## Redundancy in complete sets (2006)

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Venue: | In Proceedings 23nd Symposium on Theoretical Aspects of Computer Science |

Citations: | 6 - 6 self |

### BibTeX

@INPROCEEDINGS{Glaßer06redundancyin,

author = {Christian Glaßer and A. Pavan and Alan L. Selman and Liyu Zhang},

title = {Redundancy in complete sets},

booktitle = {In Proceedings 23nd Symposium on Theoretical Aspects of Computer Science},

year = {2006},

pages = {444--454},

publisher = {Springer}

}

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### Abstract

We show that a set is m-autoreducible if and only if it is m-mitotic. 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 m-mitotic: 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 well-studied 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 polynomialtime-bounded reducibilities between 3-tt-reducibility and Turing-reducibility: There exists a sparse set in EXP that is polynomial-time 3-tt-autoreducible, but not weakly polynomial-time T-mitotic. In particular, polynomial-time T-autoreducibility does not imply polynomial-time weak T-mitoticity, which solves an open question by Buhrman and Torenvliet. We generalize autoreducibility to define poly-autoreducibility and give evidence that NPcomplete sets are poly-autoreducible. 1

### Citations

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On isomorphisms and density of NP and other complete sets
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(Show Context)
Citation Context ...pen questions raised by Ambos-Spies [2], Buhrman, Hoene, and Torenvliet [7], and Buhrman and Torenvliet [8]. 2sOur result can also be viewed as a step towards understanding the isomorphism conjecture =-=[5]-=-. This conjecture states that all NP-complete sets are isomorphic to each other. In spite of several years of research, we do not have any concrete evidence either in support or against the isomorphis... |

105 |
Sparse complete sets for NP: Solution of a conjecture of Berman and Hartmanis
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(Show Context)
Citation Context ...hen NP-complete sets cannot be sparse. This motivated researchers to consider the question of whether complete sets for NP can be sparse. This line of research led to the beautiful results of Mahaney =-=[14]-=- and Ogiwara and Watanabe [15] who showed that complete sets for NP cannot be sparse unless P = NP. Our results show that another consequence of isomorphism, namely “NP-complete sets are m-mitotic” ho... |

49 |
Bi-immune sets for complexity classes
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- 1985
(Show Context)
Citation Context ...class C, or C-immune, if L is infinite and no infinite subset of L belongs to C. A language L is bi-immune to a complexity class C, or C-bi-immune, if both L and L are C-immune. Balcázar and Schöning =-=[3]-=- proved that for every time-constructible function T, L is DTIME(T(n))-complex if and only if L is bi-immune to DTIME(T(n)). 3 m-Autoreducibility equals m-Mitoticity It is easy to see that if a nontri... |

45 | On polynomial-time bounded truth-table reducibility of NP sets to sparse sets
- Ogiwara, Watanabe
- 1991
(Show Context)
Citation Context ... sparse. This motivated researchers to consider the question of whether complete sets for NP can be sparse. This line of research led to the beautiful results of Mahaney [14] and Ogiwara and Watanabe =-=[15]-=- who showed that complete sets for NP cannot be sparse unless P = NP. Our results show that another consequence of isomorphism, namely “NP-complete sets are m-mitotic” holds. Note that this is an unco... |

28 |
P-mitotic sets
- Ambos-Spies
- 1984
(Show Context)
Citation Context ...cible if there is an oracle Turing machine M such that A = L(M A ) and M on input x never queries x. For complexity classes like NP and PSPACE refined measures are needed. In this spirit, Ambos-Spies =-=[2]-=- defined the notion of polynomialtime autoreducibility and the more restricted form m-autoreducibility. A set A is polynomial-time autoreducible if it is autoreducible via a oracle Turing machine that... |

27 |
Mitotic recursively enumerable sets
- Ladner
- 1973
(Show Context)
Citation Context ...n more apparent—if a set A is m-autoreducible, then x and f(x) have the same information about A. A stronger form of redundancy is described by the notion of mitoticity which was introduced by Ladner =-=[13]-=- for the recursive setting and by Ambos-Spies [2] for the polynomial-time setting. A set A is m-mitotic if there is a set S ∈ P such that A, A ∩ S, and A ∩ S are polynomial-time many-one equivalent. T... |

27 |
Coherent functions and program checkers
- Yao
- 1990
(Show Context)
Citation Context ... isomorphism conjecture is false. However, we do not have a proof of this. 3s1.1 Previous Work The question of whether complete sets for various classes are autoreducible has been studied extensively =-=[17, 4, 6]-=-. Beigel and Feigenbaum [4] showed that Turing complete sets for the classes that form the polynomial hierarchy, ΣP i ,ΠPi , and ∆Pi , are Turing autoreducible. Thus, all Turing complete sets for NP a... |

