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Computational depth and reducibility

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Automata, Languages and Programming (ICALP 1993)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 700))

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Abstract

This paper investigates Bennett's notions of strong and weak computational depth (also called logical depth) for infinite binary sequences. Roughly, an infinite binary sequence x is defined to be weakly useful if every element of a non-negligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett's observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost every infinite binary sequence is weakly deep, but not strongly deep.

This research was supported in part by National Science Foundation Grant CCR-9157382, with matching funds from Rockwell International.

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Authors and Affiliations

  1. Department of Computer Science, Iowa State University, 50011, Ames, IA

    David W. Juedes, James I. Lathrop & Jack H. Lutz

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  1. David W. Juedes
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  2. James I. Lathrop
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  3. Jack H. Lutz
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Andrzej Lingas Rolf Karlsson Svante Carlsson

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© 1993 Springer-Verlag Berlin Heidelberg

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Juedes, D.W., Lathrop, J.I., Lutz, J.H. (1993). Computational depth and reducibility. In: Lingas, A., Karlsson, R., Carlsson, S. (eds) Automata, Languages and Programming. ICALP 1993. Lecture Notes in Computer Science, vol 700. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56939-1_79

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  • DOI: https://doi.org/10.1007/3-540-56939-1_79

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  • Print ISBN: 978-3-540-56939-8

  • Online ISBN: 978-3-540-47826-3

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