and it exploits the "isolation lemma" of Mulmuley, Vazirani and Vazira

We prove unlikely consequences of the existence of sparse hard sets for P under deterministic as well as one-sided error randomized truth-table reductions. Our main results are as follows. We establish that the existence of a polynomially dense hard set for P under (randomized) logspace bounded truth-table reductions implies that P ` (R)L, and that the collapse goes down to P ` (R)NC 1 in case of reductions computable in (R)NC 1 . We also prove that the existence of a quasipolynomially dense hard set for P under (randomized) polylog-space truth-table reductions using polylogarithmically many queries implies that P ` (R)SPACE[polylogn]. The randomized space complexity classes we consider are based on the multiple access randomness concept. 1 Introduction A lot of research effort in complexity theory has been spent on the sparse hard set problem for NP, i.e., the question whether there are sparse hard sets for NP under various polynomial-time reducibilities. Two major motivations ...

If there is a sparse set hard for P under bounded truth table reductions computable in LOGSPACE or NC 2 , then P = NC 2 . We give the details of the proof to this theorem. 1 Introduction Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95a], following a breakthrough by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace many-one reductions, unless P = LOGSPACE. Bounded truth table reductions are a natural extension of many-one reductions and it is natural to ask what consequences can be drawn assuming there is a sparse set hard for P under bounded truth table reductions computable in LOGSPACE. In this note we give the details of the proof of the theorem that if such a sparse set exists, then a very unlikely consequence follows, namely P = NC 2 . This theorem is even valid for bounded truth table reductions computable in NC 2 . The proof for the case of 1-truth table reductions, which already generalizes the manyone reductions, has...

We prove that there is no sparse hard set for P under logspace computable bounded truthtable reductions unless P = L. In case of reductions computable in NC 1 , the collapse goes down to P = NC 1 . We parameterize this result and obtain a generic theorem allowing to vary the sparseness condition, the space bound and the number of queries of the truth-table reduction. Another instantiation yields that there is no quasipolynomially dense hard set for P under polylogspace computable truth-table reductions using polylogarithmically many queries unless P is in polylog-space. We also apply the proof technique to NL and L. We establish that there is no sparse hard set for NL under logspace computable bounded truth-table reductions unless NL = L, and that there is no sparse hard set for L under NC 1 -computable bounded truth-table reductions unless L = NC 1 . We show that all these results carry over to the randomized setting: If we allow two-sided error randomized reductions with con...