## P-selective Self-reducible sets: A New Characterization of P (1996)

Venue: | In Proceedings of the 8th Structure in Complexity Theory Conference |

Citations: | 27 - 6 self |

### BibTeX

@INPROCEEDINGS{Buhrman96p-selectiveself-reducible,

author = {Harry Buhrman and Leen Torenvliet},

title = {P-selective Self-reducible sets: A New Characterization of P},

booktitle = {In Proceedings of the 8th Structure in Complexity Theory Conference},

year = {1996},

pages = {44--51},

publisher = {IEEE Computer Society Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

We show that any p-selective and self-reducible 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 auto-reducibility and self-reducibility differ on NP , and that there are non-p-T -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. Well-known 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...

### Citations

717 | Proof Verification and the Hardness of Approximation Problems - Arora, Lund, et al. - 1998 |

403 | Non-deterministic exponential time has two-prover interactive protocols - Babai, Fortnow, et al. - 1991 |

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On the structure of polynomial time reducibility
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Citation Context ... ( P T ), and non-adaptive machines to model truth-table reductions ( P tt ). For polynomial time bounded machines, this yields definitions equivalent to the standard definitions of reducibilities in =-=[LLS75]-=-. If the number of queries in a truth-table reduction is fixed by some constant, we call such a reduction a bounded truth-table reduction. If we can identify this constant, i.e. no more than k queries... |

230 | Languages that capture complexity classes - Immerman - 1987 |

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Structural Complexity I
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Citation Context ...r any set S, the cardinality of S is denoted by jjSjj. We assume that the reader is familiar with the standard Turing machine model and other standard notions of complexity theory, as can be found in =-=[BDG88]-=-. Nevertheless, some of the definitions that we feel may not be common knowledge are cited below. An oracle machine is a multi-tape Turing machine with an input tape, an output tape, work tapes, and a... |

84 |
P-selective sets, tally languages, and the behavior of polynomial time reducibilities on NP
- Selman
- 1979
(Show Context)
Citation Context ...xist. Unfortunately, many problems of interest are not known to be in P . Therefore, interest has shifted from P to classes `near' P , and classes of 1 sets as `near-testable' [GHJY91], `p-selective' =-=[Sel79]-=-, `self-reducible' [MP79] and such have been defined. For many of these, characterizations in terms of standard complexity classes are more or less known. For instance it is known that all self-reduci... |

73 |
On simultaneous resource bounds
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(Show Context)
Citation Context ... new insights and a `handle' to force the separation where earlier attempts have failed. Well-known 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 standard complexity classes as PSPACE , EXP and NP [Sha90, BFL90, ALM + 92]. Also, the classification of complexity classes ... |

63 |
Semirecursive sets and positive reducibility
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(Show Context)
Citation Context ...ccordingly. Selman introduced p-selective sets in [Sel79], which are a direct translation of the semi-recursive sets introduced, in the context of an attempted solution to Post's program, by Jockusch =-=[Joc68]-=-. 4 Definition 4: A set A is called p-selective iff there exists a polynomial time computable function f : \Sigma \Theta \Sigma 7! \Sigma , called a p-selector , such that for any x; y 2 \Sigma 1. f(x... |

55 |
On self-reducibility and weak-P-selectivity
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(Show Context)
Citation Context ...f-reducible sets are in PSPACE , all p-cheatable selfreducible sets are in P [GJY93], all p-cheatable and near testable sets are in P [GJY93], and that all disjunctively self-reducible sets are in NP =-=[Ko83]-=-. Furthermore it is known that disjunctively self-reducible sets that are in addition p-selective are all in P [Sel79]. In this paper, we give a generalization of that theorem. We show that if a set i... |

52 | F.Stephan: Approximable Sets
- Beigel
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(Show Context)
Citation Context ...T 0 (cf. [HIS85]). From this and the earlier cited result by Selman [Sel82] we infer the existence of a p-selective set B in NP \Gamma P such that TsP m DsP T T 0 P T B 2 Very recently, Beigel et al. =-=[BKS94]-=- have improved upon this Theorem, by showing that there exists a relativized world such that there exists a disjunctive self-reducible set A in NP \Gamma P such that A issP 2\Gammatt -reducible to som... |

