Results 1  10
of
20
Complete Sets and Structure in Subrecursive Classes
 In Proceedings of Logic Colloquium '96
, 1998
"... 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 completene ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
(Show Context)
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...
On the Structure of Low Sets
 PROC. 10TH STRUCTURE IN COMPLEXITY THEORY CONFERENCE, IEEE
, 1995
"... Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected ..."
Abstract

Cited by 12 (2 self)
 Add to MetaCart
Over a decade ago, Schöning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected topics in the area. Among the lowness properties we consider are polynomialsize circuit complexity, membership comparability, approximability, selectivity, and cheatability. Furthermore, we review some of the recent results concerning lowness for counting classes.
Partial Biimmunity, Scaled Dimension, and NPCompleteness
"... The Turing and manyone completeness notions for NP have been previously separated undermeasure, genericity, and biimmunity hypotheses on NP. The proofs of all these results rely on ..."
Abstract

Cited by 9 (4 self)
 Add to MetaCart
The Turing and manyone completeness notions for NP have been previously separated undermeasure, genericity, and biimmunity hypotheses on NP. The proofs of all these results rely on
Comparing Reductions to NPComplete Sets
"... Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: 1. Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. 2. Strong nondetermini ..."
Abstract

Cited by 8 (3 self)
 Add to MetaCart
Under the assumption that NP does not have pmeasure 0, we investigate reductions to NPcomplete sets and prove the following: 1. Adaptive reductions are more powerful than nonadaptive reductions: there is a problem that is Turingcomplete for NP but not truthtablecomplete. 2. Strong nondeterministic reductions are more powerful than deterministic reductions: there is a problem that is SNPcomplete for NP but not Turingcomplete. 3. Every problem that is manyone complete for NP is complete under lengthincreasing reductions that are computed by polynomialsize circuits.
Autoreducibility, mitoticity and immunity
 Mathematical Foundations of Computer Science: Thirtieth International Symposium, MFCS 2005
, 2005
"... We show the following results regarding complete sets. • NPcomplete sets and PSPACEcomplete sets are manyone autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are manyone autoreducible. • EXPcomplete sets are manyone mitotic. • NEXPcomplete sets are we ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
(Show Context)
We show the following results regarding complete sets. • NPcomplete sets and PSPACEcomplete sets are manyone autoreducible. • Complete sets of any level of PH, MODPH, or the Boolean hierarchy over NP are manyone autoreducible. • EXPcomplete sets are manyone mitotic. • NEXPcomplete sets are weakly manyone mitotic. • PSPACEcomplete sets are weakly Turingmitotic. • If oneway permutations and quick pseudorandom generators exist, then NPcomplete languages are mmitotic. • If there is a tally language in NP ∩ coNP − P, then, for every ɛ> 0, NPcomplete sets are not 2 n(1+ɛ)immune. These results solve several of the open questions raised by Buhrman and Torenvliet in their 1994 survey paper on the structure of complete sets. 1
Redundancy in complete sets
 In Proceedings 23nd Symposium on Theoretical Aspects of Computer Science
, 2006
"... We show that a set is mautoreducible if and only if it is mmitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al. [11], complete sets for all of the following complexity classes are mmitotic: NP, coNP, ..."
Abstract

Cited by 7 (6 self)
 Add to MetaCart
(Show Context)
We show that a set is mautoreducible if and only if it is mmitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Glaßer et al. [11], complete sets for all of the following complexity classes are mmitotic: NP, coNP, ⊕P, PSPACE, and NEXP, as well as all levels of PH, MODPH, and the Boolean hierarchy over NP. In the cases of NP, PSPACE, NEXP, and PH, this at once answers several wellstudied open questions. These results tell us that complete sets share a redundancy that was not known before. We disprove the equivalence between autoreducibility and mitoticity for all polynomialtimebounded reducibilities between 3ttreducibility and Turingreducibility: There exists a sparse set in EXP that is polynomialtime 3ttautoreducible, but not weakly polynomialtime Tmitotic. In particular, polynomialtime Tautoreducibility does not imply polynomialtime weak Tmitoticity, which solves an open question by Buhrman and Torenvliet. We generalize autoreducibility to define polyautoreducibility and give evidence that NPcomplete sets are polyautoreducible. 1
SPLITTING NPCOMPLETE SETS
, 2006
"... We show that a set is mautoreducible if and only if it is mmitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Gla6er et al., complete sets for all of the following complexity classes are mmitotic: NP, coNP, â ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
We show that a set is mautoreducible if and only if it is mmitotic. This solves a long standing open question in a surprising way. As a consequence of this unconditional result and recent work by Gla6er et al., complete sets for all of the following complexity classes are mmitotic: NP, coNP, âP, PSPACE, and NEXP, as well as all levels of PH, MODPH, and the Boolean hierarchy over NP. In the cases of NP, PSPACE, NEXP, and PH, this at once answers several wellstudied open questions. These results tell us that complete sets share a redundancy that was not known before. In particular, every NPcomplete set A splits into two NPcomplete sets A1 and A2. We disprove the equivalence between autoreducibility and mitoticity for all polynomialtimebounded reducibilities between 3ttreducibility and Turingreducibility: There exists a sparse set in EXP that is polynomialtime 3ttautoreducible, but not weakly polynomialtime Tmitotic. In particular, polynomialtime Tautoreducibility does not imply polynomialtime weak Tmitoticity, which solves an open question by Buhrman and Torenvliet.
Mitosis in computational complexity
 IN THEORY AND APPLICATIONS OF MODELS OF COMPUTATION (TAMC
, 2006
"... This expository paper describes some of the results of two recent research papers [GOP + 05,GPSZ05]. The first of these papers proves that every NPcomplete set is manyone autoreducible. The second paper proves that every manyone autoreducible set is manyone mitotic. It follows immediately that ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
This expository paper describes some of the results of two recent research papers [GOP + 05,GPSZ05]. The first of these papers proves that every NPcomplete set is manyone autoreducible. The second paper proves that every manyone autoreducible set is manyone mitotic. It follows immediately that every NPcomplete set is manyone mitotic. Hence, we have the compelling result that every NPcomplete set A splits into two NPcomplete sets A1 and A2.
Strong Reductions and Isomorphism of Complete Sets
"... We study the structure of the polynomialtime complete sets for NP and PSPACE under strong nondeterministic polynomialtime reductions (SNPreductions). We show the following results. • If NP contains a prandom language, then all polynomialtime complete sets for PSPACE are SNPisomorphic. • If NP ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We study the structure of the polynomialtime complete sets for NP and PSPACE under strong nondeterministic polynomialtime reductions (SNPreductions). We show the following results. • If NP contains a prandom language, then all polynomialtime complete sets for PSPACE are SNPisomorphic. • If NP ∩ coNP contains a prandom language, then all polynomialtime complete sets for NP are SNPisomorphic.