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.
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.
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.
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.
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.
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.
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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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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