Nieuwenhuis, R., Rubio, A. (1992b) Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, 11th CADE, LNAI 607, pages 477--491, Saratoga Springs, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....extremely useful in logic programming and theorem proving. Among such constraints, the ordering constraints can be used in expressing ordered strategies at the formula level instead of the inference level. This allows to cut further the search space, while keeping the completeness of the strategy [7]. Solving ordering constraints also allows for a nice lifting of orderings from the ground level to the terms with variables: define s t by 8 x:s t where x, the variables of s; t, range over all ground terms. This provides with more powerful orderings for termination proofs in rewriting ....

....that, if there is a solution, there is some solution which has a small size. They also use normal forms of inequations systems in which all inequations s t have a variable on the left. As we will see, it is not possible to follow this technique with the embedding relation. On the other hand, [1, 6, 7] use the linearity of the ordering in many places: the main problem is the expression of the successor function on the term level. Of course, we cannot use such a technique with the embedding which is not a linear ordering. First, we set up precisely the problem in section 2. Then we give some ....

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, Proc. 11th Int. Conf. on Automated Deduction, Saratoga Springs, NY, LNCS 607. Springer-Verlag, June 1992.

....path ordering (Lankford [8] For more details, see Dershowitz and Hoot [2] An instance of the gpo based on a particular Phi = Theta; lex ) will be noted Phi gpo . 3 gpo ordering constraints A constraint approach of orderings problems has already been proposed in several works [1, 12, 13, 11, 6]. Those works are all concerned with the satis ability problem of ordering constraints. Let s i and t i (i = 1; n) be terms of T (F; X) Given a precedence F and a conjunction of ordering constraints V n i=1 s i t i , Comon [1] and Nieuwenhuis [11] gave algorithms for deciding if ....

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proceedings 11th International Conference on Automated Deduction, Saratoga Springs (N.Y., USA), volume 607 of Lecture Notes in Computer Science, pages 477491. Springer-Verlag, 1992.

....for proving a theorem in the initial model of a presentation. This proof procedure uses an ordering on terms and atoms to apply inference rules only on maximal literals in the clauses, which reduces the search space. Let us rst brieAEy recall the saturation procedure as described for instance in [2, 16]. The three main inference rules are superposition into conclusion, superposition into premisses and equality resolution. Subsumption by another clause and elimination of tautologies are also used to eliminate redundant clauses. An ordered strategy is used for reducing the search space by using ....

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proceedings 11th International Conference on Automated Deduction, Saratoga Springs (N.Y., USA), volume 607 of Lecture Notes in Computer Science, pages 477491. Springer-Verlag, 1992.

....on a general path ordering, we will reduce the set of possibilities for instantiation of termination functions by constraint solving, until we get a particular ordering when it is possible. The problem tackled here is different from the constraint approach of ordering problems already proposed [2, 15, 16, 14, 8]. All are concerned with the satisfiability problem of ordering constraints. In our approach however, we try to find a gpo ordering for validating inequalities between terms. The algorithm, we propose, works in two steps, using a term sharing data structure for rewrite rules. The first step ....

.... Phi gpo of gpo. Restricting to ground stable instances of gpo is not critical since the instances of gpo used in practice are ground stable. 3 Solving gpo constraints using a term sharing data structure A constraint approach of ordering problems has already been proposed in several works [2, 15, 16, 14, 8]. All are concerned with the satisfiability problem of ordering constraints. INRIA Termination Proofs using gpo Ordering Constraints 7 Let s i and t i (i = 1; n) be terms of T (F; X) Given a precedence F and a conjunction of ordering constraints V n i=1 s i t i , Comon [2] and ....

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proc. 11th CADE Conf., Saratoga Springs (N.Y., USA), volume 607 of LNCS, pages 477--491. Springer-Verlag, 1992.

....clauses in S. Therefore, RC G is a redundancy criterion for clauses. RIG is a redundancy criterion for inferences, since for all inferences by I G and HG the conclusion is smaller than the maximal premise, which by definition of RI G implies that 2 RI(S) if the conclusion of is in S. 12 R. Nieuwenhuis and A. Rubio Conditions RC G (S) RC G (S [ S 0 ) and RI G (S) RI G (S [ S 0 ) of the correctness of (RCG ; RIG ) follow trivially from the definition. To prove that for every derivation S 0 ; S 1 ; RCG (S j ) RCG (S1 ) for j 0, we will prove that RCG ( S j ) RCG (S1 ) which is a stronger property. We derive a contradiction ....

Nieuwenhuis, R., Rubio, A. (1992b) Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, 11th CADE, LNAI 607, pages 477--491, Saratoga Springs, New York.

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