Robert Nieuwenhuis and Albert Rubio. Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, 11th CADE, LNAI 607, pages 477--491, Saratoga Springs, New York, 1992. Extended version to appear in Journal of Symbolic Computation.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to generality , or the possibility of determining minimal sets of uni ers which are complete, in the sense that all other uni ers are less general. In this paper we focus on testing uni ability. This is sucient in many applications (e.g. in constraint based approaches to automated deduction [9, 17, 16]) and is often simpler than computing complete sets of uni ers. Theorem 2. For any E uni cation problem S : fs 1 = t 1 ; s k = t k g with free constants in C and variables Y = fy 1 ; ym g the following are equivalent: 1) S has a solution w.r.t. E. 2) The formula 9y 1 ; ....

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proceedings of CADE-11, LNAI 607, pages 477-491. Springer Verlag, 1992.

....To be able to lift narrowing steps on terms to constrained narrowing we use the standard technique that considers only reduced ground instances on the ground level. This technique was originally used to show completeness of basic narrowing [9] and later for constrained or basic rst order calculi [4, 10]. Formally, a substitution is reduced if x is irreducible with respect to R t for all variables x. An instance is called reduced if it is obtained by a reduced substitution. We write gnd(N) for the set of ground instances of clauses in N , and rgnd R (N) for the subset of reduced ground ....

Robert Nieuwenhuis and Albert Rubio. Theorem proving with ordering constrained clauses. In Proc. 11th Int. Conf. on Automated Deduction, LNCS 607, pages 477-491, Saratoga Springs, NY, 1992. Springer.

....Then one can prove as in [6] that for any ground atom L, P L if and only if Theta(P [ HA) HCL Theta(L) However, for a better readability, in the examples below, we do not perform the transformation Theta. Let us now briefly 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. 20 SubType H: t 2 S T # S T ) t 2 T Choice H: s 2 t ) ....

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

....style of OBJ 3, but using dynamic types and sort constraints with clear set theoretic semantics. In order to transform a R or G presentation into a conditional term rewriting system, we can adapt a saturation procedure on equational Horn clauses, such as the ones described for instance in [2, 27]. The three main inference rules are superposition into conclusion, superposition into premisses and equality resolution. Application of these rules requires the existence of a well founded reduction ordering [4] on terms and literals. Superposition is performed by unifying the maximum term in an ....

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 477--491. Springer-Verlag, 1992.

....[11, 16, 21] In this setting an ordering such as rpo is used both to orient rewrite rules and to select maximal literals to perform inferences on. In order to further prune the search space the ordering requirements on conditional inferences are better handled when they are treated as constraints [12, 18]. Typically a nonorientable equation s = t will be split as two constrained rewrite rules: s t j s t and t s j t s. Such constrained rules are useless when the constraint is unsatisfiable. Therefore it is important for the efficiency of automated reasoning systems to investigate decision ....

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proceedings of 11th Conf. on Automated Deduction, Saratoga Springs, NY, 1992, volume 607 of Lecture Notes in Artificial Intelligence, pages 477--491, June 1992, Springer-Verlag.

....logic programming and, in the context of automated theorem proving, formalized by Kirchner, Kirchner and Rusinowitch [1990] and others. Completeness proofs for certain saturation based theorem proving strategies involving constrained clauses were rst obtained by Huet [1972] B urckert [1990] and Nieuwenhuis and Rubio [1992]. In moving the constraints from the meta level to the object level, and using constraint inheritance, the ordering restrictions of the ground level are represented in a precise way and in principle no information is lost. Constraints are also signi cant when nite representations of saturated ....

Nieuwenhuis R. and Rubio A. [1992], Theorem proving with ordering constrained clauses, in `Automated Deduction | CADE'11', Vol. 607 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, pp. 477-491.

....past a large variety of approaches have been presented on how additional knowledge can improve proof search. In automated theorem the proofs that speci c calculus rules can be permuted are used to cut o redundant branches of the search tree. In basic ordered paramodulation and basic superposition [4,22], for instance, paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. To implement such a strategy, we have to maintain such knowledge for each individual subterm. In tactic based theorem proving, the problem arises to monitor parts of the ....

