Article

Combining logics in simple type theory

Author:

Christoph BenzmüllerAuthors Info & Claims

CLIMA'10: Proceedings of the 11th international conference on Computational logic in multi-agent systems

Pages 33 - 48

Published: 16 August 2010 Publication History

Abstract

Simple type theory is suited as framework for combining classical and non-classical logics. This claim is based on the observation that various prominent logics, including (quantified) multimodal logics and intuitionistic logics, can be elegantly embedded in simple type theory. Furthermore, simple type theory is sufficiently expressive to model combinations of embedded logics and it has a well understood semantics. Off-the-shelf reasoning systems for simple type theory exist that can be uniformly employed for reasoning within and about combinations of logics. Combinations of modal logics and other logics are particularly relevant for multi-agent systems.

References

[1]

Andrews, P.B.: General models and extensionality. Journal of Symbolic Logic 37, 395-397 (1972).

[2]

Andrews, P.B.: General models, descriptions, and choice in type theory. Journal of Symbolic Logic 37, 385-394 (1972).

[3]

Andrews, P.B.: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof, 2nd edn. Kluwer Academic Publishers, Dordrecht (2002).

Digital Library

[4]

Andrews, P.B., Brown, C.E.: TPS: A hybrid automatic-interactive system for developing proofs. Journal of Applied Logic 4(4), 367-395 (2006).

[5]

Backes, J., Brown, C.E.: Analytic tableaux for higher-order logic with choice. In: Giesl, J., Haehnle, R. (eds.) IJCAR 2010 - 5th International Joint Conference on Automated Reasoning, Edinburgh, UK. LNCS (LNAI). Springer, Heidelberg (2010).

Digital Library

[6]

Baldoni, M.: Normal Multimodal Logics: Automatic Deduction and Logic Programming Extension. PhD thesis, Universita degli studi di Torino (1998).

[7]

Benzmüller, C.: Automating access control logic in simple type theory with LEO-II. In: Gritzalis, D., López, J. (eds.) Proceedings of 24th IFIP TC 11 International Information Security Conference, SEC 2009, Emerging Challenges for Security, Privacy and Trust, Pafos, Cyprus, May 18-20. IFIP, vol. 297, pp. 387-398. Springer, Heidelberg (2009).

[8]

Benzmüller, C., Brown, C.E., Kohlhase, M.: Higher order semantics and extensionality. Journal of Symbolic Logic 69, 1027-1088 (2004).

[9]

Benzmüller, C., Paulson, L.: Exploring Properties of Normal Multimodal Logics in Simple Type Theory with LEO-II. In: Festschrift in Honor of Peter B. Andrews on His 70th Birthday. Studies in Logic, Mathematical Logic and Foundations. College Publications (2008).

[10]

Benzmüller, C., Paulson, L.C.: Quantified Multimodal Logics in Simple Type Theory. SEKI Report SR-2009-02, SEKI Publications, DFKI Bremen GmbH, Safe and Secure Cognitive Systems, Cartesium, Enrique Schmidt Str. 5, D-28359 Bremen, Germany (2009), ISSN 1437-4447, http://arxiv.org/abs/0905.2435

[11]

Benzmüller, C., Paulson, L.C.: Multimodal and intuitionistic logics in simple type theory. The Logic Journal of the IGPL (2010).

[12]

Benzmüller, C., Paulson, L.C., Theiss, F., Fietzke, A.: LEO-II -- A Cooperative Automatic Theorem Prover for Higher-Order Logic. In: Baumgartner, P., Armando, A., Gilles, D. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 162-170. Springer, Heidelberg (2008).

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[13]

Benzmüller, C., Pease, A.: Progress in automating higher-order ontology reasoning. In: Konev, B., Schmidt, R., Schulz, S. (eds.) Workshop on Practical Aspects of Automated Reasoning (PAAR 2010), Edinburgh, UK, July 14. CEUR Workshop Proceedings (2010).

[14]

Benzmüller, C., Pease, A.: Reasoning with embedded formulas and modalities in SUMO. In: Bundy, A., Lehmann, J., Qi, G., Varzinczak, I.J. (eds.) The ECAI 2010 Workshop on Automated Reasoning about Context and Ontology Evolution (ARCOE 2010), Lisbon, Portugal, August 16-17 (2010).

[15]

Benzmüller, C., Rabe, F., Sutcliffe, G.: THF0 -- The Core TPTP Language for Classical Higher-Order Logic. In: Baumgartner, P., Armando, A., Gilles, D. (eds.) IJCAR 2008. LNCS (LNAI), vol. 5195, pp. 491-506. Springer, Heidelberg (2008).

Digital Library

[16]

Blackburn, P., Marx, M.: Tableaux for quantified hybrid logic. In: Egly, U., Fermüller, C.G. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 38-52. Springer, Heidelberg (2002).

