Skip to main content Accesibility Help
×
×
Home

Higher-order semantics and extensionality

  • Christoph Benzmüller (a1), Chad E. Brown (a2) and Michael Kohlhase (a3) (a4)
Abstract.

In this paper we re-examine the semantics of classical higher-order logic with the purpose of clarifying the role of extensionality. To reach this goal, we distinguish nine classes of higher-order models with respect to various combinations of Boolean extensionality and three forms of functional extensionality. Furthermore, we develop a methodology of abstract consistency methods (by providing the necessary model existence theorems) needed to analyze completeness of (machine-oriented) higher-order calculi with respect to these model classes.

Copyright
References
Hide All
[1]Andrews, Peter B., Resolution in type theory, this Journal, vol. 36 (1971), no. 3, pp. 414–432.
[2]Andrews, Peter B., General models and extensionality, this Journal, vol. 37 (1972), no. 2, pp. 395–397.
[3]Andrews, Peter B., General models descriptions and choice in type theory, this Journal, vol. 37 (1972), no. 2, pp. 385–394.
[4]Andrews, Peter B., letter to Roger Hindley dated 01 22, 1973.
[5]Andrews, Peter B., On connections and higher order logic, Journal of Automated Reasoning, vol. 5 (1989), pp. 257–291.
[6]Andrews, Peter B., An introduction to mathematical logic and type theory: To truth through proof, second ed., Kluwer Academic Publishers, 2002.
[7]Andrews, Peter B., Bishop, Matthew, and Brown, Chad E., TPS: A theorem proving system for type theory, Proceedings of the 17th international conference on automated deduction (Pittsburgh, USA) (McAllester, David, editor), Lecture Notes in Artifical Intelligence, no. 1831. Springer-Verlag, 2000, pp. 164–169.
[8]Andrews, Peter B., Bishop, Matthew, Issar, Sunil, Nesmith, Dan, Pfenning, Frank, and Xi, Hongwei, TPS: A theorem proving system for classical type theory, Journal of Automated Reasoning, vol. 16 (1996), no. 3, pp. 321–353.
[9]Barendregt, Henk P., The lambda calculus, North-Holland, 1984.
[10]Benzmüller, Christoph, Equality and extensionality in automated higher-order theorem proving, Ph.D. thesis, Saarland University, 1999.
[11]Benzmüller, Christoph, Extensional higher-order paramodulation and RUE-resolution, Proceedings of the 16th international Conference on Automated Deduction (Trento, Italy) (Ganzinger, Harald, editor), Lecture Notes in Artificial Intelligence, vol. 1632, Springer-Verlag, 1999, pp. 399–413.
[12]Benzmüller, Christoph, Brown, Chad E., and Kohlhase, Michael, Semantic techniques for higher-order cut-elimination, manuscript, http://www.ags.uni-sb.de/~chris/papers/R19.pdf, 2002.
[13]Benzmüller, Christoph and Kohlhase, Michael, Extensional higher order resolution, in Kirchner and Kirchner [35], pp. 56–72.
[14]Benzmüller, Christoph and Kohlhase, Michael, LEO—a higher order theorem prover, in Kirchner and Kirchner [35], pp. 139–144.
[15]Benzmüller, Christoph and Kohlhase, Michael, Model existence for higher-order logic, SEKI-Report SR-97-09, Saarland University, 1997.
[16]Bibel, Wolfgang and Schmitt, Peter (editors), Automated deduction—a basis for applications, Kluwer, 1998.
[17]Chierchia, Gennaro and Turner, Raymond, Semantics and property theory, Linguistics and Philosophy, vol. 11 (1988), pp. 261–302.
[18]Church, Alonzo, A formulation of the simple theory of types, this Journal, vol. 5 (1940), pp. 56–68.
[19]de Bruijn, Nicolaas Govert, Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with an application to the Church-Rosser theorem, Indagationes Mathematicae, vol. 34 (1972), no. 5, pp. 381–392.
[20]Demarco, Mary, Intuitionistic semantics for heriditarily harrop logic programming, Ph.D. thesis, Wesleyan University, 1999.
[21]Dowek, Gilles, Hardin, ThéRèse, and Kirchner, Claude, HOL-λσ an intentional first-order expression of higher-order logic, Mathematical Structures in Computer Science, vol. 11 (2001), no. 1, pp. 1–25.
[22]Fitting, Melvin, First-order logic and automated theorem proving, second ed., Graduate Texts in Computer Science, Springer-Verlag, 1996.
[23]Fitting, Melvin, Types, tableaus, and Gödel's God, Kluwer, 2002.
[24]Gödel, Kurt, Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte der Mathematischen Physik, vol. 38 (1931), pp. 173–198, English version in [57].
[25]Gordon, M. J. C. and Melham, T. F., Introduction to HOL—a theorem proving environment for higher order logic, Cambridge University Press, 1993.
[26]Henkin, Leon, Completeness in the theory of types, this Journal, vol. 15 (1950), no. 2, pp. 81–91.
[27]Henkin, Leon, The discovery of my completeness proofs, The Bulletin of Symbolic Logic, vol. 2 (1996), no. 2, pp. 127–158.