....and the given rewrite rules (3.2) and (3.3) 3.1. Example: Basic Ordered Paramodulation and Basic Superposition Paramodulation [25] is a theorem proving method for rst order logic with equality, which is refutationally complete if various ordering restrictions are imposed. Basic superposition [4,22] provide restrictions as paramodulation inferences are forbidden at terms introduced by substitutions from previous inference steps. Basic ordered paramodulation is another re nement, which also forbids inferences at any term positioned below a former paramodulation inference. Both re nements are ....

Robert Nieuwenhuis and Albert Rubio. Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, Proceedings of the 11th International Conference on Automated Deduction (CADE-11), volume 607 of LNAI, pages 477-491, Saratoga Springs, NY, June 1992. Springer.

....be able to do lift narrowing steps on terms to constrained narrowing we use the standard technique that considers only reduced ground instances on the ground level. This technique was originally used to show completeness of basic narrowing [9] and later for constrained or basic rst order calculi [4, 10]. Formally, a substitution is reduced if x is irreducible with respect to R t for all variables x. An instance is called reduced if it is obtained by a reduced substitution. We write gnd(N) for the set of ground instances of clauses in N , and rgnd R (N) for the subset of reduced ground ....

Robert Nieuwenhuis and Alberto Rubio. Theorem proving with ordering constrained clauses. In Proc. 11th Int. Conf. on Automated Deduction, LNCS 607, pages 477-491, Saratoga Springs, NY, 1992. Springer.

....[11, 16, 21] In this setting an ordering such as rpo is used both to orient rewrite rules and to select maximal literals to perform inferences on. In order to further prune the search space the ordering requirements on conditional inferences are better handled when they are treated as constraints [12, 18]. Typically a non orientable equation s = t will be split as two constrained rewrite rules: s t j s t and t s j t s. Such constrained rules are useless when the constraint is unsatis able. Therefore it is important for the eciency of automated reasoning systems to investigate decision ....

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proceedings of 11th Conf. on Automated Deduction, Saratoga Springs, NY, 1992, volume 607 of Lecture Notes in Articial Intelligence, pages 477-491, June 1992, Springer-Verlag.

....of OBJ 3, but using dynamic types and sort constraints with clear set theoretic semantics. In order to transform a R n or G n presentation into a conditional term rewriting system, we can adapt a saturation procedure on equational Horn clauses, such as the ones described for instance in [2, 27]. The three main inference rules are superposition into conclusion, superposition into premisses and equality resolution. Application of these rules requires the existence of a well founded reduction ordering [4] on terms and literals. Superposition is performed by unifying the maximum term in an ....

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.

.... of ordered completion described in [36, 45] and completion modulo associativity, commutativity and identity [10, 24] Completion with membership constraints is studied in [12] Refutational completeness results for deduction systems based on constrained superposition have been obtained in [9, 40, 41]. This paper attempts to present the evolution from ordered rewriting and class rewriting to rewriting with constraints in a comprehensive way. So Section 2 briefly reminds the definitions and results related to ordered rewriting, Section 3 combines this concept with equivalence classes modulo a ....

....pair lemma. This is done with so called built in constraints in [32] for instance. When the initial set of equalities is unconstrained, the basic superposition deduction rule (an instance with A = of the rule Deduce Gamma Eq Gamma Eq given in Figure 5) is the basis of a saturation process in [41, 9]. Another alternative initially proposed in [30] is to use propagation. Constraints are weakened by partially solving them and propagating the instantiations in the equality. This is formalized by the rule Propagate given in Figure 4, where is the equational form of the substitution . ....

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R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proceedings of CADE-11, volume 607 of Lecture Notes in Computer Science, pages 477--491. Springer-Verlag, 1992.

....strategies, the first one permits to restrict to left hand sides of equations, while the second one permits to block the right hand sides when they replace subterms of a clause. In this paper, we extend the basic paramodulation and superposition to AC theories. We use explicit constraints as in [NR92b] to get an homogeneous framework with ordering and unification constraints at the same level. Such ordering constraints were used by Peterson [Pet90] and Martin and Nipkow [MN90] to orient equations like the commutativity axiom : x : y y : x if x y. But, constraints are also fundamental to ....