50 |
Sparse sets
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Citation Context ... Furthermore D is sparse since for each 0 n 2 T there is only one witness of size q(n) and there are only q(n) prefixes of w. Since D is sparse there is a tally set T 0 in NP such that DsP T T 0 (cf. =-=[HIS85]-=-). From this and the earlier cited result by Selman [Sel82] we infer the existence of a p-selective set B in NP \Gamma P such that TsP m DsP T T 0 P T B 2 Very recently, Beigel et al. [BKS94] have imp... |

45 |
Tally languages and complexity classes
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(Show Context)
Citation Context ...er generalizations are probably hard to obtain (if they can be obtained at all). 4.1 Consequences of the theorems From the assumption E 6= NE , it follows that there exists a tally set in NP \Gamma P =-=[Boo74]-=-. Selman showed in [Sel82], that for any tally language T , there exists a p-selective set A such that TsP T A and AsP pos\Gammatt T . From this, it follows that E 6= NE implies the existence of a p-s... |

44 |
Reductions on NP and p-selective sets
- Selman
- 1982
(Show Context)
Citation Context ...e, we will make no significant difference between these two types of reductions. A reduction (of any of the above types) is called positive iff for any two oracles A ` B it holds that L(M;A) ` L(M;B) =-=[Sel82]-=-. The set of queries generated on input x by oracle machine M is denoted QM (x). For adaptive machines, this set may be oracle dependent, and is therefore denoted Q A M (x), if A is the oracle set. Th... |

39 | Expressibility as a complexity measure: results and directions - Immerman - 1987 |

39 | P-selective sets and reducing search to decision vs. self-reducibility
- Hemaspaandra, Naik, et al.
- 1996
(Show Context)
Citation Context ...y 2 \Sigma 1. f(x; y) 2 fx; yg and 2. A (f(x; y)) = maxfA (x); A (y)g 3 Main results Selman showed in [Sel79] that a set L is disjunctively self-reducible and pselective if and only if L is in P . In =-=[NOS93]-=-, a similar characterization of P is obtained, which shows that only the sets in P can be both p-selective and have search nonadaptively reducing to decision We obtain another characterization of P he... |

38 |
With what frequency are apparently intractable problems difficult
- Meyer, Paterson
- 1979
(Show Context)
Citation Context ...problems of interest are not known to be in P . Therefore, interest has shifted from P to classes `near' P , and classes of 1 sets as `near-testable' [GHJY91], `p-selective' [Sel79], `self-reducible' =-=[MP79]-=- and such have been defined. For many of these, characterizations in terms of standard complexity classes are more or less known. For instance it is known that all self-reducible sets are in PSPACE , ... |

28 |
P-mitotic sets
- Ambos-Spies
- 1984
(Show Context)
Citation Context ...cible, we note that Naik [Nai92] has shown under the assumption UE 6= co\GammaUE , that there exists a set in NP which is conjunctively selfreducible yet not disjunctively self-reducible. Ambos-Spies =-=[AS84]-=- has shown, that any set that is p-T-mitotic is also p-T-auto-reducible. It follows immediately from this observation that Corollary 10 There exists a non p-T-mitotic set in NP unless NEE = EE . 4.2 A... |

27 |
Mitotic recursively enumerable sets
- Ladner
- 1973
(Show Context)
Citation Context ...rollary. Corollary 8 [BF92] If A issP T -complete for NP then A is auto-reducible This corollary is also interesting seen in the light that such a fact is plainly not true for r.e. sets, since Ladner =-=[Lad73]-=- has demonstratedsT - complete r.e. sets that are not mitotic and hence not auto-reducible (for recursive reductions of course). The p-selective set in NP \Gamma P cited before [NOS93] is clearly not ... |

26 | On being incoherent without being very hard
- Beigel, Feigenbaum
- 1992
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Citation Context ... be output of the computation. 2 It is known that all sets in NP that are complete undersP T reductions have SRTD [Sel92]. Together with proposition 7, this gives the following corollary. Corollary 8 =-=[BF92]-=- If A issP T -complete for NP then A is auto-reducible This corollary is also interesting seen in the light that such a fact is plainly not true for r.e. sets, since Ladner [Lad73] has demonstratedsT ... |

