Abstract
The inverse method, due to Maslov, is a forward theorem proving method for cut-free sequent calculi that relies on the subformula property. The Logic of Bunched Implications (BI), due to Pym and O’Hearn, is a logic which freely combines the familiar connectives of intuitionistic logic with multiplicative linear conjunction and its adjoint implication. We present the first formulation of an inverse method for propositional BI without units. We adapt the sequent calculus for BI into a forward calculus. The soundness and completeness of the calculus are proved, and a canonical form for bunches is given.
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References
Andreoli, J.-M.: Logic programming with focusing proofs in linear logic. Journal of Logic and Computation 2(3), 197–347 (1992)
Armelín, P.: Logic programming with bunched logic. PhD thesis, University of London (2002)
Armelín, P.A., Pym, D.J.: Bunched logic programming. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 289–304. Springer, Heidelberg (2001)
Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 1, ch. 2, pp. 19–100. North Holland, Amsterdam (2001)
Beal, F., Méry, D., Galmiche, D.: Bill: A theorem prover for propositional bi logic
Chaudhuri, K., Pfenning, F.: Resource management for the inverse method in linear logic. Carnegie Mellon University, Unpublished Maniscript (January 2003)
Degtyarev, A., Voronkov, A.: The inverse method. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. 1, pp. 179–272. Elsevier Science and MIT Press (2001)
Galmiche, D., Méry, D., Pym, D.J.: Resource tableaux. In: Bradfield, J.C. (ed.) CSL 2002 and EACSL 2002. LNCS, vol. 2471, pp. 183–199. Springer, Heidelberg (2002)
Galmiche, D., Méry, D.: Semantic labelled tableaux for propositional bi (without bottom). Journal of Logic and Computation 13(5) (October 2003)
Girard, J.-Y.: Linear logic. Theoretical Computer Science 50, 1–102 (1987)
Harland, J., Pym, D.: Resource-distribution via boolean constraints. ACM Trans. Comput. Logic 4(1), 56–90 (2003)
Maslov, S.: The inverse method of establishing deducibility in classical predicate calculus. Soviet Mathematical Doklady 5, 1420–1424 (1964)
O’Hearn, P.W., Pym, D.J.: The logic of bunched implications. Bulletin of Symbolic Logic 5(2), 215–244 (1999)
Pfenning, F.: The inverse method. Carnegie Mellon University, Lecture Notes, Ch. 5 (February 2004)
Pym, D.J.: The Semantics and Proof Theory of the Logic of the Logic of Bunched Implications. Applied Logic Series, vol. 26. Kluwer Academic Publishers, Dordrecht (2002), Errata and Remarks maintained at: http://www.cs.bath.ac.uk/pym/BI-monograph-errata.pdf
Reynolds, J.C.: Separation logic: A logic for shared mutable data structures. In: Proceedings of the 17th Annual IEEE Symposium on Logic in Computer Science, pp. 55–74. IEEE Computer Society, Los Alamitos (2002)
Tammet, T.: Proof strategies in linear logic. Journal of Automated Reasoning 12(3), 273–304 (1994)
Tammet, T.: A resolution theorem prover for intuitionistic logic. In: McRobbie, M.A., Slaney, J.K. (eds.) CADE 1996. LNCS, vol. 1104, pp. 2–16. Springer, Heidelberg (1996)
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Donnelly, K., Gibson, T., Krishnaswami, N., Magill, S., Park, S. (2005). The Inverse Method for the Logic of Bunched Implications. In: Baader, F., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2005. Lecture Notes in Computer Science(), vol 3452. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-32275-7_31
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DOI: https://doi.org/10.1007/978-3-540-32275-7_31
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