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The intensional content of Rice's theorem

Published:07 January 2008Publication History
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Abstract

The proofs of major results of Computability Theory like Rice, Rice-Shapiro or Kleene's fixed point theorem hidemore information of what is usually expressed in theirrespective statements. We make this information explicit, allowing to state stronger, complexity theoretic-versions of all these theorems. In particular, we replace the notion of extensional set of indices of programs, by a set of indices of programs having not only the same extensional behavior but also similar complexity (Complexity Clique). We prove, under very weak complexity assumptions, that any recursive Complexity Clique is trivial, and any r.e. Complexity Clique is an extensional set (and thus satisfies Rice-Shapiro conditions). This allows, for instance, to use Rice's argument to prove that the property of having polynomial complexity is not decidable, and to use Rice-Shapiro to conclude that it is not even semi-decidable. We conclude the paper with a discussion of "complexity-theoretic" versions of Kleene's Fixed Point Theorem.

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      • Published in

        cover image ACM SIGPLAN Notices
        ACM SIGPLAN Notices  Volume 43, Issue 1
        POPL '08
        January 2008
        420 pages
        ISSN:0362-1340
        EISSN:1558-1160
        DOI:10.1145/1328897
        Issue’s Table of Contents
        • cover image ACM Conferences
          POPL '08: Proceedings of the 35th annual ACM SIGPLAN-SIGACT symposium on Principles of programming languages
          January 2008
          448 pages
          ISBN:9781595936899
          DOI:10.1145/1328438

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        • Published: 7 January 2008

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