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Confluence Framework: Proving Confluence with CONFident

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Logic-Based Program Synthesis and Transformation (LOPSTR 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13474))

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Abstract

This paper describes CONFident, a tool which is able to automatically prove and disprove confluence of variants of rewrite systems: term rewriting systems, conditional term rewriting systems (using join, oriented, or semi-equational semantics), and context-sensitive term rewriting systems. We introduce a new proof framework to generate proof trees by combining different techniques for proving confluence (including modular decompositions, checking joinability of (conditional) critical pairs, transformations, etc.). We also use external tools for proving termination and operational termination (mu-term), or feasibility (infChecker) and deducibility (Prover9).

Partially supported by grants PID2021-122830OB-C42 and PID2021-122830OB-C44 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe” and PROMETEO/2019/098.

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Notes

  1. 1.

    String rewriting systems (SRSs), which are rewrite systems where only monadic or constant function symbols are used, are also supported in CONFident.

  2. 2.

    These are called \(\textsf{LH}_{\mu }\)-critical pairs, see [21, Section 4.3] for further motivation.

  3. 3.

    The expression normal form is used in [3, Definition 2]. Although ‘normal form’ is not formally defined anywhere in [3], from the discussion in page 32 preceding [3, Definition 1], it is clear that ‘normal form’ means irreducible term. See [19] for a discussion about normal forms vs. irreducible terms in conditional rewriting.

  4. 4.

    Termination Problems Data Base, see https://www.lri.fr/~marche/tpdb/.

  5. 5.

    Confluence Problems database, see https://cops.uibk.ac.at/.

  6. 6.

    Such CS-TRSs are called \(\mu \)-orthogonal in [21, Definition 35].

  7. 7.

    Originally in [33, Theorem 4.6], this result concerns level-confluence, which implies confluence. We also implement [27, Theorem 7.4.11] as part of \({\textsf{P}_{ HL}}\).

  8. 8.

    Theorem 8 in [17] concerns transformation \({\mathscr {U}}\) and the optimized variant \({{\mathscr {U}}_{\text {opt}}}\) in [4]. Although the proof for \({{\mathscr {U}}_{\text {opt}}}\) is flawed due to a misuse of a result in [4], it holds for \({\mathscr {U}}\), though.

  9. 9.

    Roughly speaking, a DCTRS is weakly left-linear if “variables that occur more than once in the lhs of a conditional rule and the rhs’s of conditions should not occur at all in lhs’s of conditions or the rhs of the conditional rule” [7, Definition 3.17].

  10. 10.

    http://project-coco.uibk.ac.at/problems/.

  11. 11.

    http://project-coco.uibk.ac.at/.

  12. 12.

    The TPDB format, see https://www.lri.fr/~marche/tpdb/format.html, introduced in 2003 for use in the International Termination Competition (TermComp https://termination-portal.org/wiki/Termination_Competition) can also be used in CONFident.

  13. 13.

    http://project-coco.uibk.ac.at/2021/.

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Authors and Affiliations

  1. DSIC & VRAIN, Universitat Politècnica de València, Valencia, Spain

    Miguel Vítores & Salvador Lucas

  2. DLSIIS, Universidad Politécnica de Madrid, Madrid, Spain

    Raúl Gutiérrez

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    Alicia Villanueva

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Gutiérrez, R., Vítores, M., Lucas, S. (2022). Confluence Framework: Proving Confluence with CONFident. In: Villanueva, A. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2022. Lecture Notes in Computer Science, vol 13474. Springer, Cham. https://doi.org/10.1007/978-3-031-16767-6_2

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