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Symbolic Reasoning Methods in Rewriting Logic and Maude

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Logic, Language, Information, and Computation (WoLLIC 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10944))

Abstract

Rewriting logic is both a logical framework where many logics can be naturally represented, and a semantic framework where many computational systems and programming languages, including concurrent ones, can be both specified and executed. Maude is a declarative specification and programming language based on rewriting logic. For reasoning about the logics and systems represented in the rewriting logic framework symbolic methods are of great importance. This paper discusses various symbolic methods that address crucial reasoning needs in rewriting logic, how they are supported by Maude and other symbolic engines, and various applications that these methods and engines make possible. Because of the generality of rewriting logic, these methods are widely applicable: they can be used in many areas and can provide useful reasoning components for other reasoning engines.

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Notes

  1. 1.

    For a very general formulation of convergence, including conditional equations E, see [84].

  2. 2.

    In some cases, the algebra of constructors may have the more general form \(T_{\varOmega /E_{\varOmega } \uplus B_{\varOmega }}\). For example, the data elements may be sets, which, together with axioms \(B_{\varOmega }\) of associativity and commutativity (AC) of set union, may also have an idempotency equation in \(E_{\varOmega }\) which is applied modulo AC to put all set expressions in normal form. All I say applies also to this more general case, but tree automata and/or pattern methods may be more complex in the presence of both axioms and equations for constructors.

  3. 3.

    The same remarks as in Footnote 2 apply here: the signature of constructors could more generally be a convergent theory of the form \((\varOmega , E_{\varOmega } \uplus B_{\varOmega })\). Unification modulo \(E_{\varOmega }\uplus B_{\varOmega }\) can still be performed in practice to intersect constrained patterns because in many cases \((\varOmega , E_{\varOmega } \uplus B_{\varOmega })\) has the finite variant property (see Sect. 2.2).

  4. 4.

    To avoid wasteful computations we can further make the binary operator \(\wedge \) frozen in its second argument, which forbids rewriting under that second argument. See [95] for a detailed explanation of frozen arguments and [29] for its use in narrowing.

  5. 5.

    Here “standard” should be taken with a grain of salt, since extra variables with sorts in \(\varSigma _{0}\) are allowed in the rule’s righthand side v. These extra variables will be mapped by \(\theta \) to new, fresh variables.

  6. 6.

    The notation \(\{A\} \mathcal {R} \{B\}\), and the relation to Hoare logic are explained in [125].

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Acknowledgments

I thank the organizers of WoLLIC for kindly giving me the opportunity of presenting these ideas in Bogotá. As the references make clear, these ideas have been developed in joint work with a large number of collaborators and former or present students, and in dialogue with many other colleagues. I cannot mention them all and apologize in advance for this; but I would like to mention and cordially thank in particular: María Alpuente, Kyungmin Bae, Andrew Cholewa, Angel Cuenca-Ortega, Francisco Durán, Steven Eker, Santiago Escobar, Raúl Gutiérrez, Joseph Hendrix, Dorel Lucanu, Salvador Lucas, Narciso Martí-Oliet, Catherine Meadows, César A. Muñoz, Hitoshi Ohsaki, Camilo Rocha, Grigore Rosu, Vlad Rusu, Sonia Santiago, Ralf Sasse, Andrei Stefanescu, Carolyn Talcott, Prasanna Thati, and Fan Yang. 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, USA

    José Meseguer

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  1. José Meseguer
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  1. Department of Mathematics, Indiana University, Bloomington, Indiana, USA

    Lawrence S. Moss

  2. Centro de Informática, Univ Federal de Pernambuco (UFPE), Recife, Pernambuco, Brazil

    Ruy de Queiroz

  3. Universidad de los Andes, Bogotá, Colombia

    Maricarmen Martinez

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Meseguer, J. (2018). Symbolic Reasoning Methods in Rewriting Logic and Maude. In: Moss, L., de Queiroz, R., Martinez, M. (eds) Logic, Language, Information, and Computation. WoLLIC 2018. Lecture Notes in Computer Science(), vol 10944. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-57669-4_2

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