....such as the maximal positions requirement, which consists in forbidding paramodulations at non maximal positions of AC operators. Moreover, we never use the AC axioms explicitly ; they are handled though AC unification constraints and the inference rules application. Nieuwenhuis and Rubio [NR92b] have defined a complete set of inference rules based on superposition, with ordering and unification constraints. Their work deals with the empty theory. They have recently extended this result to AC theories [NR94] However, we have important refinements on the rules named contextual ....

[Article contains additional citation context not shown here]

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, Proceedings of CADE-11, volume 607 of Lecture Notes in Computer Science, pages 477--491. Springer-Verlag, 1992.

....set of equations is that it is constrained correctly with respect to E and E 0 . This is trivially true if it is unconstrained. But the de nition also allows for constrained equations. Other inference systems for Basic Completion require that the initial set of equations is unconstrained (but see [NR92b]) 5.2 Completeness For completeness, we show that in the ground case, our inference rules are able to saturate the set of equations. First we consider the case of an inference into a non variable position, which is standard. The following lemma proves that inferences at non variable positions ....

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 477-491. Springer-Verlag, 1992.

....instances of the equations involved. The notion of redundancy we use is due to Bachmair and Ganzinger [1] see also [17] and amounts to a semantic version of the wellknown subconnectedness criterion. Building on recent work on Basic Paramodulation [4, 5, 15] on constrained completion [11, 15, 16], and on various critical pair criteria (see [3] for a survey) we show how a wide variety of techniques for removing redundant equations can be combined and strengthened in a consistent framework. The tradeooes we explore are essentially renements of the techniques for weakening constraints in ....

....4 present our inference system and discuss its general features and its relationship to other critical Computer Science Department, Boston University, 111 Cummington St. Boston, MA 02215, U.S.A. lynch,snyder cs.bu.edu. 1 For a discussion of the eoeect of initial ordering constraints see [16]. 1 pair criteria. In section 5 we discuss constraint solving with irreducibility constraints. In section 6 we develop the theoretical justication for this system and rigorously prove completeness. In section 7, we show how it is possible to apply our completion process without recalculating ....

[Article contains additional citation context not shown here]

R. Nieuwenhuis and A. Rubio. Theorem Proving with Ordering Constrained Clauses. Proc. 11th Conference on Automated Deduction, Springer-Verlag LNAI 607, Berlin (1992) pp. 477491.

.... of symbolic model representation techniques, respectively: BGW93a] BGW93b] DG79] FS93] FLTZ93] GH95] Lei95] Pel95] and [Sal91] Equational Calculi: BG91] BG94] BGLS92] Hue80] HKLR92] HR91] NR94] and [Pla93] Implementation of superposition calculi: BG94] [NR92], NN93] and [Sal91] 4 Aims of the Project The general aim consists in working out a firm mathematical basis for model building with equality. From this basis efficient algorithms should be developed REFERENCES 7 and eventually be implemented. This can be achieved by systematically combining ....

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, 11 th International Conference on Automated Deduction, pages 477--491, Saratoga Springs, New York, USA, 1992. Springer Verlag, LNAI 607.

....and, in the context of automated theorem proving, been suggested by Kirchner, Kirchner Rusinowitch (1990) among others. Completeness proofs for certain saturation based theorem proving strategies involving constrained clauses have first been obtained by Huet (1972) Burckert (1990) and Nieuwenhuis Rubio (1992). By moving the constraints from the meta level to the object level, and through constraint inheritance, the ordering restrictions of the ground level are represented in a precise way and no information is lost a priori. The significance of constraint notations also becomes apparent when finite ....

Nieuwenhuis, R. & Rubio, A. (1992), Theorem proving with ordering constrained clauses, in `Automated Deduction --- CADE'11', Vol. 607 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, pp. 477--491.

No context found.

Robert Nieuwenhuis and Albert Rubio. Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, 11th CADE, LNAI 607, pages 477--491, Saratoga Springs, New York, 1992. Extended version to appear in Journal of Symbolic Computation.

