Automatic Decidability and Combinability Revisited
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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- 1.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)CrossRefGoogle Scholar
- 3.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)CrossRefGoogle Scholar
- 4.Dershowitz, N., Jouannaud, J.-P.: Handbook of Theoretical Computer Science. In: Rewrite Systems, ch. 6, vol. B, pp. 244–320. Elsevier, North-Holland (1990)Google Scholar
- 6.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)Google Scholar