L. Bachmair, H. Ganzinger, C. Lynch, and W. Snyder. Basic paramodulation. Inform. and Computat. , 121(2):172--192, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....this approach is that semantic equality tests are possible. 3.3 Saturation The following inference rules form a sound and refutationally complete calculus for Horn clause sets consisting of declarations and sorted (dis)equations. They are mainly an adaption of basic superposition with selection (Bachmair, Ganzinger, Lynch Snyder 1995, Nieuwenhuis Rubio 1995) to the particular form of the Horn clauses considered here, where the sort constraints are subject to the basic restriction and are solved by a particular selection strategy. This strategy is expressed by the rule Sort Constraint Resolution, see below. As usual, we ....

Bachmair, L., Ganzinger, H., Lynch, C. & Snyder, W. (1995), `Basic paramodulation', Information and Computation 121(2), 172--192.

....languages combining constraint solvers over different data types in truly modular fashion a la Clear OBJ tradition. This needs further 17 In a remotely similar way to the theory resolution of (Stickel, 1985) Also, recent advances in making paramodulation based techniques more effcient (see (Bachmair et al. 1995), for example) have to be incorporated in any system implementing constraint paramodulation. Constraint Logic 31 development of the topic of Section 7.2 in conjunction with constraint paramodulation as operational semantics. Another important research direction is the study of ECLP over ....

Bachmair, L., Ganzinger, H., Lynch, C., and Snyder, W. (1995). Basic paramodulation. Information and Computation, 121(2):172--192.

....previous results by the model generation method. It suffices to modify the rule generation by requiring, when a rule l ) r is generated, that both l and r are irreducible by RC , instead of only l as before, and to adapt the proof of Theorem 5.6 accordingly, which is straightforward. We refer to [Bachmair, Ganzinger, Lynch and Snyder 1995] for a deeper discussion of this form of basic paramodulation. 5.4. Saturation for constrained clauses In this section the redundancy notions for constrained clauses and inferences are defined. The idea is similar to how it was done for unconstrained clauses with variables, except that here, as ....

Bachmair L., Ganzinger H., Lynch C. and Snyder W. [1995], `Basic paramodulation', Information and Computation 121(2), 172--192.

....this approach is that semantic equality tests are possible. 3.3 Saturation The following inference rules form a sound and refutationally complete calculus for Horn clause sets consisting of declarations and sorted (dis)equations. They are mainly an adaption of basic superposition with selection (Bachmair, Ganzinger, Lynch Snyder 1995, Nieuwenhuis Rubio 1995) to the particular form of the Horn clauses considered here, where the sort constraints are subject to the basic restriction and are solved by a particular selection strategy. This strategy is expressed by the rule Sort Constraint Resolution, see below. As usual, we ....

Bachmair, L., Ganzinger, H., Lynch, C. & Snyder, W. (1995), `Basic paramodulation', Information and Computation 121(2), 172-192.

....the notion of deduction with constraints spread, with constraint logic programming (Ja ar and Lassez, 1987) and then constraint programming. The counterpart in theorem proving is deduction with constraints (Kirchner et al. 1990) and complete constraint saturation processes (Vigneron, 1995; Bachmair et al. 1995; Nieuwenhuis and Rubio, 1994) This gives a generalization of the concept of building in speci c theories, it allows speci c treatment of the constraints and at last it generalizes to various constraint languages. Besides this idea of building in part of the equality in general refutation ....

Bachmair, L., H. Ganzinger, C. Lynch, and W. Snyder: 1995, `Basic Paramodulation '. Information and Computation 121(2), 172-192.

....constraints we need slightly more complicated notions of redundancy for clauses and inferences, which consider only reduced instances of clauses. An instance Coe of a clause C is called reduced with respect to a rewrite system R if oe is irreducible with respect to R (Nieuwenhuis and Rubio 1992, Bachmair, Ganzinger, Lynch and Snyder 1993). The ground instance Coe with maximal term t is called redundant in N (with respect to ZMod) if for any rewriting system R such that oe is reduced with respect to ZMod [ R there exist ground instances C 1 oe 1 ; C k oe k of clauses C 1 ; C k in N which are reduced with respect to ....

Bachmair, L., Ganzinger, H., Lynch, C. and Snyder, W. (1993). Basic paramodulation. Technical Report MPI-I-93-236, Max-Planck-Institut fur Informatik, Saarbrucken.

.... paramodulation (Robinson and Wos, 1969) and its refinements, like rewriting and Knuth Bendix completion based techniques in pure equational reasoning, and ordered (Hsiang and Rusinowitch, 1991; Bachmair and Ganzinger, 1994) and basic paramodulation and superposition (Nieuwenhuis and Rubio, 1995; Bachmair et al. 1995) for general clauses. These techniques for building in the equality predicate apply to any set of clauses, but again special treatments for some equational subset of the axioms are usually worthwhile. Historically, these special treatments were motivated by the fact that equations like the ....

