M. Rusinowitch. Theorem-proving with Resolution and Superposition. JSC, 11(1&2):2150, January /February 1991.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... : fc 1 ; c 2 ; c 3 g (where c 1 ; c 2 ; c 3 are constants) It is easy to check that attening preserves the satis ability of the Fairness means that if some inference is possible, it will be performed at some step unless one of the parent clauses gets simpli ed or deleted (see, e.g. Rus91] for a formal de nition) set S of ground literals and that it returns sets of literals which are O(n) where n is the length of the string obtained by concatenating the literals in S written in pre x notation) Some subtlety is required to perform attening in O(n) time (see [DST80] for the ....

M. Rusinowitch. Theorem-proving with Resolution and Superposition. JSC, 11(1&2):2150, January/February 1991.

.... Ganzinger, Lynch, et al. 1992, Nieuwenhuis and Rubio 1992) Traditionally completion procedures were formulated for sets of equations (unit clauses) but ordered completion has been generalized to arbitrary non unit clauses, resulting in several variants of paramodulation called superposition (Rusinowitch 1991, Zhang 1988, Bachmair and Ganzinger 1990, Pais and Peterson 1991) and the basic strategy has actually first been developed for first order clauses. Associativity and commutativity have been built into ordered paramodulation (Paul 1992, Rusinowitch and Vigneron 1991) a calculus (Hsiang and ....

....1991, Zhang 1988, Bachmair and Ganzinger 1990, Pais and Peterson 1991) and the basic strategy has actually first been developed for first order clauses. Associativity and commutativity have been built into ordered paramodulation (Paul 1992, Rusinowitch and Vigneron 1991) a calculus (Hsiang and Rusinowitch 1991) that does not generalize completion, but includes similar rewrite techniques; and Wertz (1992) designed an associative commutative superposition calculus. Unfortunately, the completeness proofs proposed for these calculi are technically involved and quite complicated. The calculus described in ....

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M. Rusinowitch, 1991. Theorem proving with resolution and superposition: An extension of the Knuth and Bendix completion procedure as a complete set of inference rules. J. Symbolic Computation, Vol. 11, pp. 21--49.

....apply replacements only into maximal sides of equations, and has been extensively used for term rewriting and completion. But, the completeness of inference systems representing this strategy has been a longstanding open problem. For completeness, either some deductions using non maximal literals [24], or some replacements into minimal sides of equations [3] were needed. Bachmair and Ganzinger [2] have proved the completeness in the empty theory of the entire superposition strategy by adding two Equational Factoring rules (one for negative and one for positive literals) Defining a particular ....

M. Rusinowitch. Theorem-proving with Resolution and Superposition. Journal of Symbolic Computation, 11:21--49, 1991.

....Q(a; b) The subterm a of the deduced clause is blocked (framed) to show that no more paramodulation step is allowed in it. A refinement of this strategy is to block in addition the replacing term in the deduced clause (i.e. b in P (b) This refinement is not valid for the superposition strategy [Rus91]. In the following, we refine and extend this strategy by associating to each clause a set of constraints as in [KKR90] Then, when a new clause is generated, it inherits the constraints of its ancestors and also the AC unification constraints (not computed) and ordering constraints produced by ....

....Q(x; y) j[ x=AC a y =AC b ]j. These strategies have been implemented in our theorem prover DATAC, and experiments with nontrivial examples are successful. The method used here to prove completeness of inference systems in AC theories has been adapted to other strategies such as superposition [Rus91]. But, an essential point to be studied and implemented is the propagation of ordering constraints. Acknowledgments : I wish to thank Michael Rusinowitch and Claude and H el ene Kirchner for their pertinent remarks and their support for this work. Especially, I would like to thank Robert ....

M. Rusinowitch. Theorem-proving with resolution and superposition. Journal of Symbolic Computation, 11:21--49, 1991.

.... Remarks on Transfinite E Semantic Trees and Superposition Georg Moser Technische Universitat Wien Abstract We prove the refutational completeness of P mep by proof techniques employed in establishing the completeness of weak superposition [9]. By giving a counter example we show that the same approach is impossible wrt. P eqf . Hence, this result shows a semantic differences between P mep and P eqf . We apply the result to Automated Model Building. 1 Introduction We study the relationship between two possible instances of the ....

