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33
Computing Solutions Uniquely Collapses the Polynomial Hierarchy
 SIAM Journal on Computing
, 1993
"... Is there a singlevalued NP function that, when given a satisfiable formula as input, outputs a satisfying assignment? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterm ..."
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Cited by 39 (24 self)
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Is there a singlevalued NP function that, when given a satisfiable formula as input, outputs a satisfying assignment? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to its second level. As the existence of such a function is known to be equivalent to the statement "every multivalued NP function has a singlevalued NP refinement," our result provides the strongest evidence yet that multivalued NP functions cannot be refined. We prove our result via theorems of independent interest. We say that a set A is NPSVselective (NPMVselective) if there is a 2ary partial function in NPSV (NPMV, respectively) that decides which of its inputs (if any) is "more likely" to belong to A; this is a nondeterministic analog of the recursiontheoretic notion of the semirecursive sets and the extant complexitythe...
PolynomialTime Membership Comparable Sets
, 1994
"... This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m j ..."
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Cited by 31 (4 self)
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This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m jg), outputs b 2 f0; 1g m such that (A(x 1 ); \Delta \Delta \Delta ; A(xm )) 6= b. The following is a list of major results proven in the paper. 1. Polynomialtime membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomialtime membership comparable sets have polynomialsize circuits. 3. For any function f and for any constant c ? 0, if a set is p f(n)tt reducible to a Pselective set, then the set is polynomialtime (1 + c) log f(n)membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD 2 P; MOD 3 P; \Delta \Delta \Deltag, if C ` Pmc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...
Pselective Selfreducible sets: A New Characterization of P
 In Proceedings of the 8th Structure in Complexity Theory Conference
, 1996
"... We show that any pselective and selfreducible set is in P . As the converse is also true, we obtain a new characterization of the class P . A generalization and several consequences of this theorem are discussed. Among other consequences, we show that under reasonable assumptions autoreducibi ..."
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Cited by 27 (6 self)
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We show that any pselective and selfreducible set is in P . As the converse is also true, we obtain a new characterization of the class P . A generalization and several consequences of this theorem are discussed. Among other consequences, we show that under reasonable assumptions autoreducibility and selfreducibility differ on NP , and that there are nonpT mitotic sets in NP . 1 Introduction Separating complexity classes is a very popular, but rarely won game in complexity theory. Frustrated by misfortune, computer scientists have often turned to attempts of characterizing complexity classes in a different way. The hopes are, that the new characterization of the complexity class may provide new insights and a `handle' to force the separation where earlier attempts have failed. Wellknown examples of this are the many ways to define the class of sets for which there exist small circuits [Pip79], and the identification of various forms of interactive proof systems with stan...
Separation of NPcompleteness notions
 SIAM Journal on Computing
, 2001
"... Abstract. We use hypotheses of structural complexity theory to separate various NPcompleteness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P Tcomplete but not ¡ P ttcomplete. We provide fairly thorough analyses of the hypotheses that we introduc ..."
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Cited by 25 (12 self)
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Abstract. We use hypotheses of structural complexity theory to separate various NPcompleteness notions. In particular, we introduce an hypothesis from which we describe a set in NP that is ¡ P Tcomplete but not ¡ P ttcomplete. We provide fairly thorough analyses of the hypotheses that we introduce. Key words. Turing completeness, truthtable completeness, manyone completeness, pselectivity, pgenericity AMS subject classifications. 1. Introduction. Ladner, Lynch, and Selman [LLS75] were the first to compare the strength of polyno), truth), that mialtime reducibilities. They showed, for the common polynomialtime reducibilities, ( ¢ Turing P T ( ¢ table P tt), bounded truthtable ( ¢ P btt), and manyone ( ¢ P m
On ResourceBounded Instance Complexity
 Theoretical Computer Science A
, 1995
"... The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is the length of the shortest program for A that runs in time t, decides x correctly, and makes no mistakes on other strings (where "don't know" answers are permitted). The Instance Complexity Conjecture of ..."
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Cited by 19 (9 self)
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The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is the length of the shortest program for A that runs in time t, decides x correctly, and makes no mistakes on other strings (where "don't know" answers are permitted). The Instance Complexity Conjecture of Ko, Orponen, Schoning, and Watanabe states that for every recursive set A not in P and every polynomial t there is a polynomial t 0 and a constant c such that for infinitely many x, ic t (x : A) C t 0 (x) \Gamma c, where C t 0 (x) is the t 0 time bounded Kolmogorov complexity of x. In this paper the conjecture is proved for all recursive tally sets and for all recursive sets which are NPhard under honest reductions, in particular it holds for all natural NPhard problems. The method of proof also yields the polynomialspace bounded and the exponentialtime bounded versions of the conjecture in full generality. On the other hand, the conjecture itself turns out to be oracl...
