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Realizability in Cyclic Proof: Extracting Ordering Information for Infinite Descent

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Automated Reasoning with Analytic Tableaux and Related Methods (TABLEAUX 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10501))

Abstract

In program verification, measures for proving the termination of programs are typically constructed using (notions of size for) the data manipulated by the program. Such data are often described by means of logical formulas. For example, the cyclic proof technique makes use of semantic approximations of inductively defined predicates to construct Fermat-style infinite descent arguments. However, logical formulas must often incorporate explicit size information (e.g. a list length parameter) in order to support inter-procedural analysis.

In this paper, we show that information relating the sizes of inductively defined data can be automatically extracted from cyclic proofs of logical entailments. We characterise this information in terms of a graph-theoretic condition on proofs, and show that this condition can be encoded as a containment between weighted automata. We also show that under certain conditions this containment falls within known decidability results. Our results can be viewed as a form of realizability for cyclic proof theory.

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Notes

  1. 1.

    Note this is a stronger property than local soundness, which only requires the conclusion to be valid whenever all of the premises are.

  2. 2.

    Excepting certain instances of the (\(=\)L) rule, e.g. \({\text {P}}{x}, {\text {P}}{x} \vdash \varDelta \Rightarrow {\text {P}}{x}, {\text {P}}{y}, x = y \vdash \varDelta \). However, note that one can check whether any given instance of (\(=\)L) satisfies the injectivity property, and exclude proofs containing such instances from consideration.

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Acknowledgements

We extend thanks to Radu Grigore, Carsten Fuhs, and the PPLV group at UCL for useful discussions and invaluable comments. We are grateful to Alexandra Silva for suggesting to investigate weighted automata. This work was supported primarily by EPSRC grant EP/K040049/1, and also by EPSRC grant EP/N028759/1.

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Authors and Affiliations

  1. School of Computing, University of Kent, Canterbury, UK

    Reuben N. S. Rowe

  2. Department of Computer Science, University College London, London, UK

    James Brotherston

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  1. Reuben N. S. Rowe
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Correspondence to Reuben N. S. Rowe .

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Editors and Affiliations

  1. University of Manchester, Manchester, United Kingdom

    Renate A. Schmidt

  2. University of Brasília, Brasília D.F., Brazil

    Cláudia Nalon

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Rowe, R.N.S., Brotherston, J. (2017). Realizability in Cyclic Proof: Extracting Ordering Information for Infinite Descent. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_18

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  • DOI: https://doi.org/10.1007/978-3-319-66902-1_18

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