....two possible instances of the superposition calculus [1] the equality factoring fragment, denoted by P eqf , and the merging paramodulation fragment, P mep . We establish refutational completeness for P mep by methods extending the techniques used in the completeness proof for weak superposition [9]. From this result we conclude that the model I H generated from the initial clause set C in the completeness proof for superposition [1] coincides exactly with the model I P described by the (infinite) right most maximal path P in a transfinite E semantic tree corresponding to C, where C is a ....

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M. Rusinowitch. Theorem-proving with resolution and superposition. Journal of Symbolic Computation, pages 21--49, 1991. 102

....calculus for equational clausal logic, for short BFP. Its outstanding property is that it does neither require inferences into right hand sides of equations nor an additional rule on positive literals (positive or equational factoring) like most of the existing Basic Paramodulation calculi [1, 12, 15]. Instead, Basic Factored Paramodulation uses a related inference rule, factored overlap, which applies inferences on positive equations in a very homogeneous way and thus allows for a straight simulation argument. The BFP calculus consists of reflection and factored positive overlap (cf. Section ....

M. Rusinowitch. Theorem-Proving with Resolution and Superposition. Journal of Symbolic Computation, 11(1-2):21--50, 1991.

....in S. In the following, equations are always written in the form s t, such that t s holds. 3.2 An Inference System for Equational Logic In the following we define a superposition inference system I for ground clauses. There exist more refined superposition calculi, for instance [BG] [Ru], our intention, however, is to provide a completeness proof without going too much into the details of the inference system. The system I consists of the following rules: Equality resolution: A, t t B A B where t #t is a maximal occurrence in A B. Superposition left: A B, s ....

Rusinowitch, M. Theorem Proving with resolution and superposition: An extension of the Knuth and Bendix procedure as a complete set of inference rules. J. of Symbolic Computation 11 (1991).

....Brand (1975) was the first to prove that the functional reflexive axioms are not needed, and his proof requires only a very limited form of paramodulation into a variable. The first proof that paramodulation into a variable is not needed at all was given by Peterson (1983) while Hsiang and Rusinowitch (1991) were the first to explicitly put some ordering restrictions on paramodulation. The more restrictive superposition calculus described in (Zhang and Kapur 1988, Zhang 1988) turned out to be incomplete; for a counterexample see Bachmair and Ganzinger (1990) The superposition calculus with equality ....

.... calculus described in (Zhang and Kapur 1988, Zhang 1988) turned out to be incomplete; for a counterexample see Bachmair and Ganzinger (1990) The superposition calculus with equality factoring was introduced by Bachmair and Ganzinger (1990) There are also alternatives to equality factoring: Rusinowitch (1991) weakens some ordering constraints, while merging paramodulation is proposed in (Bachmair and Ganzinger 1990) see also (Pais and Peterson 1991) More recently new paramodulation and superposition calculi have been proposed by Bachmair, Ganzinger, Lynch and Snyder (1995b) and by Nieuwenhuis and ....

M. Rusinowitch, 1991. Theorem proving with resolution and superposition: An extension of the Knuth and Bendix completion procedure as a complete set of inference rules. J. Symbolic Computation 11: 21--49.

....logic programming was made by methods using refinements of inference in the paramodulation based calculus via ordering strategies and deletion of redundant clauses. This technique appeared [KB 70, Sla 74, Lan 75] and has been further developed in [Hul 80, Pet 83, HR 86, Hol 88, ZK 88, BG 90, Rus 91, Soc 92, NR 92b, BG 94] In logic programming, this technique was introduced in many papers, including [Fri 84, Hus 85, Red 85, VE 87, Yam 87, Kap 87, GM 88, Pad 88, FHS 89, GHR 92] especially in the framework of integration of logic and functional programming languages. In this paper we propose ....

M. Rusinowitch. Theorem proving with resolution and superposition: an extension of the Knuth and Bendix completion procedure as a complete set of inference rules. Journal of Symbolic Computations, 11:21--49, 1991.