26 | On being incoherent without being very hard
- Beigel, Feigenbaum
- 1992
(Show Context)
Citation Context ... isomorphism conjecture is false. However, we do not have a proof of this. 3s1.1 Previous Work The question of whether complete sets for various classes are autoreducible has been studied extensively =-=[17, 4, 6]-=-. Beigel and Feigenbaum [4] showed that Turing complete sets for the classes that form the polynomial hierarchy, ΣP i ,ΠPi , and ∆Pi , are Turing autoreducible. Thus, all Turing complete sets for NP a... |

24 | Pseudo-random generators and structure of complete degrees
- Agrawal
- 2002
(Show Context)
Citation Context ...milarly, output polynomially many distinct strings such that none of them are in L. Below, we show that if one-way permutations exist, then we can achieve this task. We start with a result by Agrawal =-=[1]-=-. Definition 5.4 Let f be a many-one reduction from A to B. We say f is g(n)-sparse, if for every n, no more than g(n) strings of length n are mapped to a single string via f. Lemma 5.5 ([1]) If one-w... |

24 | Using autoreducibility to separate complexity classes
- Buhrman, Fortnow, et al.
(Show Context)
Citation Context ...r NP cannot be sparse unless P = NP. Our results show that another consequence of isomorphism, namely “NP-complete sets are m-mitotic” holds. Note that this is an unconditional result. Buhrman et al. =-=[6]-=- and Buhrman and Torenvliet [9, 10] argue that it is critical to study the notions of autoreducibility and mitoticity. They showed that resolving questions regarding autoreducibility of complete sets ... |

22 | On the structure of complete sets
- Buhrman, Torenvliet
- 1994
(Show Context)
Citation Context ...ly polynomialtime T-mitotic. In particular, polynomial-time T-autoreducible does not imply polynomial-time weakly T-mitotic. This result settles another open question raised by Buhrman and Torenvliet =-=[8]-=-. Our main result relates local redundancy to global redundancy in the following sense. If a set A is m-autoreducible, then x and f(x) contain the same information about A. This can be viewed as local... |

16 | Splittings, robustness, and structure of complete sets
- Buhrman, Hoene, et al.
- 1998
(Show Context)
Citation Context ...levels of PH are m-mitotic. Thus they all contain redundant information in a strong sense. This resolves several long standing open questions raised by Ambos-Spies [2], Buhrman, Hoene, and Torenvliet =-=[7]-=-, and Buhrman and Torenvliet [8]. 2sOur result can also be viewed as a step towards understanding the isomorphism conjecture [5]. This conjecture states that all NP-complete sets are isomorphic to eac... |

11 | Properties of NP-complete sets
- Glaßer, Pavan, et al.
- 2004
(Show Context)
Citation Context ...rse set, from which it follows NP = P [14]. Therefore, L ∈ P and so L is trivially not E-immune. Hence, we assume that L − T �= ∅. Then L − T is NP-complete by Theorem 3.1 of a paper by Glaßer et al. =-=[12]-=-. Hence, L ∪ T is NP-complete since L − T ≤ p m L ∪ T. By assumption, L ∪ T is poly-autoreducible. So there exists a polynomial-time computable function f such that for every input x, 1. f(x) is a set... |

10 | Separating complexity classes using structural properties
- Buhrman, Torenvliet
- 2004
(Show Context)
Citation Context ...= NP. Our results show that another consequence of isomorphism, namely “NP-complete sets are m-mitotic” holds. Note that this is an unconditional result. Buhrman et al. [6] and Buhrman and Torenvliet =-=[9, 10]-=- argue that it is critical to study the notions of autoreducibility and mitoticity. They showed that resolving questions regarding autoreducibility of complete sets leads to unconditional separation r... |

9 |
On autoreducibility. Dokl. Akad. Nauk SSSR
- Trakhtenbrot
- 1970
(Show Context)
Citation Context ...n of how much redundancy is contained in complete sets of complexity classes. There are several ways to measure “redundancy”. We focus on the two notions autoreducibility and mitoticity. Trakhtenbrot =-=[16]-=- defined a set A to be autoreducible if there is an oracle Turing machine M such that A = L(M A ) and M on input x never queries x. For complexity classes like NP and PSPACE refined measures are neede... |

7 |
A Post’s program for complexity theory
- Buhrman, Torenvliet
(Show Context)
Citation Context ...= NP. Our results show that another consequence of isomorphism, namely “NP-complete sets are m-mitotic” holds. Note that this is an unconditional result. Buhrman et al. [6] and Buhrman and Torenvliet =-=[9, 10]-=- argue that it is critical to study the notions of autoreducibility and mitoticity. They showed that resolving questions regarding autoreducibility of complete sets leads to unconditional separation r... |