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Algorithmic Correspondence for Relevance Logics, Bunched Implication Logics, and Relation Algebras via an Implementation of the Algorithm PEARL

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 13027)

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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Notes

  1. 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 (xyz), Rxyz iff \(x \preceq o\) for some \(o \in O\) and \(x \preceq y\).

  2. 2.

    Also called an order type (e.g. [19]) or monotonicity type (e.g. [20]).

  3. 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. 4.

    In [7] these are treated set-theoretically and are called there ‘quasi-inclusions’.

  5. 5.

    This is an optimised version of the post-processing procedure outlined in [7].

  6. 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.

References

  1. Badia, G.: On Sahlqvist formulas in relevant logic. J. Philos. Log. 47(4), 673–691 (2018)

    MathSciNet  CrossRef  Google Scholar 

  2. van Benthem, J.: Modal correspondence theory. Ph.D. thesis, Mathematisch Instituut & Instituut voor Grondslagenonderzoek, University of Amsterdam (1976)

    Google Scholar 

  3. van Benthem, J.: A note on dynamic arrow logic. Technical report LP-92-11, ILLC, University of Amsterdam (1992)

    Google Scholar 

  4. van Benthem, J.: Correspondence theory. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic, vol. 3, 2nd edn., pp. 325–408. Springer, Dordrecht (2001). https://doi.org/10.1007/978-94-017-0454-0_4

  5. Bimbó, K., Dunn, J.M., Maddux, R.D.: Relevance logics and relation algebras. Rev. Symb. Log. 2(1), 102–131 (2009)

    MathSciNet  CrossRef  Google Scholar 

  6. Conradie, W., Ghilardi, S., Palmigiano, A.: Unified correspondence. In: Baltag, A., Smets, S. (eds.) Johan van Benthem on Logic and Information Dynamics. OCL, vol. 5, pp. 933–975. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06025-5_36

    CrossRef  Google Scholar 

  7. Conradie, W., Goranko, V.: Algorithmic correspondence for relevance logics I. The algorithm PEARL. In: Düntsch, I., Mares, E. (eds.) Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs, pp. 163–209. Springer, Heidelberg (2021). https://www2.philosophy.su.se/goranko/papers/PEARL.pdf

  8. Conradie, W., Goranko, V., Jipsen, P.: Algorithmic correspondence for relevance logics, bunched implication logics, and relation algebras: the algorithm PEARL and its implementation (2021). Technical report https://arxiv.org/abs/2108.06603

  9. Conradie, W., Goranko, V., Vakarelov, D.: Algorithmic correspondence and completeness in modal logic, I. The core algorithm SQEMA. Log. Methods Comput. Sci. 2(1:5), 1–26 (2006)

    Google Scholar 

  10. Conradie, W., Palmigiano, A.: Algorithmic correspondence and canonicity for distributive modal logic. Ann. Pure Appl. Logic 163(3), 338–376 (2012)

    MathSciNet  CrossRef  Google Scholar 

  11. Conradie, W., Palmigiano, A.: Algorithmic correspondence and canonicity for non-distributive logics. Ann. Pure Appl. Logic 170(9), 923–974 (2019). https://doi.org/10.1016/j.apal.2019.04.003

    MathSciNet  CrossRef  MATH  Google Scholar 

  12. Dahlqvist, F., Pym, D.: Coalgebraic completeness-via-canonicity for distributive substructural logics. J. Log. Algebr. Methods Program. 93, 1–22 (2017). https://doi.org/10.1016/j.jlamp.2017.07.002

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. de Rijke, M., Venema, Y.: Sahlqvist’s theorem for Boolean algebras with operators with an application to cylindric algebras. Stud. Log. 54(1), 61–78 (1995). https://doi.org/10.1007/BF01058532

    MathSciNet  CrossRef  MATH  Google Scholar 

  14. Doumane, A., Pous, D.: Non axiomatisability of positive relation algebras with constants, via graph homomorphisms. In: Konnov, I., Kovács, L. (eds.) Proceedings of CONCUR 2020. LIPIcs, vol. 171, pp. 29:1–29:16. Schloss Dagstuhl (2020)

    Google Scholar 

  15. Dunn, J.M.: Arrows pointing at arrows: arrow logic, relevance logic, and relation algebras. In: Baltag, A., Smets, S. (eds.) Johan van Benthem on Logic and Information Dynamics. OCL, vol. 5, pp. 881–894. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-06025-5_34

    CrossRef  Google Scholar 

  16. Dunn, J.M., Gehrke, M., Palmigiano, A.: Canonical extensions and relational completeness of some substructural logics. J. Symb. Logic 70, 713–740 (2005)

    MathSciNet  CrossRef  Google Scholar 

  17. Dunn, J., Restall, G.: Relevance logic. In: Gabbay, D., Guenthner, F. (eds.) Handbook of Philosophical Logic. HALO, vol. 6, 2nd edn., pp. 1–128. Springer, Dordrecht (2002). https://doi.org/10.1007/978-94-017-0460-1_1