....clause. But the other premise can be a condition of another program clause. The inference against the goal is not until the end of the proof. A related inference system is one where only a maximal literal in a clause can be used in an inference [16] This was extended to the rstorder case in [25, 13, 23, 3]. It is not always possible to choose an ordering such that this reduces to SLDresolution in the nonequational case. For this strategy, the relationship between refutational completeness and completeness with respect to answer substitutions has been investigated in [21] None of the known proof ....

....It seems to be the natural way to test the other convergence conditions anyway. Another advantage of our method is that there is an algorithm to create the condition for completeness of Basic Conditional Narrowing: saturate the set by Superposition. This is also the case for the methods of [8, 19, 4, 6, 16, 25, 13, 23, 3], which are special cases of our general framework. The following example from [18] illustrates why a set saturated by Superposition is dioeerent from a convergent set: 1. a b 2. a c 3. x b; x c ) b c 4. b c ) This set is convergent but, as [18] shows, Conditional Narrowing is ....

M. Rusinowitch. Theorem-proving with resolution and superposition: an extension of Knuth and Bendix procedure to a complete set of inference rules. In Proceedings of the International Conference on Fifth Generation Computer Systems, 1988.

....rewrite rules, then there is a decision procedure for the conclusion of the clause to be proved: it can be proved or disproved by computing a closed rewrite tree for it, i.e. by giving normal forms for the left and right hand side of the conclusion and comparing them syntactically. Rusinowitch [Rus91] presented a refutationally complete set of inference rules for first order logic with equality based on resolution and superposition, where equations are oriented by a well founded ordering. When restricted to equational logic his strategy reduces to a Knuth Bendix procedure. When restricted to ....

M. Rusinowitch. Theorem-proving with Resolution and Superposition. Journal of Symbolic Computation, 11:21--49, 1991.

....main influence on the contemporary technique of equational logic programming was made by methods using refinements of inference in the paramodulation based calculus via ordering strategies and deletion of redundant clauses. This technique appeared in [33, 48, 35] and has been further developed in [29, 44, 26, 54, 46, 49, 3]. In logic programming, this technique was introduced in many papers, including [50, 53, 32, 21, 17, 23] especially in the framework of integration of logic and functional programming languages. In this paper we propose a new method which uses all three ideas in a new way. We try to make all ....

M. Rusinowitch. Theorem proving with resolution and superposition: an extension of the knuth and bendix completion procedure as a complete set of inference rules. Journal of Symbolic Computations, 11:21--49, 1991.

.... 1993) Calculi and proof procedures for full first order order restricted equational theorem proving were proposed in (Bachmair, 1991; Bachmair and Ganzinger, 1990a; Bachmair and Ganzinger, 1990b; Zhang and Kapur, 1988; Hsiang and Rusinowitch, 1986; Hsiang and Rusinowitch, 1987; Kounalis and Rusinowitch, 1991; Rusinowitch, 1991; Kirchner et al. 1990; Socher Ambrosius, 1990) These systems are basically paramodulation like, however the possible inferences are highly restricted by orderings, which have to be given as an input parameter. More precisely, the given ordering must satisfy the following ....

.... and proof procedures for full first order order restricted equational theorem proving were proposed in (Bachmair, 1991; Bachmair and Ganzinger, 1990a; Bachmair and Ganzinger, 1990b; Zhang and Kapur, 1988; Hsiang and Rusinowitch, 1986; Hsiang and Rusinowitch, 1987; Kounalis and Rusinowitch, 1991; Rusinowitch, 1991; Kirchner et al. 1990; Socher Ambrosius, 1990) These systems are basically paramodulation like, however the possible inferences are highly restricted by orderings, which have to be given as an input parameter. More precisely, the given ordering must satisfy the following properties ( complete ....

Rusinowitch, M. (1991). Theorem-proving with Resolution and Superposition. Journal of Symbolic Computation, 11:21--49.

....modern technique of equational logic programming was made by equational theorem proving methods using refinements of inference in the paramodulation based calculus via ordering strategies and deletion of redundant clauses. This technique appeared in [24, 35, 25] and has been further developed in [20, 32, 37, 34, 3]. In logic programming, it was introduced in many papers, including [14, 22, 31, 15, 17, 1, 30] especially in the framework of the integration of logic and functional programming languages. Further references may be found in [8] In [10] we introduced a new method called equality elimination. ....