....inferences can be obtained in both frameworks, but that our version of the model generation method provides simpler proofs than the semantic tree method. Furthermore, the known extensions to the model generation method for constrained paramodulation like redex orderings and variable abstraction (Bachmair et al. 1995) can also be smoothly incorporated here. Regarding simplification and other redundancy methods, the abstract redundancy notions given here express sharp bounds on the existing concrete redundancy methods (like the ones given in Vigneron, 1994 , which indeed fit into our abstract ones) 2. Basic ....

[Article contains additional citation context not shown here]

Bachmair, L., Ganzinger, H., Lynch, C., Snyder, W. (1995). Basic paramodulation. Information and Computation, 121(2):172--192.

.... unification rules were derived (cf. also the techniques for dealing with shallow equations and rewrite rules given in [Christian, 1992, Domenjoud, 1993] Here we proceed in a completely different way, namely by a careful termination analysis of basic paramodulation [Nieuwenhuis and Rubio, 1995, Bachmair et al. 1995] for Horn clauses with equality. Basicness means in this context that no inferences are needed on certain blocked terms of the clauses, typically the terms created in unifiers of previous inferences, as in basic narrowing . Completeness results have been given for several basic strategies. The ....

....a variable. The equality constraints express in a natural way the basicness restriction: no inferences take place on subterms introduced by unifiers of previous inferences. In this paper, assume that in every clause C having at least one negative literal, one such negative literal is selected [Bachmair et al. 1995], i.e. all inferences with C as a premise are paramodulations and equality resolutions on this literal. This implies for the Horn case that in all paramodulation inferences the left premise is a positive unit clause (hence the part C 0 of the left premise is empty) Furthermore, depending on the ....

[Article contains additional citation context not shown here]

Bachmair, L., Ganzinger, H., Lynch, C., and Snyder, W. (1995). Basic paramodulation. Information and Computation, 121(2):172--192.

....As a consequence, the usual simplification methods (reduction, subsumption) need to be somewhat restricted. On the other hand, a clause can be eliminated if its substitution part is reducible by eqtp.tex; 19 02 1998; 17:22; p.38 EQUATIONAL REASONING 39 another equation in the system. We refer to (Bachmair et al. 1995; Nieuwenhuis and Rubio, 1995) for a detailed exposition of the subject. In (Bachmair et al. 1995) there is also a discussion of variable abstraction techniques that allow one to expand the substitution part of clauses. Redex orderings can be used to mark positions in the conclusion of a ....

....On the other hand, a clause can be eliminated if its substitution part is reducible by eqtp.tex; 19 02 1998; 17:22; p.38 EQUATIONAL REASONING 39 another equation in the system. We refer to (Bachmair et al. 1995; Nieuwenhuis and Rubio, 1995) for a detailed exposition of the subject. In (Bachmair et al. 1995) there is also a discussion of variable abstraction techniques that allow one to expand the substitution part of clauses. Redex orderings can be used to mark positions in the conclusion of a superposition inference that precede (in the redex ordering) the position at which the inference takes ....

[Article contains additional citation context not shown here]

Bachmair, L., H. Ganzinger, C. Lynch, and W. Snyder: 1995, `Basic Paramodulation '. Information and Computation 121(2), 172--192.

....us to do that. When terms contain variables, each inference involves a renaming and a constraint. It is still true that all of the equations created are formed from pieces of the initial ones, but some of the equations have renamings and constraints applied to them. We use the Basic Completion [2, 3] inference rules. The idea of Basic Completion is that whenever we perform an inference, instead of applying the most general unifier to the conclusion of the inference, we save the unification problem as an equational constraint. This is the first idea necessary for the SOUR graph philosophy. ....

....in S if there exist equations C 1 ; Delta Delta Delta ; Cn 2 S such that 1. for all i, C i C , 2. C is true whenever C 1 ; Delta Delta Delta ; Cn are true. For a non ground term, we give a sufficient definition of redundancy, which is an instance of the one defined by Bachmair et al. [2]. If C [ is a non ground term, we say that C [ is redundant in S if for all instances Coe of C [ where oe = mgs( there exist C 1 [ 1 ] Delta Delta Delta ; Cn [ n ] 2 S and instances C i oe i of C i [ i ] where oe i mgs( i ) for each i such that 1. for all i, C ....

[Article contains additional citation context not shown here]

Bachmair, L., Ganzinger, H., Lynch, C. and Snyder, W. (1995). Basic paramodulation. Information and Computation, 121(2):172--192.

....us to do that. When terms contain variables, each inference involves a renaming and a constraint. It is still true that all of the equations created are formed from pieces of the initial ones, but some of the equations have renamings and constraints applied to them. We use the Basic Completion [2, 3] inference rules. The idea of Basic Completion is that whenever we perform an inference, instead of applying the most general unifier to the conclusion of the inference, we save the unification problem as an equational constraint. This is the first idea necessary for the SOUR graph philosophy. ....

....of ground equations, then C is redundant in S if there exist equations C 1 # ####C n2S such that 1. for all i, C i #C, 2. C is true whenever C 1 # ####C n are true. For a non ground term, we give a sufficient definition of redundancy, which is an instance of the one defined by Bachmair et al.[2]. IfC ## # ## is a non ground term, we say that C ## # ## is redundant in S if for all instances C# of C ## # ## where # # mgs###,thereexistC 1 ## # 1 ### ####C n ### n ## 2S and instances C i # i of C i ## # i ## where # i mgs## i # for each i such that 1. for all i, C i # i #C#, 2. C# is true ....

