Skip to main content

Unified Classical Logic Completeness

A Coinductive Pearl

  • Conference paper
Automated Reasoning (IJCAR 2014)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 8562))

Included in the following conference series:

Abstract

Codatatypes are absent from many programming and specification languages. We make a case for their importance by revisiting a classical result: the completeness theorem for first-order logic established through a Gentzen system. The core of the proof establishes an abstract property of possibly infinite derivation trees, independently of the concrete syntax or inference rules. This separation of concerns simplifies the presentation. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems as well as various flavors of first-order logic. The corresponding Isabelle/HOL formalization demonstrates the recently introduced support for codatatypes and the Haskell code generator.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Bell, J.L., Machover, M.: A Course in Mathematical Logic. North-Holland (1977)

    Google Scholar 

  2. Berghofer, S.: First-order logic according to Fitting. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs (2007), http://afp.sf.net/entries/FOL-Fitting.shtml

  3. Bertot, Y.: Filters on coinductive streams, an application to Eratosthenes’ sieve. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 102–115. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  4. Blanchette, J.C., Popescu, A.: Mechanizing the metatheory of Sledgehammer. In: Fontaine, P., Ringeissen, C., Schmidt, R.A. (eds.) FroCoS 2013. LNCS, vol. 8152, pp. 245–260. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  5. Blanchette, J.C., Hölzl, J., Lochbihler, A., Panny, L., Popescu, A., Traytel, D.: Truly modular (co)datatypes for Isabelle/HOL. In: Klein, G., Gamboa, R. (eds.) ITP 2014. LNCS, Springer (2014)

    Google Scholar 

  6. Blanchette, J.C., Popescu, A., Traytel, D.: Abstract completeness. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs (2014), http://afp.sf.net/entries/Abstract_Completeness.shtml

  7. Blanchette, J.C., Popescu, A., Traytel, D.: Formal development associated with this paper (2014), http://www21.in.tum.de/~traytel/compl_devel.zip

  8. Ciaffaglione, A., Gianantonio, P.D.: A certified, corecursive implementation of exact real numbers. Theor. Comput. Sci. 351(1), 39–51 (2006)

    Article  MATH  Google Scholar 

  9. Fitting, M.: First-Order Logic and Automated Theorem Proving, 2nd edn. Graduate Texts in Computer Science. Springer (1996)

    Google Scholar 

  10. Francez, N.: Fairness. Texts and Monographs in Computer Science, Springer (1986)

    Google Scholar 

  11. Gallier, J.H.: Logic for Computer Science: Foundations of Automatic Theorem Proving. Computer Science and Technology. Harper & Row (1986)

    Google Scholar 

  12. Gödel, K.: Über die Vollständigkeit des Logikkalküls. Ph.D. thesis, Universität Wien (1929)

    Google Scholar 

  13. Gordon, M.J.C., Melham, T.F. (eds.): Introduction to HOL: A Theorem Proving Environment for Higher Order Logic. Cambridge University Press (1993)

    Google Scholar 

  14. Haftmann, F., Nipkow, T.: Code generation via higher-order rewrite systems. In: Blume, M., Kobayashi, N., Vidal, G. (eds.) FLOPS 2010. LNCS, vol. 6009, pp. 103–117. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  15. Hähnle, R.: Tableaux and related methods. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, pp. 100–178. Elsevier (2001)

    Google Scholar 

  16. Harrison, J.: Formalizing basic first order model theory. In: Grundy, J., Newey, M. (eds.) TPHOLs 1998. LNCS, vol. 1479, pp. 153–170. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  17. Ilik, D.: Constructive Completeness Proofs and Delimited Control. Ph.D. thesis, École Polytechnique (2010)

    Google Scholar 

  18. Jacobs, B., Rutten, J.: A tutorial on (co)algebras and (co)induction. Bull. Eur. Assoc. Theor. Comput. Sci. 62, 222–259 (1997)

