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Complexity Analysis for Term Rewriting by Integer Transition Systems

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Frontiers of Combining Systems (FroCoS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10483))

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Abstract

We present a new method to infer upper bounds on the innermost runtime complexity of term rewrite systems (TRSs), which benefits from recent advances on complexity analysis of integer transition systems (ITSs). To this end, we develop a transformation from TRSs to a generalized notion of ITSs with (possibly non-tail) recursion. To analyze their complexity, we introduce a modular technique which allows us to use existing tools for standard ITSs in order to infer complexity bounds for our generalized ITSs. The key idea of our technique is a summarization method that allows us to analyze components of the transition system independently. We implemented our contributions in the tool AProVE, and our experiments show that one can now infer bounds for significantly more TRSs than with previous state-of-the-art tools for term rewriting.

Supported by DFG grant GI 274/6-1 and the Air Force Research Laboratory (AFRL).

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Notes

  1. 1.

    In this way, the complexity of \(\mathsf{gt}\) is 0, whereas comparisons have complexity 1 with the slightly more complicated encoding from [8]. Since this difference does not affect the asymptotic complexity of Example 2, we use the simpler encoding for the sake of readability.

  2. 2.

    Termination Problems Data Base, the collection of examples used at the annual Termination and Complexity Competition, see http://termination-portal.org.

  3. 3.

    See [29] for an adaption of an amortized analysis as in [27] to term rewriting. However, [29] is not automated, and it is restricted to ground rewriting with orthogonal rules.

  4. 4.

    See http://termination-portal.org/wiki/Termination_Competition/.

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Acknowledgments

We thank A. Flores-Montoya for his help with CoFloCo and the anonymous reviewers for their suggestions and comments.

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Authors and Affiliations

  1. LuFG Informatik 2, RWTH Aachen University, Aachen, Germany

    Matthias Naaf, Florian Frohn & Jürgen Giesl

  2. Microsoft Research, Cambridge, UK

    Marc Brockschmidt

  3. Birkbeck, University of London, London, UK

    Carsten Fuhs

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  1. Matthias Naaf
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  5. Jürgen Giesl
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Correspondence to Jürgen Giesl .

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Editors and Affiliations

  1. University of Liverpool, Liverpool, United Kingdom

    Clare Dixon

  2. University of Sao Paulo, Sao Paulo, Brazil

    Marcelo Finger

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Naaf, M., Frohn, F., Brockschmidt, M., Fuhs, C., Giesl, J. (2017). Complexity Analysis for Term Rewriting by Integer Transition Systems. In: Dixon, C., Finger, M. (eds) Frontiers of Combining Systems. FroCoS 2017. Lecture Notes in Computer Science(), vol 10483. Springer, Cham. https://doi.org/10.1007/978-3-319-66167-4_8

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