## Using Autoreducibility to Separate Complexity Classes (1995)

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Venue: | In Proceedings of the 36th IEEE Symposium on Foundations of Computer Science |

Citations: | 24 - 11 self |

### BibTeX

@INPROCEEDINGS{Buhrman95usingautoreducibility,

author = {Harry Buhrman and Lance Fortnow and Leen Torenvliet},

title = {Using Autoreducibility to Separate Complexity Classes},

booktitle = {In Proceedings of the 36th IEEE Symposium on Foundations of Computer Science},

year = {1995},

pages = {520--527},

publisher = {IEEE}

}

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### Abstract

A language is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate exponential space from doubly exponential space by showing that all Turing-complete sets for exponential space are autoreducible but there exists some Turingcomplete set for doubly exponential space that is not. We immediately also get a separation of logarithmic space from polynomial space. Although we already know how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's Program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular if we could settle the question as to whether all complete sets for doubly exponential time were autoreducible we would separate polynomial time from either logarithmic space or polynomial space. We also show several other theorems about autore...

### Citations

241 |
On the structure of polynomial time reducibility
- LADNER
- 1975
(Show Context)
Citation Context ...ns ( p T ) and nonadaptive 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]-=-. 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), where A is the oracle set.... |

190 |
On the computational complexity of algorithms
- Hartmanis, Stearns
- 1965
(Show Context)
Citation Context ...SPACE that is not autoreducible. As an immediate corollary we get that EXPSPACE 6= EEXPSPACE and thus that L 6= PSPACE. Although we have known these separations via the usual space hierarchy theorems =-=[HS65]-=- our proof does not rely on diagonalization, rather separates the classes by showing that the classes have different structural properties. Moreover our approach does not relativize. ffl There exists ... |

171 |
Structural Complexity I
- Balcázar, Díaz, et al.
- 1988
(Show Context)
Citation Context ... and are denoted by capital letters A; B; C; S; : : :. 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 multitape Turing machine with an input tape, an output tape, work tapes, and a... |

74 | On the random-self-reducibility of complete sets
- Feigenbaum, Fortnow
- 1993
(Show Context)
Citation Context ...or B and will cause M (x) to reject. 2 Similar though simpler proofs yield the following corollary: Corollary 3.4 All Turing-complete sets for PSPACE and EXP are autoreducible. Feigenbaum and Fortnow =-=[FF93]-=- define the following concept of #P-robustness: A set L is #P-robust if P #P L = P L Theorem 3.5 Every #P-robust language is nonuniformly autoreducible. Proof Sketch: Feigenbaum and Fortnow [FF93] sho... |

44 | Classical Recursion Theory, volume 125 - Odifreddi - 1989 |

28 |
P-mitotic sets
- Ambos-Spies
- 1984
(Show Context)
Citation Context ...autoreducibility in both the recursion theory and resource-bounded models. Ladner [Lad73] showed that there existed Turing-complete recursively enumerable sets that are not autoreducible. Ambos-Spies =-=[AS84]-=- first transferred the notion of autoreducibility to the polynomial-time settings. More recently, Yao [Yao90] and Beigel and Feigenbaum [BF92] have studied a probabilistic variant of autoreducibility ... |

27 | Selective self-reducible sets: A new characterization of P
- Buhrman, Helden, et al.
- 1993
(Show Context)
Citation Context ...lso finds it counterpart for non-adaptive autoreductions. Theorem 5.3 Allsp tt -complete sets for NP aresp tt autoreducible with respect to nonuniform reductions. Proof Sketch: We use techniques from =-=[BvHT93]-=-. Let L be asp tt -complete set for NP. On input x one can compute relative to L a witness y x that witnesses that x 2 L. Whenever x is queried in this computation answer 1 (x 2 L). If a witness y x i... |

27 |
Mitotic recursively enumerable sets
- Ladner
- 1973
(Show Context)
Citation Context ...s "autoreducibility." A set L is autoreducible if there is a polynomial-time oracle Turing machine M that accepts L using L as an oracle with the caveat that M (x) may not query whether x 2 =-=L. Ladner [Lad73]-=- looked at autoreducibility in the recursion theory environment where he showed that there existed Turing-complete recursively enumerable sets that are not autoreducible. Ambos-Spies [AS84] CWI. PO Bo... |

26 | On being incoherent without being very hard
- Beigel, Feigenbaum
- 1992
(Show Context)
Citation Context ...y enumerable sets that are not autoreducible. Ambos-Spies [AS84] first transferred the notion of autoreducibility to the polynomial-time settings. More recently, Yao [Yao90] and Beigel and Feigenbaum =-=[BF92] have stud-=-ied a probabilistic variant of autoreducibility known as "coherence. " In this paper, we ask for what complexity classes do all the complete sets have the autoreducibility property. In parti... |

18 | Resource Bounded Reductions
- Buhrman
- 1993
(Show Context)
Citation Context ...shows that the coding in Theorem 4.1 can not be done via asp 2\Gammatt reduction. It also shows a structural difference betweensp 2\GammaT -complete sets andsp 2\Gammatt -complete sets. Corollary 5.6 =-=[BST93]-=- There exists asp 2\GammaT -complete set for EXPSPACE that is notsp 2\Gammatt -complete. Proof: Use A in Theorem 5.2 together with Corollary 5.5. 2 6 Conclusions We believe that this research may lead... |

16 | Splittings, robustness, and structure of complete sets - Buhrman, Hoene, et al. - 1998 |

7 |
Complexity Classes of Alternating Machines with Oracles
- Orponen
(Show Context)
Citation Context ...e Turing-complete set for EXPSPACE is autoreducible. Proof: We use the following characterization of EXPSPACE that extends the oracle characterization of the exponential-time hierarchy due to Orponen =-=[Orp83]-=- and the alternating characterization of PSPACE due to Chandra, Kozen and Stockmeyer [CKS81]. Let p be a polynomial and M an oracle Turing machine running in time p(n). Let us also have two arbitrary ... |