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Generalized Rewrite Theories and Coherence Completion

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Rewriting Logic and Its Applications (WRLA 2018)

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Abstract

A new notion of generalized rewrite theory suitable for symbolic reasoning and generalizing the standard notion in [3] is motivated and defined. Also, new requirements for symbolic executability of generalized rewrite theories that extend those in [8] for standard rewrite theories, including a generalized notion of coherence, are given. Finally, symbolic executability, including coherence, is both ensured and made available for a wide class of such theories by automatable theory transformations.

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Notes

  1. 1.

    If \(B = B_{0} \uplus U\), with \(B_{0}\) associativity and/or commutativity axioms, and U identity axioms, the B-preregularity notion can be broadened by requiring only that: (i) \(\varSigma \) is \(B_{0}\)-preregular in the standard sense, so that \( ls (u\rho )= ls (v\rho )\) for all \(u=v \in B_{0}\) and substitutions \(\rho \); and (ii) the axioms U oriented as rules \(\vec {U}\) are sort-decreasing in the sense that \(u=v \in U \Rightarrow ls (u\rho ) \geqslant ls (v\rho )\) for each \(\rho \). Maude automatically checks B-preregularity of an OS signature \(\varSigma \) in this broader sense [4].

  2. 2.

    See [24] for the more general definition of both convergence and the relation \(\rightarrow _{\vec {E},B}\) when \(\varSigma \) is B-preregular in the broader sense of Footnote 1.

  3. 3.

    This is supported in Maude by the frozen operator attribute, which forbids rewrites below the specified argument positions. For example, when giving a rewriting semantics to a CCS-like process calculus, the process concatenation operator \(\_{\cdot }\_\), appearing in process expressions like \(a \cdot P\), will typically be frozen in its second argument.

  4. 4.

    By definition this means that there is no function symbol f and position q such that: (i) \(p=q \cdot i \cdot q'\), (ii) \(u'|_{q}=f(u_{1},\ldots ,u_{n})\), and (iii) \(i \in \phi (f_{[s]}^{[s_{1}]\ldots [s_{n}]})\). Intuitively this means that the frozenness restrictions \(\phi \) do not block rewriting at position p in \(u'\).

  5. 5.

    Admittedly, it is possible to allow more general rules with additional “rewrite conditions” of the form \(l \rightarrow r \; if \; \varphi \wedge \bigwedge _{i=1\ldots n} u_{i} \rightarrow v_{i}\) in a generalized rewrite theory. Then, generalized rewrite theories would specialize to standard rewrite theories whose rules also allow rewrite conditions. I leave this further generalization as future work.

  6. 6.

    Recall that the strongly deterministic and convergent rules \(\vec {E}\) may be conditional. We are therefore using Definition 3 in its straightforward generalization to the conditional case.

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Acknowledgments

I thank the referees for their constructive criticism and valuable suggestions to improve the paper. This work has been partially supported by NRL under contract number N00173-17-1-G002.

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  1. Department of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    José Meseguer

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Correspondence to José Meseguer .

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  1. Inria, Lille, France

    Vlad Rusu

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Meseguer, J. (2018). Generalized Rewrite Theories and Coherence Completion. In: Rusu, V. (eds) Rewriting Logic and Its Applications. WRLA 2018. Lecture Notes in Computer Science(), vol 11152. Springer, Cham. https://doi.org/10.1007/978-3-319-99840-4_10

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