Abstract
We define a semantic criterion ensuring termination of the hyperresolution calculus, which allows us to prove the decidability of certain classes of clause sets. We also define an algorithm for deciding – in polynomial time – whether a given clause set satisfies the proposed criterion. Comparisons with existing works on hyperresolution-based decision procedures are provided, showing evidence of the interest of our approach.
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References
Baader, F., Snyder, W.: Unification theory. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, vol. I, Chapt. 8, pp. 445–532. Elsevier, Amsterdam (2001)
Baumgartner, P., Furbach, U., Niemelä, I.: Hyper-tableaux. In: Logics in AI, JELIA'96. Springer, Berlin Heidelberg New York (1996)
Bry, F., Yahya, A.: Positive unit hyperresolution. J. Autom. Reason. 25(1), 35–82 (2000)
De Schreye, D., Decorte, D.: Termination of logic programs: The neverending story. J. Log. Program. 19, 199–260 (1993)
Fermüller, C., Leitsch, A.: Hyperresolution and automated model building. J. Log. Comput. 6(2), 173–203 (1996)
Fermüller, C., Leitsch, A., Tammet, T., Zamov, N.: Resolution Methods for the Decision Problem, LNAI 679. Springer, Berlin Heidelberg New York (1993)
Fermüller, C.G., Leitsch, A., Hustadt, U., Tammet, T.: Resolution Decision Procedures. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, Chapt. 25, pp. 1791–1849. North-Holland, Amsterdam (2001)
Fitting, M.: First-Order Logic and Automated Theorem Proving, Texts and Monographs in Computer Science. Springer, Berlin Heidelberg New York (1990)
Ganzinger, H., de Nivelle, H.: A superposition decision procedure for the guarded fragment with equality. In: Proc. 14th IEEE Symposium on Logic in Computer Science. IEEE Computer Society, Los Alamitos, California (1999)
Georgieva, L., Hustadt, U., Schmidt, R.: A new clausal class decidable by hyperresolution. In: Voronkov, A. (ed.) Automated Deduction–CADE-18, vol. 2392 of LNCS, pp. 258–272. Springer, Berlin Heidelberg New York (2002)
Georgieva, L., Hustadt, U., Schmidt, R.A.: Hyperresolution for guarded formulae. In: Baumgartner, P., Zhang, H. (eds.) Proc. Third International Workshop on First-Order Theorem Proving (FTP 2000), vol. 5/2000 of Fachberichte Informatik, pp. 101–112. Institut für Informatik, Universität Koblenz-Landau, Koblenz, Germany (2000)
Georgieva, L., Hustadt, U., Schmidt, R.A.: Hyperresolution for guarded formulae. J. Symb. Comput. 36(1–2), 163–192 (2003)
Hustadt, U., Schmidt, R.A.: Using resolution for testing modal satisfiability and building models. J. Autom. Reason. 28(2), 205–232 (2002)
Jouannaud, J., Kirchner, C.: Solving equations in abstract algebras: A rule-based survey of unification. In: Lassez, J.-L., Plotkin, G. (eds.) Essays in Honor of Alan Robinson, pp. 91–99. MIT, Cambridge, Massachusetts (1991)
Joyner, W.: Resolution strategies as decision procedures. J. ACM 23, 398–417 (1976)
Langholm, T.: A strong version of Herbrand's theorem for introvert sentences. J. Symb. Log. 63, 555–569 (1998)
Leitsch, A.: Deciding clause classes by semantic clash resolution. Fundam. Inform. 18, 163–182 (1993)
Loveland, D.W.: Automated Theorem Proving: A Logical Basis, vol. 6 of Fundamental Studies in Computer Science. North-Holland, Amsterdam (1978)
Lutz, C., Sattler, U., Tobies, S.: A suggestion of an \(n\)-ary description logic. In: Proc. DL'99. Linköping University (1999)
Manthey, R., Bry, F.: SATCHMO: A Theorem Prover Implemented in Prolog. In: Proc. CADE-9. LNCS 310, pp. 415–434. Springer, Berlin Heidelberg New York (1988)
Martelli, A., Montanari, U.: An efficient unification algorithm. ACM Trans. Program. Lang. Syst. 4(2), 258–282 (1982)
de Nivelle, H.: A resolution decision procedure for the guarded fragment. In: Automated Deduction–CADE-15, vol. 1421 of LNCS (1998)
Peltier, N.: Constructing decision procedures in equational clausal logic. Fundam. Inform. 54(1), 17–65 (2003)
Robinson, J.A.: Automatic deduction with hyperresolution. Int. J. Comput. Math. 1, 227–234 (1965a)
Robinson, J.A.: A machine-oriented logic based on the resolution principle. J. ACM 12, 23–41 (1965b)
Rudlof, T.: SHR tableaux–A framework for automated model generation. J. Logic Comput. 10(6), 107–155 (2000)
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Peltier, N. Some Techniques for Proving Termination of the Hyperresolution Calculus. J Autom Reasoning 35, 391–427 (2005). https://doi.org/10.1007/s10817-006-9028-z
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DOI: https://doi.org/10.1007/s10817-006-9028-z