    CrossRef  Google Scholar 

  18. Galatos, N., Jipsen, P., Kowalski, T., Ono, H.: Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  19. Gehrke, M., Nagahashi, H., Venema, Y.: A Sahlqvist theorem for distributive modal logic. Ann. Pure Appl. Logic 131, 65–102 (2005)

    MathSciNet  CrossRef  Google Scholar 

  20. Gehrke, M., Jónsson, B.: Bounded distributive lattice expansions. Mathematica Scandinavica 94(1), 13–45 (2004)

    Google Scholar 

  21. Goranko, V., Vakarelov, D.: Sahlqvist formulae in hybrid polyadic modal languages. J. Log. Comput. 11(5), 737–754 (2001)

    CrossRef  Google Scholar 

  22. Goranko, V., Vakarelov, D.: Sahlqvist formulas unleashed in polyadic modal languages. In: Wolter, F., Wansing, H., de Rijke, M., Zakharyaschev, M. (eds.) Advances in Modal Logic, vol. 3, pp. 221–240. World Scientific, Singapore (2002)

    CrossRef  Google Scholar 

  23. Goranko, V., Vakarelov, D.: Elementary canonical formulae: extending Sahlqvist’s theorem. Ann. Pure Appl. Logic 141(1–2), 180–217 (2006)

    MathSciNet  CrossRef  Google Scholar 

  24. Hirsch, R., Mikulás, S.: Positive fragments of relevance logic and algebras of binary relations. Rev. Symb. Log. 4(1), 81–105 (2011)

    MathSciNet  CrossRef  Google Scholar 

  25. Jónsson, B.: On the canonicity of Sahlqvist identities. Studis Logica 53(4), 473–491 (1994). https://doi.org/10.1007/BF01057646

    MathSciNet  CrossRef  MATH  Google Scholar 

  26. Kowalski, T.: Relevant logic and relation algebras. In: Galatos, N., Kurz, A., Tsinakis, C. (eds.) TACL 2013. Sixth International Conference on Topology, Algebra and Categories in Logic. EPiC Series in Computing, vol. 25, pp. 125–128 (2014)

    Google Scholar 

  27. Maddux, R.: Some varieties containing relation algebras. Trans. Am. Math. Soc. 272, 501–526 (1982)

    MathSciNet  CrossRef  Google Scholar 

  28. Maddux, R.D.: Relevance logic and the calculus of relations. Rev. Symb. Log. 3(1), 41–70 (2010). https://doi.org/10.1017/S1755020309990293

  29. Pratt, V.R.: Top down operator precedence. In: Fischer, P.C., Ullman, J.D. (eds.) Conference Record of the ACM Symposium on Principles of Programming Languages, Boston, Massachusetts, USA, October 1973, pp. 41–51. ACM Press (1973)

    Google Scholar 

  30. Pym, D.: The Semantics and Proof Theory of the Logic of Bunched Implications. APLS, vol. 26. Springer, Dordrecht (2002). https://doi.org/10.1007/978-94-017-0091-7

    CrossRef  MATH  Google Scholar 

  31. Routley, R., Meyer, R., Plumwood, V., Brady, R.: Relevant Logics and its Rivals (Volume I). Ridgeview, CA (1982)

    Google Scholar 

  32. Sahlqvist, H.: Correspondence and completeness in the first and second-order semantics for modal logic. In: Kanger, S. (ed.) Proceedings of the 3rd Scandinavian Logic Symposium, Uppsala 1973, pp. 110–143. Springer, Amsterdam (1975). https://doi.org/10.1016/S0049-237X(08)70728-6

  33. Sambin, G., Vaccaro, V.: A new proof of Sahlqvist’s theorem on modal definability and completeness. J. Symb. Log. 54(3), 992–999 (1989)

    MathSciNet  CrossRef  Google Scholar 

  34. Seki, T.: A Sahlqvist theorem for relevant modal logics. Stud. Logica. 73(3), 383–411 (2003)

    MathSciNet  CrossRef  Google Scholar 

  35. Suzuki, T.: Canonicity results of substructural and lattice-based logics. Rev. Symb. Log. 4(1), 1–42 (2011). https://doi.org/10.1017/S1755020310000201

    MathSciNet  CrossRef  MATH  Google Scholar 

  36. Suzuki, T.: A Sahlqvist theorem for substructural logic. Rev. Symb. Log. 6(2), 229–253 (2013). https://doi.org/10.1017/S1755020313000026

    MathSciNet  CrossRef  MATH  Google Scholar 

  37. Urquhart, A.: Duality for algebras of relevant logics. Studia Logica 56(1/2), 263–276 (1996). https://doi.org/10.1007/BF00370149

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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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