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Optimizing Maude Programs via Program Specialization

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Analysis, Verification and Transformation for Declarative Programming and Intelligent Systems

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13160))

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Abstract

We develop an automated specialization framework for rewrite theories that model concurrent systems. A rewrite theory \(\mathscr {R}=(\Sigma ,E\uplus B,R)\) consists of two main components: an order-sorted equational theory \(\mathscr {E}=(\Sigma ,E\uplus B)\) that defines the system states as terms of an algebraic data type and a term rewriting system R that models the concurrent evolution of the system as state transitions. Our main idea is to partially evaluate the underlying equational theory \(\mathscr {E}\) to the specific calls required by the rewrite rules of R in order to make the system computations more efficient. The specialization transformation relies on folding variant narrowing, which is the symbolic operational engine of Maude’s equational theories. We provide three instances of our specialization scheme that support distinct classes of theories that are relevant for many applications. The effectiveness of our method is finally demonstrated in some specialization examples.

This work has been partially supported by the EC H2020-EU grant agreement No. 952215 (TAILOR), grants RTI2018-094403-B-C32 and PID2021-122830OB-C42 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, by Generalitat Valenciana under grant PROMETEO/2019/098, and by the Department Strategic Plan (PSD) of the University of Udine—Interdepartmental Project on Artificial Intelligence (2021-25).

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Notes

  1. 1.

    For example, assuming a commutative binary operator \(*\), the term \(s(0)*0\) matches within the term \(X*s(Y)\) modulo the commutativity of symbol \(*\) with matching substitution \(\{X/0,Y/0\}\).

  2. 2.

    A variant [22] of a term t in the theory \(\mathcal{E}\) is the canonical (i.e., irreducible) form of \(t \sigma \) in \(\mathcal{E}\) for a given substitution \(\sigma \); in symbols, it is represented as the pair \((t\sigma \!\!\downarrow _{\vec {E},B},\sigma )\).

  3. 3.

    In [40], natural numbers are encoded by using two constants 0 and 1 and an ACU operator + so that a natural number is either the constant 0 or a finite sequence 1 + 1 ... + 1.

  4. 4.

    Besides the topmost assumption for \(\mathscr {R}\), we also consider the classical executability restriction that the set R of rules is coherent with E modulo B (intuitively, this ensures that a rewrite step with R can always be postponed in favor of deterministically rewriting with E modulo B).

  5. 5.

    In an order-sorted setting, multiple equations are actually used to cover any possible sort in \(\mathscr {R}\).

  6. 6.

    For example, by using \(\epsilon \), the term \(s(0)*0\,=\!?\!=\,U*s(V)\) FV-narrows to tt (modulo commutativity of \(*\)), and the computed narrowing substitution does coincide with the unifier modulo commutativity of the two argument terms, i.e., \(\{U\mapsto 0,V\mapsto 0\}\).

  7. 7.

    For simplicity, we assume that Q is normalized w.r.t. the equational theory \(\mathscr {E}\). If this were not the case, for each \(t\in Q\) that is not in canonical form such that \(t\!\downarrow _{\vec {E},B}=C(\overline{t_{i}})\), where C() is the (possibly empty) constructor context of \(t\!\downarrow _{\vec {E},B}\) and \(\overline{t_{i}}\) are the maximal calls in \(t\!\downarrow _{\vec {E},B}\), we would replace t in Q with the normalized terms \(\overline{t_{i}}\) and add a suitable “bridge” equation \(t =C(\overline{t_{i}})\) to the resulting specialization.

  8. 8.

    The case when \(\mathscr {E}\) satisfies SC but not the FVP is not considered because there is no technique to compute the finite set of most general constructor variants in this case, which is a matter for future research.

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Author information

Authors and Affiliations

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

    María Alpuente, Santiago Escobar & Julia Sapiña

  2. DMIF, Università degli Studi di Udine, Udine, Italy

    Demis Ballis

  3. University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Jose Meseguer

Authors
  1. María Alpuente
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  2. Demis Ballis
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  3. Santiago Escobar
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  4. Jose Meseguer
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  5. Julia Sapiña
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Corresponding author

Correspondence to Julia Sapiña .

Editor information

Editors and Affiliations

  1. IMDEA Software Institute, Pozuelo de Alarcón, Madrid, Spain

    Pedro Lopez-Garcia

  2. Roskilde University, Roskilde, Denmark

    John P. Gallagher

  3. Università di Verona, Verona, Italy

    Roberto Giacobazzi

Appendices

A Full Specification of the Bank Account System

figure l

B Specialization of the Bank Account System \(\mathscr {R}_b\)

figure m

C Specialization of the Bank Account System \(\mathscr {R}_b\) with Compression

figure n

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Alpuente, M., Ballis, D., Escobar, S., Meseguer, J., Sapiña, J. (2023). Optimizing Maude Programs via Program Specialization. In: Lopez-Garcia, P., Gallagher, J.P., Giacobazzi, R. (eds) Analysis, Verification and Transformation for Declarative Programming and Intelligent Systems. Lecture Notes in Computer Science, vol 13160. Springer, Cham. https://doi.org/10.1007/978-3-031-31476-6_2

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