.... practical applications of ordering constraints gave rise to the distinction between fixed signature semantics (solutions are built over a given signature F) and extended signature semantics (new symbols are allowed to appear in solutions) Satisfiability under extended signatures was considered in [16] as the adequate semantics for the computation of saturated sets of ordering constrained clauses Both authors partially supported by the Spanish Cicyt project HEMOSS, ref. TIC98 0949C02 01. that can be used for deduction with other clauses containing arbitrary new (e.g. Skolem) symbols. The ....

Robert Nieuwenhuis and Albert Rubio. Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, 11th International Conference on Automated Deduction (CADE), LNAI 607, pages 477--491, Saratoga Springs, New York, 1992. 24

....semantics (solutions are built over a given signature Both authors partially supported by the ESPRIT Basic Research WG CCL II, ref. 22457. 2 F) and extended signature semantics (new symbols are allowed to appear in solutions) Satisfiability under extended signatures was considered in [NR92] as the adequate semantics for the computation of saturated sets of ordering constrained clauses that can be used for deduction with other clauses containing arbitrary new (e.g. Skolem) symbols. The satisfiability problem for ordering constraints was first shown decidable for fixed signatures ....

Robert Nieuwenhuis and Albert Rubio. Theorem proving with ordering constrained clauses. In Deepak Kapur, editor, 11th International Conference on Automated Deduction (CADE), LNAI 607, pages 477--491, Saratoga Springs, New York, 1992. 27

....is sometimes possible to compute a finite saturated set (not containing the empty clause) for a given input. In this case its satisfiability has been proved. This kind of satisfiability proving has of course many applications and is also closely related to inductive theorem proving (see [Comon and Nieuwenhuis 2000] and Chapter [chapter with id Comon] of this handbook) The Spass system [Weidenbach 1997] has successfully applied saturation to prove satisfiability for all problems in the corresponding category of the 1997 CADE theorem proving competition [Sutcliffe and Suttner 1998] Theorem proving in ....

....Section 6. 2 (see also [Rubio 1996] for builtin semigroups, i.e. associative theories) This approach is considered as well for arbitrary regular theories in [Vigneron 1996] Recent research concerns algebraic structures richer than abelian semigroups, like abelian groups [Stuber 1998a, Godoy and Nieuwenhuis 2000], cancellative abelian monoids [Ganzinger and Waldmann 1996] commutative rings [Stuber 1998b] or divisible torsion free abelian groups [Waldmann 1998] 52 R. Nieuwenhuis and A. Rubio 6.1. E compatible reduction orderings Many results in the literature on ordered paramodulation and ....

Nieuwenhuis R. and Rubio A. [1992b], Theorem proving with ordering constrained clauses, in D. Kapur, ed., `11th International Conference on Automated Deduction (CADE)', LNAI 607, Saratoga Springs, New York, pp. 477--491.

No context found.

Robert Nieuwenhuis and Albert Rubio. Theorem Proving with Ordering Constrained Clauses. In D. Kapur, editor, Proceedings of the 11th International Conference on Automated Deduction (CADE-11), pages 477--491. Springer-Verlag, June 1992. LNAI 607.

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R. Nieuwenhuis and A. Rubio, Theorem proving with ordering constrained clauses, in: D. Kapur, ed., Proc. 11th. Int. Conf. on Automated Deduction, Saratoga Springs, NY (LNAI 607, Springer-Verlag, 1992) 477--491.

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Nieuwenhuis, R. and A. Rubio: 1992b, `Theorem proving with ordering constrained clauses'. In: Automated Deduction --- CADE'11, Vol. 607 of Lecture Notes in Computer Science. Berlin, pp. 477--491.

No context found.

R. Nieuwenhuis and A. Rubio. Theorem proving with ordering constrained clauses. In D. Kapur, editor, 11th International Conference on Automated Deduction (CADE), volume 607 of Lecture Notes in Artificial Intelligence, pages 477--491, Saratoga Springs, NY, USA, June 1992. Springer Verlag.

No context found.

Nieuwenhuis, R. & Rubio, A. (1992b), Theorem proving with ordering constrained clauses, in `Automated Deduction --- CADE'11', Vol. 607 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, pp. 477-- 491.

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