Abstract
In first-order logic, forward search using a complete strategy such as the inverse method can get stuck deriving larger and larger consequence sets when the goal query is unprovable. This is the case even in trivial theories where backward search strategies such as tableaux methods will fail finitely. We propose a general mechanism for bounding the consequence sets by means of finite approximations of infinite types. If the inverse method also implements forward subsumption and globalization, then the search space under this approximation is finite. We therefore obtain a type-directed iterative refinement algorithm for disproving queries.
The method has been implemented for intuitionistic first-order logic, and we discuss its performance on a variety of problems.
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References
Bachmair, L., Ganzinger, H.: Rewrite-based equational theorem proving with selection and simplification. J. of Logic and Computation 3(4) (1994)
Bachmair, L., Ganzinger, H.: Resolution theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, chapter 2, pp. 19–99. Elsevier Science, New York (2001)
Bonacina, M.P., Lynch, C., de Moura, L.M.: On deciding satisfiability by theorem proving with speculative inferences. J. of Automated Reasoning 47(2), 161–189 (2011)
Chaudhuri, K.: The Focused Inverse Method for Linear Logic. PhD thesis, Carnegie Mellon University, Technical report CMU-CS-06-162, December 2006
Chaudhuri, K.: Magically constraining the inverse method using dynamic polarity assignment. In: Fermüller, C.G., Voronkov, A. (eds.) LPAR-17. LNCS, vol. 6397, pp. 202–216. Springer, Heidelberg (2010)
Chaudhuri, K., Pfenning, F., Price, G.: A logical characterization of forward and backward chaining in the inverse method. J. of Automated Reasoning 40(2–3), 133–177 (2008)
Chihani, Z., Miller, D., Renaud, F.: Foundational proof certificates in first-order logic. In: Bonacina, M.P. (ed.) CADE 2013. LNCS, vol. 7898, pp. 162–177. Springer, Heidelberg (2013)
Claessen, K., Sorensson, N.: New techniques that improve MACE-style finite model finding. In: Baumgartner, P., Fermueller, C. (eds.) Proceedings of the CADE-19 Workshop: Model Computation - Principles, Algorithms, Applications, Miami, USA (2003)
Degtyarev, A., Voronkov, A.: The inverse method. In: Robinson, J.A., Voronkov, A. (eds.) Handbook of Automated Reasoning (in 2 volumes), pp. 179–272. Elsevier and MIT Press (2001)
Liang, C., Miller, D.: Focusing and polarization in linear, intuitionistic, and classical logics. Theoretical Computer Science 410(46), 4747–4768 (2009)
Lynch, C.: Unsound theorem proving. In: Marcinkowski, J., Tarlecki, A. (eds.) CSL 2004. LNCS, vol. 3210, pp. 473–487. Springer, Heidelberg (2004)
McCune, W.: Mace4 reference manual and guide. Technical Report cs.SC/0310055 (2003)
McLaughlin, S., Pfenning, F.: Imogen: Focusing the polarized focused inverse method for intuitionistic propositional logic. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 174–181. Springer, Heidelberg (2008)
McLaughlin, S., Pfenning, F.: Efficient intuitionistic theorem proving with the polarized inverse method. In: Schmidt, R.A. (ed.) CADE 2009. LNCS, vol. 5663, pp. 230–244. Springer, Heidelberg (2009)
Raths, T., Otten, J., Kreitz, C.: The ILTP problem library for intuitionistic logic. Journal of Automated Reasoning 38(1), 261–271 (2007)
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Brock-Nannestad, T., Chaudhuri, K. (2015). Disproving Using the Inverse Method by Iterative Refinement of Finite Approximations. In: De Nivelle, H. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2015. Lecture Notes in Computer Science(), vol 9323. Springer, Cham. https://doi.org/10.1007/978-3-319-24312-2_11
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DOI: https://doi.org/10.1007/978-3-319-24312-2_11
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