24 | A comparison of polynomial time reducibilities. Theoret - Ladner, Lynch, et al. - 1975 |

18 |
Some comments on functional self-reducibility and the NP hierarchy
- Borodin, Demers
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Citation Context ...d that M , on input x, queries only strings y for which y 6= x. Selfreducibility is strongly related to a property that is shared by some sets in NP . This notion was introduced by Borodin and Demers =-=[BD76]-=- as functional self-reducibility . A set A is called functional self-reducible if a proof for membership in A for a string x, can be generated in polynomial time using A as an oracle. We will adopt mo... |

14 |
Near-testable sets
- Goldsmith, Hemachandra, et al.
- 1991
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Citation Context ...ch feasible algorithms exist. Unfortunately, many problems of interest are not known to be in P . Therefore, interest has shifted from P to classes `near' P , and classes of 1 sets as `near-testable' =-=[GHJY91]-=-, `p-selective' [Sel79], `self-reducible' [MP79] and such have been defined. For many of these, characterizations in terms of standard complexity classes are more or less known. For instance it is kno... |

11 |
Emde Boas: Twenty Questions to a P-selector
- Buhrman, van
- 1993
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Citation Context ...re exists a relativized world such that there exists a disjunctive self-reducible set A in NP \Gamma P such that A issP 2\Gammatt -reducible to some p-selective set. On the other hand, Buhrman et al. =-=[BTB93]-=- proved that any set that is positively Turing reducible to a p-selective set is itself p-selective. Therefore it follows directly that Theorem 11 also holds with 1-tt reductions replaced by positive ... |

10 |
Promise problems complete for complexity classes
- Selman
- 1988
(Show Context)
Citation Context ... (0 n ; w). We call this string the witness for 0 n . Let D = prefix(T ) = f!0 n ; y? j y is a prefix of w and R T (0 n ; w)g. It is not hard to see that D is disjunctively self-reducible (cf. Selman =-=[Sel88]-=-). Also, TsP m D since 0 n 2 T iff !0 n ; ? 2 D. Now as T 2 NP \Gamma P it follows that also D 2 NP \Gamma P . Furthermore D is sparse since for each 0 n 2 T there is only one witness of size q(n) and... |

6 |
Using self-reducibility to characterize polynomial time
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- 1993
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Citation Context ...se, characterizations in terms of standard complexity classes are more or less known. For instance it is known that all self-reducible sets are in PSPACE , all p-cheatable selfreducible sets are in P =-=[GJY93]-=-, all p-cheatable and near testable sets are in P [GJY93], and that all disjunctively self-reducible sets are in NP [Ko83]. Furthermore it is known that disjunctively self-reducible sets that are in a... |

6 | Analogues of semirecursive sets and effecitve reducibilities to the study of NP complexity - Selman - 1982 |

4 | Analogues of semicursive sets and effective reducibilities to the study of NP complexity - Selman - 1982 |

2 |
Languages that are easier to verify than their proofs
- Beigel, Bellare, et al.
- 1991
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Citation Context ...freducible. As it does have SRTD, and hence is auto-reducible, we conclude: Corollary 9 There exists a set in NP that is auto-reducible but not selfreducible unless NE " co\GammaNE = E. Beigel et=-= al. [BBFG91]-=- have shown the existence of a set (under the assumption that NEE 6= EE ) for which search does not reduce to decision. It is straightforward to see that this set is also not auto-reducible. If NEE 6=... |

1 |
Topics on Tournaments. Selected Topics in Mathematics
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Citation Context ...wo strings x i and x j in V , either x isf x j or x jsf x i or both, and in time polynomial in jx i j + jx j j we can decide which of the first two cases hold. Next we play a knock-out tournament(cf. =-=[Moo68]-=- page 48) among the n strings in V , where we say that x beats y if xsf y. (If xsf y is established, then ysf x is not examined, so a draw may end in an arbitrary winner, which is o.k., since 5 in tha... |