[28]Hindley, Roger J. and Seldin, Jonathan P., Introduction to combinators and lambda-calculs, Cambridge University Press, Cambridge, 1986.
[29]Hintikka, K. J. J., Form and content in quantification theory, Acta Philosophica Fennica, vol. 8 (1955), pp. 7–55.
[30]Honsell, Furio and Lenisa, Marina, Coinductive characterizations of applicative structures, Mathematical Structures in Computer Science, vol. 9 (1999), pp. 403–435.
[31]Honsell, Furio and Sannella, Donald, Pre-logical relations, Proceedings of computer science logic (CSL ’99), Lecture Notes in Computer Science, vol. 1683, Springer-Verlag, 1999, pp. 546–561.
[32]Huet, Gérard P., Constrained resolution: A complete method for higher order logic, Ph. D. thesis, Case Western Reserve University, 1972.
[33]Huet, Gérard P., A mechanization of type theory, Proceedings of the 3rd international joint conference on artificial intelligence (Walker, Donald E. and Norton, Lewis, editors), 1973, pp. 139–146.
[34]Jensen, D. C. and Pietrzykowski, Thomasz, A complete mechanization of(ω)-order type theory, Proceedings of the ACM annual conference, vol. 1, 1972, pp. 82–92.
[35]Kirchner, Claude and Kirchner, Hélène (editors), Proceedings of the 15th Conference on Automated Deduction, Lecture Notes in Artificial Intelligence, vol. 1421, Springer-Verlag, 1998.
[36]Kohlhase, Michael, A mechanization of sorted higher-order logic based on the resolution principle, Ph. D. thesis, Saarland University, 1994.
[37]Kohlhase, Michael, Higher-order tableaux, Theorem proving with analytic tableaux and related methods (Baumgartner, Peter, Hähnle, Reiner, and Posegga, Joachim, editors), Lecture Notes in Artificial Intelligence, vol. 918, Springer-Verlag, 1995, pp. 294–309.
[38]Kohlhase, Michael and Scheja, Ortwin, Higher-order multi-valued resolution, Journal of Applied Non-Classical Logics, vol. 9 (1999), no. 4, pp. 155–178.
[39]Lappin, Shalom and Pollard, Carl, Strategies for hyperintensional semantics, manuscript, King's College, London and Ohio State University, 2000.
[40]Lappin, Shalom and Pollard, Carl, A higher-order fine-grained logic for intensional semantics, manuscript, 2002.
[41]Larson, Richard and Segal, Gabriel, Knowledge of meaning, MIT Press, 1995.
[42]Miller, Dale, Proofs in higher-order logic, Ph. D. thesis, Carnegie-Mellon University, 1983.
[43]Miller, Dale, A logic programming language with lambda-abstraction, function variables, and simple unification, Journal of Logic and Computation, vol. 4 (1991), no. 1, pp. 497–536.
[44]Mitchell, John C., Foundations for programming languages, Foundations of Computing, MIT Press, 1996.
[45]Nadathur, Gopalan and Miller, Dale, Higher-order logic programming, Technical Report CS-1994-38, Department of Computer Science, Duke University, 1994.
[46]Nipkow, Tobias, Paulson, Lawrence C., and Wenzel, Markus, Isabelle/HOL—a proof assistant for higher-order logic, Lecture Notes in Computer Science, vol. 2283, Springer-Verlag, 2002.
[47]Robinson, J. Alan and Voronkov, Andrei, Handbook of automated reasoning, MIT Press, 2001.
[48]Schröder, L. and Mossakowski, T., Hascasl: towards integrated specification and development of functional programs, Algebraic methodology and software technology, Lecture Notes in Computer Science, vol. 2422, Springer-Verlag, 2002, pp. 99–116.
[49]Schröder, Lutz, Henkin models for the partial λ-calculus, manuscript, http://www.informatik.uni-bremen.de/~lschrode/hascasl/henkin.ps, 2002.
[50]Schütte, Kurt, Semantical and syntactical properties of simple type theory, this Journal, vol. 25 (1960), pp. 305–326.
[51]Siekmann, Jörg, Benzmüller, Christoph, et al., Proof development with OMEGA, Proceedings of the 18th international conference on automated deduction (Copenhagen, Denmark) (Voronkov, Andrei, editor), Lecture Notes in Artificial Intelligence, vol. 2392, Springer-Verlag, 2002, pp. 144–149.
[52]Smullyan, Raymond M., A unifying principle for quantification theory, Proceedings of the National Academy of Sciences, vol. 49 (1963), pp. 828–832.
[53]Smullyan, Raymond M., First-order logic, Springer-Verlag, 1968.
[54]Takahashi, Moto-o, Cut-elimination in simple type theory with extensionality, Journal of the Mathematical Society of Japan, vol. 19 (1967), pp. 399–410.
[55]Takeuti, Gaisi, Proof theory, North-Holland, 1987.
[56]Tomason, R., A model theory for proposistional attitudes, Linguistics and Philosophy, vol. 4 (1980), pp. 47–70.
[57]van Heijenoort, Jean, From Frege to Gödel: a source book in mathematical logic 1879–1931, 3rd printing, 1997 ed., Source books in the history of the sciences series, Harvard University Press, Cambridge, MA, 1967.
[58]Wolfram, David A., A semantics for λ-PROLOG, Theoretical Computer Science, vol. 136 (1994), no. 1, pp. 277–289.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed