Abstract
The non-deterministic algorithmic procedure PEARL (acronym for ‘Propositional variables Elimination Algorithm for Relevance Logic’) has been recently developed for computing first-order equivalents of formulas of the language of relevance logics \(\mathcal {L}_R\) in terms of the standard Routley-Meyer relational semantics. It succeeds on a large class of axioms of relevance logics, including all so called inductive formulas. In the present work we re-interpret PEARL from an algebraic perspective, with its rewrite rules seen as manipulating quasi-inequalities interpreted over Urquhart’s relevant algebras, and report on its recent Python implementation. We also show that all formulae on which PEARL succeeds are canonical, i.e., preserved under canonical extensions of relevant algebras. This generalizes the “canonicity via correspondence” result in [37]. We also indicate that with minor modifications PEARL can also be applied to bunched implication algebras and relation algebras.
\(^1\)Visiting professorship affiliation of the 2nd author.
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- 1.
The definition of Routley-Meyer frames takes the relation R and subset O as primary and defines the pre-order \(\preceq \) in terms of them. This does not restrict the pre-orders that can occur within Routley-Meyer frames. Indeed, given an upward closed subset \(O \subseteq W\) and a pre-order \(\preceq \) on W one can define a respective ternary relation \(R \subseteq W^3\) by specifying that, for all triples (x, y, z), Rxyz iff \(x \preceq o\) for some \(o \in O\) and \(x \preceq y\).
- 2.
- 3.
These rules can be seen as instantiations of the rules of the general-purpose algorithm \(\mathsf {ALBA}\) [11] in the context of perfect relevant algebras. However, the fact that the latter are distributive lattice expansions allows us to present simpler formulations of these rules closer to those in [10] and, to some extent, [9]. The approximation rules presented in [11] allow for the extraction of subformulas deep from within the consequents of quasi-inequalities, subject to certain conditions, rather than the connective-by-connective style of our presentation. Although the former style of rule is also sound in the present setting, we opted for the latter as we believe it is simpler to present since the formulation requires significantly fewer auxiliary notions.
- 4.
In [7] these are treated set-theoretically and are called there ‘quasi-inclusions’.
- 5.
This is an optimised version of the post-processing procedure outlined in [7].
- 6.
While this equational basis for relation algebras appears to be quite long, it can be shown that axioms 3–7 are redundant. Hence, it is comparable in length to the original axiomatization of relation algebras.
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Conradie, W., Goranko, V., Jipsen, P. (2021). Algorithmic Correspondence for Relevance Logics, Bunched Implication Logics, and Relation Algebras via an Implementation of the Algorithm PEARL. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_8
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