M. Rusinowitch. Theorem proving with resolution and superposition: an extension of the Knuth and Bendix completion procedure as a complete set of inference rules. Journal of Symbolic Computations, 11:21--49, 1991.

.... Ganzinger, Lynch, et al. 1992, Nieuwenhuis and Rubio 1992) Traditionally completion procedures were formulated for sets of equations (unit clauses) but ordered completion has been generalized to arbitrary non unit clauses, resulting in several variants of paramodulation called superposition (Rusinowitch 1991, Zhang 1988, Bachmair and Ganzinger 1990, Pais and Peterson 1991) and the basic strategy has actually first been developed for first order clauses. Associativity and commutativity have been built into ordered paramodulation (Paul 1992, Rusinowitch and Vigneron 1991) a calculus (Hsiang and ....

....1991, Zhang 1988, Bachmair and Ganzinger 1990, Pais and Peterson 1991) and the basic strategy has actually first been developed for first order clauses. Associativity and commutativity have been built into ordered paramodulation (Paul 1992, Rusinowitch and Vigneron 1991) a calculus (Hsiang and Rusinowitch 1991) that does not generalize completion, but includes similar rewrite techniques; and Wertz (1992) designed an associativecommutative superposition calculus. Unfortunately, the completeness proofs proposed for these calculi are technically involved and quite complicated. The calculus described in ....

[Article contains additional citation context not shown here]

M. Rusinowitch, 1991. Theorem proving with resolution and superposition: An extension of the Knuth and Bendix completion procedure as a complete set of inference rules. J. Symbolic Computation, Vol. 11, pp. 21--49.

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M. Rusinowitch. Theorem-Proving with Resolution and Superposition. J. of Symbolic Computation, (1-2), 1991.

....the clausal context fc 1 [c 0 1 ] c n [c 0 n ]g. Clausal contexts induce an ordering relation on clauses that we call context subsumption since it is an extension of the classical subsumption ordering. It can be viewed as a generalization of the functional subsumption rule de ned in [32] and is useful for redundancy elimination in rst order theorem proving. De nition 5.5 (context subsumption) The clause C ) contextually subsumes C 0 if there exists a clausal context c and a substitution such that C 0 ) c[ Note that the strict part of this ordering is ....

M. Rusinowitch. Theorem-proving with resolution and superposition. Journal of Symbolic Computation, 11(1&2), 1991.

.... ; u = t 0 ) u) 6 (t) u) 6 ( u = t) 6 (fu 0 = t 0 g [ Table 1. Inference rules of SP We will make use of a superposition calculus, SP , comprising the inference rules of Table 1 and the simpli cation rules of Table 2. SP is taken from [NR01] It extends the system from [Rus91] by the equality factoring rule [BG94] so that more ordering restrictions are possible (in the non Horn case) The relation is a reduction ordering [DJ90] which is total on ground terms. is extended to literals in the following way: a . b) c . d) if fa; bg fc; dg, where is the ....

M. Rusinowitch. Theorem-proving with Resolution and Superposition. JSC, 11(1&2):21-50, January/February 1991.

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M. Rusinowitch. Theorem-proving with Resolution and Superposition. JSC, 11(1&2):2150, January /February 1991.

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M. Rusinowitch. Theorem-proving with Resolution and Superposition. J. Symb. Comp., 11(1&2):21--50, Jan./Feb. 1991.

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M. Rusinowitch. Theorem proving with resolution and superposition: an extension of the Knuth and Bendix completion procedure as a complete set of inference rules. Journal of Symbolic Computations, 11:21--49, 1991.

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Rus91 M. Rusinowitch. Theorem proving with resolution and superposition: an extension of the Knuth and Bendix completion procedure as a complete set of inference rules. Journal of Symbolic Computations, 11:21--49, 1991.

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M. Rusinowitch. Theorem proving with resolution and superposition: an extension of the Knuth and Bendix completion procedure as a complete set of inference rules. Journal of Symbolic Computations, 11:21--49, 1991.

No context found.

M. Rusinowitch. Theorem proving with resolution and superposition: an extension of the Knuth and Bendix completion procedure as a complete set of inference rules. Journal of Symbolic Computations, 11:21--49, 1991.

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