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Collapsing degrees via strong computation

  • Complexity (Session 9)
  • Conference paper
  • First Online:
Automata, Languages and Programming (ICALP 1991)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 510))

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Abstract

An equivalence class with respect to a given type of reduction is called a degree. It is natural to expect that using more flexible reduction types will increase the size of a degree. When using more flexible reduction types fails to increase the size of a degree, this is described as a collapsing degree; for example, it is known that all many-one-complete sets for exponential time are indeed equivalent with respect to many-one length-increasing reductions.

Over the years, powerful techniques have been developed to collapse non-deterministic degrees at and above nondeterministic linear space, and Immerman and Szelepcsényi have provided techniques that collapse even sublinear nondeterministic space classes. However, it has remained an open problem whether any collapses could be proven for sublinear nondeterministic space degrees. This paper provides the first such collapses. For nondeterministic space classes C above NL, we show that all ≤ 1−Lm -complete sets for C collapseto a single ≤ 1−L1li degree (i.e., all ≤ 1−Lm -complete sets for C are ≤ 1−L1li -equivalent), and that all ≤ 1−NLm -complete sets for C are NL-isomorphic (and thus P-isomorphic). Our techniques also improve previous results for PSPACE.

(Extended Abstract)

Research supported in part by the National Science Foundation under research grants CCR-8957604 and CCR-8996198.

Research supported by a Deutsche Forschungsgemeinschaft Postdoktorandenstipendium.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Rochester, 14627, Rochester, NY, USA

    Lane A. Hemachandra

  2. Fachbereich Informatik, Technische Universität Berlin, D-1000, Berlin 10, Germany

    Albrecht Hoene

Authors
  1. Lane A. Hemachandra
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  2. Albrecht Hoene
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Editor information

Javier Leach Albert Burkhard Monien Mario Rodríguez Artalejo

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© 1991 Springer-Verlag Berlin Heidelberg

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Hemachandra, L.A., Hoene, A. (1991). Collapsing degrees via strong computation. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds) Automata, Languages and Programming. ICALP 1991. Lecture Notes in Computer Science, vol 510. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-54233-7_150

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  • DOI: https://doi.org/10.1007/3-540-54233-7_150

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  • Print ISBN: 978-3-540-54233-9

  • Online ISBN: 978-3-540-47516-3

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