[Article contains additional citation context not shown here]

Bachmair, L., Ganzinger, H., Lynch, C. and Snyder, W. (1995). Basic paramodulation. Information and Computation, 121(2):172--192.

....B S , or simply B (the Basic chaining calculus) 3.3 Refutational Completeness for Horn Clauses Clauses with at most one positive literal are called Horn clauses. The calculi B S are refutationally complete for such clauses. For the completeness proof we adapt the model construction approach of Bachmair and Ganzinger (1990) (see also Bachmair and Ganzinger 1994b) Given a set N of ground clauses, we define a corresponding Herbrand interpretation I using induction on . More precisely, we define for each clause C an interpretation I C , intended to be a model for clauses smaller than C, and a set EC that is ....

....the concrete set of built in axioms L is clear from the context. For inference systems in which the first premise of a binary ground inference is smaller than the second, the the definition of redundancy given here is equivalent to notions of redundancy that we have proposed in previous work (Bachmair and Ganzinger 1990, Bachmair and Ganzinger 1994b) With the present definition we may admit larger classes of clause orderings. For a refutationally complete inference system we expect that if N is saturated up to redundancy then N is unsatisfiable if and only if the empty clause is in N . That leaves us with the ....

[Article contains additional citation context not shown here]

L. Bachmair, H. Ganzinger, C. Lynch and W. Snyder, 1995b. Basic paramodulation.

....may not always be removed. But they may be restricted from being involved in inferences with goal clauses. We give conditions showing when they are allowed and examples illustrating why they are not always allowed. Also, we show that our inference system can be combined with a Basic strategy [12, 5, 6, 20, 22] so that goal clauses are solved without allowing inferences into substitution positions. Another inference system for equational Horn clauses is Conditional Narrowing [18] This inference system is the combination of SLDresolution and narrowing. Since no completion is involved, this is only ....

L. Bachmair, H. Ganzinger, C. Lynch, and W. Snyder. Basic Paramodulation. In Proc. 11th Int. Conf. on Automated Deduction, Lect. Notes in Artiøcial Intelligence, vol. 607, pp. 462476, Berlin, 1992. Springer-Verlag.

....in C [ We assume a well founded reduction ordering , total on ground terms, identi ed with its multiset extension. We call a literal A maximal in C [ if A 2 C and there is a solution oe of such that Aoe Boe for all B 2 C. We now present the Basic Paramodulation inference rules [4, 13]. They dioeer from the standard Paramodulation inference rules in that the most general uni er is not applied to the conclusion of the inference. Instead, it is saved as an equational constraint in the conclusion of the inference. This is more restrictive than the standard inference rules, ....

....C and a function f , we say that the length of I over C is bounded by f if for all S 0 2 C, the longest Itheorem proving derivation from S 0 has length less than or equal to f(n) where n is the number of symbols in S 0 . An inference procedure I decides a class C if the length of I over 4 See [4] for a de nition of reduced relative to. C is bounded by some function f and there is some recursive procedure to calculate each S i 1 from S i for each such sequence. An inference procedure I polynomially decides C if the length of I over C is bounded by a polynomial f and S i 1 can be ....

[Article contains additional citation context not shown here]

L. Bachmair, H. Ganzinger, C. Lynch, and W. Snyder. Basic Paramodulation. In Proc. 11th Int. Conf. on Automated Deduction, Lect. Notes in Artiøcial Intelligence, vol. 607, pp. 462476, Berlin, 1992. Springer-Verlag. To appear in Journal of Information and Computation.

.... is heavily indebted, may be found in [Nutt, R ety and Smolka 1989] In the next two sections we present further refinements which may also be found in [Bockmayr, Krischer and Werner 1992] and [Nutt et al. 1989] For a comprehensive study of basic inference systems, the reader is referred to [Bachmair, Ganzinger, Lynch and Snyder 1995] and to [Chapter paramodulation] 4.4.3. Redex orderings and variable abstraction One of the useful properties of convergent systems mentioned above is that any strategy which can find a redex in a reducible term is sufficient for reducing terms to normal form, and hence for generating rewrite ....

....Reduce rule, we observe that in the context of B, Reduce may instantiate terms into r that are known to be reduced; Propagation can remove these again. The combination of Reduction with Eager Propagation effectively gives us the more complex form of basic simplification described for example in [Bachmair et al. 1995] and [Nutt et al. 1989] see also [Chapter paramodulation] 4.4.4. Failure rules Unlike our presentation of the calculus U , we have chosen here not to present failure rules from the outset, in order to highlight the essential issues first. The conditions under which sequences may fail are of ....

Bachmair L., Ganzinger H., Lynch C. and Snyder W. [1995], `Basic paramodulation', Information and Computation 121(2), 172--192.

No context found.

L. Bachmair, H. Ganzinger, C. Lynch, and W. Snyder. Basic paramodulation. Inform. and Computat. , 121(2):172--192, 1995.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC