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Soundness and Completeness Proofs by Coinductive Methods

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Abstract

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion.

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Notes

  1. Given formulas \(\psi _1,\ldots ,\psi _k\), we let \({\textsf {Conj}}\;\psi _1\;\ldots \;\psi _k\) denote \({\textsf {Conj}}\;\psi _1\;({\textsf {Conj}}\;\psi _2\;(\ldots \psi _n)\ldots )\). In particular, when \(k = 0\) it denotes the “true” formula \(\top \), defined in a standard way, e.g., as \({\textsf {Imp}}\;a\;a\) for some atom a.

  2. This is acceptable here, since we employ finitary Horn clauses and the language is countable. Different assumptions may require larger ordinals.

  3. The definition of \(P_{p,\_}\) works with the original clauses \(\chi \in {\textsf {ind}}_p\), whereas here we apply it to the “copies” \(\chi '\) of \(\chi \) guaranteed to have their variables fresh for \(\Gamma \) and \(\Delta \), as stipulated in the \(p_{\mathrm{split}}\) rule. This is unproblematic, since it is easy to verify that the definition of \(P_{p,\_}\) is invariant under bijective renaming of variables in the clauses \(\chi \).

  4. Goodness is decidable for cyclic trees in logics where rule application is decidable, such as FOL\(_{\textsf {ind}}\) [16].

  5. In the proof system from Example 2, \({\textsf {eff}}\) is not deterministic due to the rule All R. It can be made deterministic by refining the rule with a systematic choice of the fresh variable y.

  6. And Kripke’s degree of rigor in this early article is not far from today’s state of the art in proof theory; see, e.g., Troelstra and Schwichtenberg [51].

  7. This is the only error we found in this otherwise excellent chapter on tableaux.

References

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

    MATH  Google Scholar 

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

  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 (2005)

  4. Blanchette, J.C., Böhme, S., Popescu, A., Smallbone, N.: Encoding monomorphic and polymorphic types. In: Piterman, N., Smolka, S. (eds.) TACAS 2013, LNCS, vol. 7795, pp. 493–507. Springer (2013)

  5. Blanchette, J.C., Fleury, M., Weidenbach, C.: A verified SAT solver framework with learn, forget, restart, and incrementality. In: Olivetti, N., Tiwari, A. (eds.) IJCAR 2016, LNCS, vol. 9706. Springer (2016)

  6. 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, vol. 8558, pp. 93–110. Springer (2014)

  7. 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 (2013)

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

  9. Blanchette, J.C., Popescu, A., Traytel, D.: Unified classical logic completeness—a coinductive pearl. In: Demri, S., Kapur, D., Weidenbach, C. (eds.) IJCAR 2014, LNCS, vol. 8562, pp. 46–60. Springer (2014)

  10. Blanchette, J.C., Popescu, A., Traytel, D.: Formal development associated with this paper. http://people.inf.ethz.ch/trayteld/compl-journal-devel.tgz (2015)

  11. Blanchette, J.C., Popescu, A., Traytel, D.: Foundational extensible corecursion: a proof assistant perspective. In: Fisher, K., Reppy, J.H. (eds.) ICFP 2015, pp. 192–204. ACM (2015)

  12. Brotherston, J.: Cyclic proofs for first-order logic with inductive definitions. In: Beckert, B. (ed.) TABLEAUX 2005, LNCS, vol. 3702, pp. 78–92. Springer (2005)

  13. Brotherston, J.: Sequent calculus proof systems for inductive definitions. Ph.D. thesis, University of Edinburgh (2006)

  14. Brotherston, J., Bornat, R., Calcagno, C.: Cyclic proofs of program termination in separation logic. In: Necula, G.C., Wadler, P. (eds.) POPL 2008, pp. 101–112. ACM (2008)

  15. Brotherston, J., Distefano, D., Petersen, R.L.: Automated cyclic entailment proofs in separation logic. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE-23, LNCS, vol. 6803, pp. 131–146. Springer (2011)

  16. Brotherston, J., Gorogiannis, N., Petersen, R.L.: A generic cyclic theorem prover. In: Jhala, R., Igarashi, A. (eds.) APLAS 2012, LNCS, vol. 7705, pp. 350–367. Springer (2012)

  17. Brotherston, J., Simpson, A.: Complete sequent calculi for induction and infinite descent. In: LICS 2007, pp. 51–62. IEEE Computer Society (2007)

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Diaconescu, R.: Institution-Independent Model Theory. Studies in Universal Logic. Birkhäuser, Basel (2008)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Google Scholar 

