Reductions between disjoint NP-Pairs

Glaíer, Christian; Selman, Alan L.; Sengupta, Samik

doi:10.1016/j.ic.2005.03.003

Cited by 20 publications

(9 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Namely, Q1 is equivalent to the statement that every disjoint NP-pair is easy to separate, while Q2 is equivalent to the problem, whether the class of disjoint NP-pairs (and its generalizations) possess complete elements. In the context of NP-pairs, this question was posed by Razborov [45], and it has been intensively studied during the last years [18,19,5,4,3]. Our characterizations restate and unify some of these recent results in terms of nondeterministic functions.…”

supporting

confidence: 53%

See 1 more Smart Citation

Nondeterministic functions and the existence of optimal proof systems

Beyersdorff

Köbler

Messner

2009

Theoretical Computer Science

View full text Add to dashboard Cite

a b s t r a c tWe provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements? Q2: Do nondeterministic function classes contain complete functions?We show that Q1 for the class NPMV t is equivalent to the question whether the standard proof system for SAT is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for SAT, we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class NPMV t .An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint NP-pairs and its generalizations to tuples of sets from NP and coNP with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class NPSV. Question Q1 for NPSV is equivalent to the question of whether every disjoint NP-pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint NP-pairs, and we show how different interpolation properties can be modeled by NP-pairs associated with the underlying proof system.

show abstract

supporting

confidence: 53%

“…Disjoint NP-pairs have recently been intensively studied [42,[18][19][20][21][22]49,5,4,3], mainly, because they are suitable objects to model the security of cryptosystems [23,33], and further, because they are intimately connected to propositional proof systems [45,42,20,22,4].…”

mentioning

confidence: 99%

Nondeterministic functions and the existence of optimal proof systems

Beyersdorff

Köbler

Messner

2009

Theoretical Computer Science

View full text Add to dashboard Cite

show abstract

“…While the separation property was investigated rather comprehensively (see e.g. [14,13]), the shrinking property has not been considered systematically so far. In this respect, Blass and Gurevich [3] and Selivanov [33,38] show some first results and identify open questions.…”

Section: Introductionmentioning

confidence: 99%

The shrinking property for NP and coNP

Glaβer

Reitwieβner

Selivanov

2011

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

a b s t r a c tWe study the shrinking and separation properties (two notions well-known in descriptive set theory) for NP and coNP and show that under reasonable complexity-theoretic assumptions, both properties do not hold for NP and the shrinking property does not hold for coNP. In particular we obtain the following results. 1. NP and coNP do not have the shrinking property unless PH is finite. In general, Σ P n and Π P n do not have the shrinking property unless PH is finite. This solves an open question posed by Selivanov (1994) [33]. 2. The separation property does not hold for NP unless UP ⊆ coNP. 3. The shrinking property does not hold for NP unless there exist NP-hard disjoint NP-pairs (existence of such pairs would contradict a conjecture of Even et al. (1984) [6]). 4. The shrinking property does not hold for NP unless there exist complete disjoint NPpairs.Moreover, we prove that the assumption NP ̸ = coNP is too weak to refute the shrinking property for NP in a relativizable way. For this we construct an oracle relative to which P = NP ∩ coNP, NP ̸ = coNP, and NP has the shrinking property. This solves an open question posed by Blass and Gurevich (1984) [3] who explicitly ask for such an oracle.

show abstract

“…Razborov [37] later established a deep connection between disjoint NP pairs and propositional proof systems, associating with each propositional proof system a canonical disjoint NP pair. Glaßer, Selman, Sengupta, and Zhang [10,9,11,12] investigated this connection further, and it is now known that the degree structure of propositional proof systems under the natural notion of proof simulation is identical to the degree structure of disjoint NP pairs under reducibility of separators. See [8] for a survey of this and related results and [4] for more recent work.…”

Section: Introductionmentioning

confidence: 99%

Dimension in complexity classes

Lutz

Proceedings 15th Annual IEEE Conference on Computational Complexity

119

View full text Add to dashboard Cite

We prove three results on the dimension structure of complexity classes.1. The Point-to-Set Principle, which has recently been used to prove several new theorems in fractal geometry, has resource-bounded instances. These instances characterize the resource-bounded dimension of a set X of languages in terms of the relativized resourcebounded dimensions of the individual elements of X, provided that the former resource bound is large enough to parameterize the latter. Thus for example, the dimension of a class X of languages in EXP is characterized in terms of the relativized p-dimensions of the individual elements of X.2. Every language that is ≤ P m -reducible to a p-selective set has p-dimension 0, and this fact holds relative to arbitrary oracles. Combined with a resource-bounded instance of the Point-to-Set Principle, this implies that if NP has positive dimension in EXP, then no quasipolynomial time selective language is ≤ P m -hard for NP. 3. If the set of all disjoint pairs of NP languages has dimension 1 in the set of all disjoint pairs of EXP languages, then NP has positive dimension in EXP.

show abstract

Reductions between disjoint NP-Pairs

Cited by 20 publications

References 22 publications

Nondeterministic functions and the existence of optimal proof systems

Nondeterministic functions and the existence of optimal proof systems

The shrinking property for NP and coNP

Dimension in complexity classes

Contact Info

Product

Resources

About