Results 1  10
of
14
On Membership Comparable Sets
 Journal of Computer and System Sciences
, 1999
"... 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 Unique ..."
Abstract

Cited by 15 (1 self)
 Add to MetaCart
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.
Sparse Hard Sets for P Yield SpaceEfficient Algorithms
, 1995
"... In 1978, Hartmanis conjectured that there exist no sparse complete sets for P under logspace manyone reductions. In this paper, in support of the conjecture, it is shown that if P has sparse hard sets under logspace manyone reductions, then P ` DSPACE[log 2 n]. The result is derived from a more ..."
Abstract

Cited by 11 (1 self)
 Add to MetaCart
In 1978, Hartmanis conjectured that there exist no sparse complete sets for P under logspace manyone reductions. In this paper, in support of the conjecture, it is shown that if P has sparse hard sets under logspace manyone 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 polylogarithmic spacecomputable manyone reductions, then P ` DSPACE[polylog]. 1 Introduction In 1978, Hartmanis conjectured that no Pcomplete sets under logspace manyone reductions can be polynomially sparse; i.e., for any Pcomplete 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 Pcomplete. 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...
Online learning and resourcebounded dimension: Winnow yields new lower bounds for hard sets
 SIAM Journal on Computing
, 2007
"... We establish a relationship between the online mistakebound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work ..."
Abstract

Cited by 8 (4 self)
 Add to MetaCart
We establish a relationship between the online mistakebound model of learning and resourcebounded dimension. This connection is combined with the Winnow algorithm to obtain new results about the density of hard sets under adaptive reductions. This improves previous work
Resolution of Hartmanis' Conjecture for NLHard Sparse Sets
 Theoretical Computer Science
, 1995
"... 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 vertexpairs (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 ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
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 vertexpairs (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
Reducing P to a Sparse Set using a Constant Number of Queries Collapses P to L
 In Proceedings of the 11th Conference on Computational Complexity
, 1996
"... 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 generalize this result by parameterizing the sparseness condition, the space bound and the number ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
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 generalize this result by parameterizing the sparseness condition, the space bound and the number of queries of the reduction, apply the proof technique to NL and L, and extend all these theorems to twosided error randomized reductions in the multiple access model, for which we also obtain new results for NP.
Sparse Hard Sets for P: Resolution of a Conjecture of Hartmanis
"... 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 manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE. ..."
Abstract

Cited by 5 (0 self)
 Add to MetaCart
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 manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE. We further prove that if P has a sparse hard set under manyone reductions computable in NC 1 , then P collapses to NC 1 .
On Reductions of P Sets to Sparse Sets
, 1995
"... We prove unlikely consequences of the existence of sparse hard sets for P under deterministic as well as onesided error randomized truthtable reductions. Our main results are as follows. We establish that the existence of a polynomially dense hard set for P under (randomized) logspace bounded trut ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
We prove unlikely consequences of the existence of sparse hard sets for P under deterministic as well as onesided error randomized truthtable reductions. Our main results are as follows. We establish that the existence of a polynomially dense hard set for P under (randomized) logspace bounded truthtable 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) polylogspace truthtable 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 polynomialtime reducibilities. Two major motivations ...
A Moment of Perfect Clarity I: The Parallel Census Technique
, 2000
"... We discuss the history and uses of the parallel census techniquean elegant tool in the study of certain computational objects having polynomially bounded census functions. A sequel [GH] will discuss advances (including [CNS95] and Glaer [Gla00]), some related to the parallel census technique and ..."
Abstract

Cited by 3 (3 self)
 Add to MetaCart
We discuss the history and uses of the parallel census techniquean elegant tool in the study of certain computational objects having polynomially bounded census functions. A sequel [GH] will discuss advances (including [CNS95] and Glaer [Gla00]), some related to the parallel census technique and some due to other approaches, in the complexityclass collapses that follow if NP has sparse hard sets under reductions weaker than (full) truthtable reductions.
Bounded Truth Table Reductions of P
, 1995
"... 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]. I ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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 manyone reductions, unless P = LOGSPACE. Bounded truth table reductions are a natural extension of manyone 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 1truth table reductions, which already generalizes the manyone reductions, has...
Deterministic and Randomized Bounded Truthtable Reductions of P, NL, and L to Sparse Sets
 Journal of Computer and System Sciences
, 1998
"... 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 ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
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 truthtable reduction. Another instantiation yields that there is no quasipolynomially dense hard set for P under polylogspace computable truthtable reductions using polylogarithmically many queries unless P is in polylogspace. 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 truthtable reductions unless NL = L, and that there is no sparse hard set for L under NC 1 computable bounded truthtable reductions unless L = NC 1 . We show that all these results carry over to the randomized setting: If we allow twosided error randomized reductions with con...