Digital Library

[17]

Braüner, T.: Natural deduction for first-order hybrid logic. Journal of Logic, Language and Information 14(2), 173-198 (2005).

Digital Library

[18]

Church, A.: A formulation of the simple theory of types. Journal of Symbolic Logic 5, 56-68 (1940).

[19]

Fitting, M.: Interpolation for first order S5. Journal of Symbolic Logic 67(2), 621- 634 (2002).

[20]

Gabbay, D., Kurucz, A., Wolter, F., Zakharyaschev, M.: Many-dimensional modal logics: theory and applications. Studies in Logic, vol. 148. Elsevier Science, Amsterdam (2003).

[21]

Garg, D., Abadi, M.: A Modal Deconstruction of Access Control Logics. In: Amadio, R.M. (ed.) FOSSACS 2008. LNCS, vol. 4962, pp. 216-230. Springer, Heidelberg (2008).

Digital Library

[22]

Garson, J.: Modal logic. In: Zalta, E.N. (ed.) The Stanford Encyclopedia of Philosophy (Winter 2009).

[23]

Gödel, K.: Eine interpretation des intuitionistischen aussagenkalküls. Ergebnisse eines Mathematischen Kolloquiums 8, 39-40 (1933); Also published in Gödel {24}, pp. 296-302.

[24]

Gödel, K.: Collected Works, vol. I. Oxford University Press, Oxford (1986).

[25]

Goldblatt, R.: Logics of Time and Computation. Center for the Study of Language and Information - Lecture Notes, vol. 7. Leland Stanford Junior University (1992).

Digital Library

[26]

Henkin, L.: Completeness in the theory of types. Journal of Symbolic Logic 15, 81-91 (1950).

[27]

Kaminski, M., Smolka, G.: Terminating tableau systems for hybrid logic with difference and converse. Journal of Logic, Language and Information 18(4), 437-464 (2009).

Digital Library

[28]

McKinsey, J.C.C., Tarski, A.: Some theorems about the sentential calculi of lewis and heyting. Journal of Symbolic Logic 13, 1-15 (1948).

[29]

Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL - A Proof Assistant for Higher-Order Logic. LNCS, vol. 2283. Springer, Heidelberg (2002).

Digital Library

[30]

Randell, D.A., Cui, Z., Cohn, A.G.: A spatial logic based on regions and connection. In: Proceedings 3rd International Conference on Knowledge Representation and Reasoning, pp. 165-176 (1992).

Digital Library

[31]

Segerberg, K.: Two-dimensional modal logic. Journal of Philosophical Logic 2(1), 77-96 (1973).

[32]

Sutcliffe, G.: TPTP, TSTP, CASC, etc. In: Diekert, V., Volkov, M.V., Voronkov, A. (eds.) CSR 2007. LNCS, vol. 4649, pp. 6-22. Springer, Heidelberg (2007).

Digital Library

[33]

Sutcliffe, G.: The TPTP problem library and associated infrastructure. J. Autom. Reasoning 43(4), 337-362 (2009).

Digital Library

[34]

Sutcliffe, G., Benzmüller, C.: Automated reasoning in higher-order logic using the TPTP THF infrastructure. Journal of Formalized Reasoning 3(1), 1-27 (2010).

[35]

Sutcliffe, G., Benzmüller, C., Brown, C., Theiss, F.: Progress in the development of automated theorem proving for higher-order logic. In: Schmidt, R.A. (ed.) Automated Deduction - CADE-22. LNCS, vol. 5663, pp. 116-130. Springer, Heidelberg (2009).

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Cited By

Benzmüller C(2010)Verifying the modal logic cube is an easy taskVerification, induction termination analysis10.5555/1986659.1986666(117-128)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.5555/1986659.1986666
Benzmüller C(2010)Verifying the modal logic cube is an easy taskVerification, induction termination analysis10.5555/1980681.1980688(117-128)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.5555/1980681.1980688

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Published In

cover image Guide Proceedings

CLIMA'10: Proceedings of the 11th international conference on Computational logic in multi-agent systems

August 2010

256 pages

ISBN:3642149766

Editors:
Jürgen Dix
Clausthal University of Technology, Clausthal-Zellerfeld, Germany
,
João Leite
CENTRIA, Universidade Nova de Lisboa, Caparica, Portugal
,
Guido Governatori
NICTA, St Lucia, QLD, Australia
,
Wojtek Jamroga
University of Luxembourg, Luxembourg, Luxembourg

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 16 August 2010

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Cited By

Benzmüller C(2010)Verifying the modal logic cube is an easy taskVerification, induction termination analysis10.5555/1986659.1986666(117-128)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.5555/1986659.1986666
Benzmüller C(2010)Verifying the modal logic cube is an easy taskVerification, induction termination analysis10.5555/1980681.1980688(117-128)Online publication date: 1-Jan-2010
https://dl.acm.org/doi/10.5555/1980681.1980688

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