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

    MATH  Google Scholar 

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

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

    MATH  Google Scholar 

  25. 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 (2010)

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

    Google Scholar 

  27. Harrison, J.: Formalizing basic first order model theory. In: Grundy, J., Newey, M.C. (eds.) TPHOLs ’98, LNCS, vol. 1479, pp. 153–170. Springer (1998)

  28. Ilik, D.: Constructive completeness proofs and delimited control. Ph.D. thesis, École polytechnique (2010)

  29. 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 

  30. Kaplan, D.: Review of Kripke (1959) [32]. J. Symb. Log. 31, 120–122 (1966)

  31. Kleene, S.C.: Mathematical Logic. Wiley, London (1967)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  33. 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  Google Scholar 

  34. Margetson, J., Ridge, T.: Completeness theorem. In: Klein, G., Nipkow, T., Paulson, L. (eds.) Archive of Formal Proofs. http://www.isa-afp.org/entries/Completeness.shtml (2004)

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

    Article  MathSciNet  MATH  Google Scholar 

  36. 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 (2011)

  37. 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, London (2009)

    Google Scholar 

  38. Nipkow, T., Klein, G.: Concrete Semantics: With Isabelle/HOL. Springer, Berlin (2014)

    Book  MATH  Google Scholar 

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

  40. Petria, M.: An institutional version of Gödel’s completeness theorem. In: CALCO 2007, pp. 409–424 (2007)

  41. Pfenning, F.: Review of “Jean H. Gallier: Logic for Computer Science, Harper & Row, New York 1986” [22]. J. Symb. Log. 54(1), 288–289 (1989)

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

  43. Roşu, G.: Equality of streams is a \(\Pi _2^0\)-complete problem. In: Reppy, J.H., Lawall, J.L. (eds.) ICFP ’06. ACM (2006)

  44. 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 (2007)

  45. Rutten, J.J.M.M.: Automata and coinduction (an exercise in coalgebra). In: Sangiorgi, D., de Simone, R. (eds.) CONCUR ’98, LNCS, vol. 1466, pp. 194–218. Springer (1998)

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

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

    Article  MATH  Google Scholar 

  48. Schlichtkrull, A.: Formalization of the resolution calculus for first-order logic. In: Blanchette, J.C., Merz, S. (eds.) ITP 2016, LNCS, vol. 9807. Springer (2016)

  49. Schlöder, J.J., Koepke, P.: The Gödel completeness theorem for uncountable languages. Formaliz. Math. 20(3), 199–203 (2012)

    MATH  Google Scholar 

  50. Traytel, D., Popescu, A., Blanchette, J.C.: Foundational, compositional (co)datatypes for higher-order logic: category theory applied to theorem proving. In: LICS 2012, pp. 596–605. IEEE Computer Society (2012)

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

    Book  MATH  Google Scholar 

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Acknowledgments

Tobias Nipkow made this work possible. Mark Summerfield and the anonymous reviewers suggested many textual improvements to earlier versions of this article. The reviewers read the submitted paper carefully and made useful and insightful comments and suggestions. Blanchette was partially supported by the Deutsche Forschungsgemeinschaft (DFG) project Hardening the Hammer (Grant NI 491/14-1). Popescu was partially supported by the EPSRC project Verification of Web-based Systems (VOWS, Grant EP/N019547/1) and by the DFG project Security Type Systems and Deduction (Grant NI 491/13-3). Traytel was supported by the DFG program Program and Model Analysis (PUMA, Doctorate Program 1480). The authors are listed alphabetically.

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Authors and Affiliations

  1. Inria Nancy and LORIA, Villers-lès-Nancy, France

    Jasmin Christian Blanchette

  2. Max-Planck-Institut für Informatik, Saarbrücken, Germany

    Jasmin Christian Blanchette

  3. Department of Computer Science, School of Science and Technology, Middlesex University, London, UK

    Andrei Popescu

  4. Institute of Information Security, Department of Computer Science, ETH Zurich, Zurich, Switzerland

    Dmitriy Traytel

  5. Institute of Mathematics Simion Stoilow of the Romanian Academy, Bucharest, Romania

    Andrei Popescu

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  1. Jasmin Christian Blanchette
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Blanchette, J.C., Popescu, A. & Traytel, D. Soundness and Completeness Proofs by Coinductive Methods. J Autom Reasoning 58, 149–179 (2017). https://doi.org/10.1007/s10817-016-9391-3

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