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Theorem Proving with Bounded Rigid E-Unification

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Automated Deduction - CADE-25 (CADE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9195))

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Abstract

Rigid E-unification is the problem of unifying two expressions modulo a set of equations, with the assumption that every variable denotes exactly one term (rigid semantics). This form of unification was originally developed as an approach to integrate equational reasoning in tableau-like proof procedures, and studied extensively in the late 80 s and 90s. However, the fact that simultaneous rigid E-unification is undecidable has limited practical adoption, and to the best of our knowledge there is no tableau-based theorem prover that uses rigid E-unification. We introduce simultaneous bounded rigid E-unification (BREU), a new version of rigid E-unification that is bounded in the sense that variables only represent terms from finite domains. We show that (simultaneous) BREU is NP-complete, outline how BREU problems can be encoded as propositional SAT-problems, and use BREU to introduce a sound and complete sequent calculus for first-order logic with equality.

This work was partly supported by the Microsoft PhD Scholarship Programme and the Swedish Research Council.

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Notes

  1. 1.

    http://user.it.uu.se/~petba168/breu/.

References

  1. Bachmair, L., Tiwari, A., Vigneron, L.: Abstract congruence closure. J. Autom. Reasoning 31(2), 129–168 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baumgartner, P., Furbach, U., Pelzer, B.: Hyper tableaux with equality. In: Pfenning, F. (ed.) CADE 2007. LNCS (LNAI), vol. 4603, pp. 492–507. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  3. Beckert, B.: Equality and other theories. In: D’Agostino, M., Gabbay, D., Hähnle, R., Posegga, J. (eds.) Handbook of Tableau Methods. Kluwer, Dordrecht (1999)

    Google Scholar 

  4. Degtyarev, A., Voronkov, A.: Simultaneous rigid E-Unification is undecidable. In: Kleine Büning, H. (ed.) CSL 1995. LNCS, vol. 1092, pp. 178–190. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  5. Degtyarev, A., Voronkov, A.: What you always wanted to know about rigid E-Unification. J. Autom. Reasoning 20(1), 47–80 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Degtyarev, A., Voronkov, A.: Equality reasoning in sequent-based calculi. In: Handbook of Automated Reasoning (in 2 volumes). Elsevier and MIT Press (2001)

    Google Scholar 

  7. Degtyarev, A., Voronkov, A.: Kanger’s Choices in Automated Reasoning. Springer, The Netherlands (2001)

    Google Scholar 

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

    Book  Google Scholar 

  9. Gallier, J.H., Raatz, S., Snyder, W.: Theorem proving using rigid e-unification equational matings. In: LICS. pp. 338–346. IEEE Computer Society (1987)

    Google Scholar 

  10. Giese, M.A.: Incremental closure of free variable tableaux. In: Goré, R.P., Leitsch, A., Nipkow, T. (eds.) IJCAR 2001. LNCS (LNAI), vol. 2083, pp. 545–560. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  11. Giese, M.A.: A model generation style completeness proof for constraint tableaux with superposition. In: Egly, U., Fermüller, C. (eds.) TABLEAUX 2002. LNCS (LNAI), vol. 2381, pp. 130–144. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  12. Giese, M.: Simplification rules for constrained formula tableaux. In: TABLEAUX, pp. 65–80 (2003)

    Google Scholar 

  13. Kanger, S.: A simplified proof method for elementary logic. In: Siekmann, J., Wrightson, G. (eds.) Automation of Reasoning 1: Classical Papers on Computational Logic 1957–1966, pp. 364–371. Springer, Heidelberg (1983). originally appeared in 1963

    Chapter  Google Scholar 

  14. Rümmer, P.: A constraint sequent calculus for first-order logic with linear integer arithmetic. In: Cervesato, I., Veith, H., Voronkov, A. (eds.) LPAR 2008. LNCS (LNAI), vol. 5330, pp. 274–289. Springer, Heidelberg (2008)

    Chapter  Google Scholar 

  15. Rümmer, P.: E-matching with free variables. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18 2012. LNCS, vol. 7180, pp. 359–374. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  16. Tiwari, A., Bachmair, L., Rueß, H.: Rigid E-Unification revisited. In: CADE. pp. 220–234, CADE-17. Springer-Verlag, London (2000)

    Google Scholar 

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Acknowledgements

We would like to thank Christoph M. Wintersteiger for comments on this paper, and the anonymous referees for helpful feedback.

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Authors and Affiliations

  1. Uppsala University, Uppsala, Sweden

    Peter Backeman & Philipp Rümmer

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  1. Peter Backeman
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  2. Philipp Rümmer
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Correspondence to Philipp Rümmer .

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Editors and Affiliations

  1. University of Ottawa, Ottawa, Canada

    Amy P. Felty

  2. University of Innsbruck, Innsbruck, Austria

    Aart Middeldorp

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Backeman, P., Rümmer, P. (2015). Theorem Proving with Bounded Rigid E-Unification. In: Felty, A., Middeldorp, A. (eds) Automated Deduction - CADE-25. CADE 2015. Lecture Notes in Computer Science(), vol 9195. Springer, Cham. https://doi.org/10.1007/978-3-319-21401-6_39

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  • DOI: https://doi.org/10.1007/978-3-319-21401-6_39

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21400-9

  • Online ISBN: 978-3-319-21401-6

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