Abstract
We present an inference system for clauses with ordering constraints, called Schematic Paramodulation. Then we show how to use Schematic Paramodulation to reason about decidability and stable infiniteness of finitely presented theories. We establish a close connection between the two properties: if Schematic Paramodulation for a theory halts then the theory is decidable; and if, in addition, Schematic Paramodulation does not derive the trivial equality X = Y then the theory is stably infinite. Decidability and stable infiniteness of component theories are conditions required for the Nelson-Oppen combination method. Schematic Paramodulation is loosely based on Lynch-Morawska’s meta-saturation but it differs in several ways. First, it uses ordering constraints instead of constant constraints. Second, inferences into constrained variables are possible in Schematic Paramodulation. Finally, Schematic Paramodulation uses a special deletion rule to deal with theories for which Lynch-Morawska’s meta-saturation does not halt.
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Armando, A., Bonacina, M.P., Ranise, S., Schulz, S.: On a Rewriting Approach to Satisfiability Procedures: Extension, Combination of Theories and an Experimental Appraisal. In: Gramlich, B. (ed.) Frontiers of Combining Systems. LNCS (LNAI), vol. 3717, pp. 65–80. Springer, Heidelberg (2005)
Armando, A., Ranise, S., Rusinowitch, M.: A Rewriting Approach to Satisfiability Procedures. Info. and Comp. 183(2), 140–164 (2003)
Bonacina, M.P., Ghilardi, S., Nicolini, E., Ranise, S., Zucchelli, D.: Decidability and Undecidability Results for Nelson-Oppen and Rewrite-Based Decision Procedures. In: Furbach, U., Shankar, N. (eds.) IJCAR 2006. LNCS (LNAI), vol. 4130, pp. 513–527. Springer, Heidelberg (2006)
Dershowitz, N., Jouannaud, J.-P.: Handbook of Theoretical Computer Science. In: Rewrite Systems, ch. 6, vol. B, pp. 244–320. Elsevier, North-Holland (1990)
Kirchner, H., Ranise, S., Ringeissen, C., Tran, D.-K.: Automatic Combinability of Rewriting-Based Satisfiability Procedures. In: Hermann, M., Voronkov, A. (eds.) LPAR 2006. LNCS (LNAI), vol. 4246, pp. 542–556. Springer, Heidelberg (2006)
Lynch, C., Morawska, B.: Automatic decidability. In: Proc. of 17th IEEE Symposium on Logic in Computer Science, Copenhagen, Copenhagen, Denmark, pages 7. IEEE Computer Society Press, Los Alamitos (2002)
Nelson, G., Oppen, D.C.: Simplification by cooperating decision procedures. ACM Trans. on Programming Languages and Systems 1(2), 245–257 (1979)
Nieuwenhuis, R., Rubio, A.: Paramodulation-based theorem proving. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, ch. 7, vol. I, pp. 371–443. Elsevier, North-Holland (2001)
van Dalen, D.: Logic and Structure, 2nd edn. Springer, Heidelberg (1989)
Weidenbach, C.: Spass version 0.49. Journal of Automated Reasoning 14(2), 247–252 (1997)
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Lynch, C., Tran, DK. (2007). Automatic Decidability and Combinability Revisited. In: Pfenning, F. (eds) Automated Deduction – CADE-21. CADE 2007. Lecture Notes in Computer Science(), vol 4603. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73595-3_22
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DOI: https://doi.org/10.1007/978-3-540-73595-3_22
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