On Using Oracles That Compute Values
 In Proc. 10th Annual Symp. on Theoret. Aspects of Computer Science, Lecture Notes in Computer Science
, 1993
"... This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomialtime reducibilities, it is shown th ..."
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Cited by 17 (6 self)
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This paper focuses on complexity classes of partial functions that are computed in polynomial time with oracles in NPMV, the class of all multivalued partial functions that are computable nondeterministically in polynomial time. Concerning deterministic polynomialtime reducibilities, it is shown that 1. A multivalued partial function is polynomialtime computable with k adaptive queries to NPMV if and only if it is polynomialtime computable via 2 k \Gamma 1 nonadaptive queries to NPMV. 2. A characteristic function is polynomialtime computable with k adaptive queries to NPMV if and only if it is polynomialtime computable with k adaptive queries to NP. 3. Unless the Boolean hierarchy collapses, for every k, k adaptive (nonadaptive) queries to NPMV is different than k + 1 adaptive (nonadaptive) queries to NPMV. Nondeterministic reducibilities, lowness and the difference hierarchy over NPMV are also studied. The difference hierarchy for partial functions does not collapse unless the...
Optimal Advice
 Theoretical Computer Science
, 1994
"... Ko [Ko83] proved that the Pselective sets are in the advice class P/quadratic. We prove that the Pselective sets are in NP=linear T coNP=linear. We show this to be optimal in terms of the amount of advice needed. 1 Introduction Selective sets are sets for which there is a "selector function," u ..."
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Cited by 15 (12 self)
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Ko [Ko83] proved that the Pselective sets are in the advice class P/quadratic. We prove that the Pselective sets are in NP=linear T coNP=linear. We show this to be optimal in terms of the amount of advice needed. 1 Introduction Selective sets are sets for which there is a "selector function," usually a polynomialtime deterministic or nondeterministic function, that selects which of any two given inputs is logically no less likely than the other to belong to the given set. Definition 1.1 [HNOS94] Let FC be any class of functions (possibly multivalued or partial). A set A is FCselective if there is a function f 2 FC such that for every x and y, it holds that f(x; y) ` fx; yg, and if fx; yg " A 6= ;, then f(x; y) 6= ; and f(x; y) ` A. Let FCsel denote the class of sets that are FCselective. The class that would be notated FP single\Gammavalued; total sel according to the definition above was defined directly by Selman in 1979 [Sel79]. Henceforward, we refer to these sets ...
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 ..."
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Cited by 15 (1 self)
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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.
The Power of Adaptiveness and Additional Queries in RandomSelfReductions
 Computational Complexity
, 1994
"... We look at relationships between adaptive and nonadaptive randomselfreductions. We also look at what happens to randomselfreductions if we restrict the number of queries they are allowed to make. We show the following results: ffl There exist sets that are adaptively randomselfreducible but n ..."
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Cited by 13 (2 self)
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We look at relationships between adaptive and nonadaptive randomselfreductions. We also look at what happens to randomselfreductions if we restrict the number of queries they are allowed to make. We show the following results: ffl There exist sets that are adaptively randomselfreducible but not nonadaptively randomselfreducible. Under reasonable assumptions, there exist such sets in NP: ffl There exists a function that has a nonadaptive (k + 1)randomselfreduction but does not have an adaptive krandomselfreduction. ffl For any countable class of functions C and any unbounded function k(n), there exists a function that is nonadaptively kuniformlyrandom selfreducible but is not in C=poly. These results were presented in preliminary form at the 7th IEEE Structure in Complexity Theory Conference, Boston MA, June 1992. 1 Introduction Informally, a function f is randomselfreducible if the evaluation of f at any given instance x can be reduced in polynomial time to the...
Reductions between Disjoint NPPairs
 Information and Computation
, 2004
"... We prove that all of the following assertions are equivalent: There is a manyone complete disjoint NPpair; there is a strongly manyone complete disjoint NPpair; there is a Turing complete disjoint NPpair such that all reductions are smart reductions; there is a complete disjoint NPpair for one ..."
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Cited by 13 (3 self)
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We prove that all of the following assertions are equivalent: There is a manyone complete disjoint NPpair; there is a strongly manyone complete disjoint NPpair; there is a Turing complete disjoint NPpair such that all reductions are smart reductions; there is a complete disjoint NPpair for onetoone, invertible reductions; the class of all disjoint NPpairs is uniformly enumerable. Let A, B, C, and D be nonempty sets belonging to NP. A smart reduction between the disjoint NPpairs (A, B) and (C, D) is a Turing reduction with the additional property that if D. We prove under the reasonable assumption UP coUP has a Pbiimmune set that there exist disjoint NPpairs (A, B) and (C, D) such that (A, B) is truthtable reducible to (C, D), but there is no smart reduction between them. This paper contains several additional separations of reductions between disjoint NPpairs. We exhibit an oracle relative to which DisjNP has a truthtablecomplete disjoint NPpair, but has no manyonecomplete disjoint NPpair.