    MATH  Google Scholar 

  19. Kaplan, D.: Review of Kripke (1959) [21]. J. Symb. Log. 31(1966), 120–122 (1966)

    Google Scholar 

  20. Kleene, S.C.: Mathematical Logic. John Wiley & Sons (1967)

    Google Scholar 

  21. Kripke, S.: A completeness theorem in modal logic. J. Symb. Log. 24(1), 1–14 (1959)

    Article  MATH  MathSciNet  Google Scholar 

  22. Krivine, J.L.: Une preuve formelle et intuitionniste du théorème de complétude de la logique classique. Bull. Symb. Log. 2(4), 405–421 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. Margetson, J., Ridge, T.: Completeness theorem. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs (2004), http://afp.sf.net/entries/Completeness.shtml

  24. Mayr, R., Nipkow, T.: Higher-order rewrite systems and their confluence. Theor. Comput. Sci. 192(1), 3–29 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  25. Nakata, K., Uustalu, T., Bezem, M.: A proof pearl with the fan theorem and bar induction: Walking through infinite trees with mixed induction and coinduction. In: Yang, H. (ed.) APLAS 2011. LNCS, vol. 7078, pp. 353–368. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  26. Negri, S.: Kripke completeness revisited. In: Primiero, G., Rahman, S. (eds.) Acts of Knowledge: History, Philosophy and Logic: Essays Dedicated to Göran Sundholm, pp. 247–282. College Publications (2009)

    Google Scholar 

  27. Nipkow, T., Klein, G.: Concrete Semantics: A Proof Assistant Approach. Springer (to appear), http://www.in.tum.de/~nipkow/Concrete-Semantics

  28. Nipkow, T., Paulson, L.C., Wenzel, M.: Isabelle/HOL. LNCS, vol. 2283. Springer, Heidelberg (2002)

    Book  MATH  Google Scholar 

  29. Pfenning, F.: Review of “Jean H. Gallier: Logic for Computer Science. J. Symb. Log. 54(1), 288–289 (1989)

    Article  Google Scholar 

  30. Ridge, T., Margetson, J.: A mechanically verified, sound and complete theorem prover for first order logic. In: Hurd, J., Melham, T. (eds.) TPHOLs 2005. LNCS, vol. 3603, pp. 294–309. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  31. Roşu, G.: Equality of streams is a \(\Pi_2^0\)-complete problem. In: Reppy, J.H., Lawall, J.L. (eds.) ICFP 2006. ACM (2006)

    Google Scholar 

  32. Roşu, G.: An effective algorithm for the membership problem for extended regular expressions. In: Seidl, H. (ed.) FoSSaCS 2007. LNCS, vol. 4423, pp. 332–345. Springer, Heidelberg (2007)

    Google Scholar 

  33. Rutten, J.J.M.M.: Automata and coinduction (an exercise in coalgebra). In: Sangiorgi, D., de Simone, R. (eds.) CONCUR 1998. LNCS, vol. 1466, pp. 194–218. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  34. Rutten, J.J.M.M.: Regular expressions revisited: A coinductive approach to streams, automata, and power series. In: Backhouse, R., Oliveira, J.N. (eds.) MPC 2000. LNCS, vol. 1837, pp. 100–101. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  35. Rutten, J.J.M.M.: Elements of stream calculus (an extensive exercise in coinduction). Electr. Notes Theor. Comput. Sci. 45, 358–423 (2001)

    Article  Google Scholar 

  36. Schlöder, J.J., Koepke, P.: The Gödel completeness theorem for uncountable languages. Formalized Mathematics 20(3), 199–203 (2012)

    MATH  Google Scholar 

  37. Troelstra, A.S., Schwichtenberg, H.: Basic Proof Theory, 2nd edn. Cambridge University Press (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Blanchette, J.C., Popescu, A., Traytel, D. (2014). Unified Classical Logic Completeness. In: Demri, S., Kapur, D., Weidenbach, C. (eds) Automated Reasoning. IJCAR 2014. Lecture Notes in Computer Science(), vol 8562. Springer, Cham. https://doi.org/10.1007/978-3-319-08587-6_4

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-08587-6_4

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-08586-9

  • Online ISBN: 978-3-319-08587-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics