L. Bachmair and H. Ganzinger. A theory of resolution. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier Science and MIT Press, 2000. To appear.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[6] When the initial set of clauses is Horn, our strati ed resolution with redundancies becomes SLDresolution. The possibility for arbitrary selection for Horn clauses, and even in the case of equational logic was proved in [7] For one of our proofs we used a renaming technique introduced in [2, 3]. Acknowledgments The idea of this paper has appeared due to discussions of the second author during CADE 16 with Harald Ganzinger and Tanel Tammet. Harald Ganzinger explained that the Saturate system, tries to use an ordering under which the heads of de nition like clauses are the greatest ....

....we write L L if L L or L = L and write L C if for every literal M 2 C we have L M . In this paper we assume that always denotes a xed well founded ordering. We will now de ne a notion of selection function which makes a slight deviation from the standard notions (see e.g. [3]. However, the resulting inference system will be standard, and several following de nitions will become simpler. We call a selection function any function on the set of clauses such that (i) C) is a set of literals in C, ii) C) is nonempty whenever C is nonempty, and (iii) if A 2 (C) and ....

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L. Bachmair and H. Ganzinger. A theory of resolution. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier Science and MIT Press, 2000. To appear.

....can be lifted to literals by associating with every positive literal A the multiset fAg and with every negative literal :A the multiset fA; Ag, and comparing these by the multiset extension of 0 . We denote the resulting ordering by . The ordering is an admissible ordering in the sense of Bachmair and Ganzinger (1997). Thus, the following two inference rules provide a sound and complete calculus for first order logic in clausal form: Res true ) C A true ) D :B true ) C D)oe where (i) CA and D:B are simple or modal clauses, ii) oe is the most general unifier of A and B, iii) Aoe is strictly maximal ....

Bachmair, L. and Ganzinger, H. (1997). A theory of resolution. Research report MPII -97-2-005, Max-Planck-Institut fur Informatik, Saarbrucken, Germany. To appear in J. A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning.

.... ) h i h i [ h i h i h i [ for 2 f ; g 4. First order resolution We brie y describe the general ordered resolution calculus R (with selection and simpli cation) of Bachmair and Ganzinger [5, 6, 7, 8]. In the calculus inference rules are parameterised by an admissible ordering on literals and a selection function S. Essentially, an admissible ordering is a total (well founded) strict ordering on the ground Resolution for Testing Modal Satis ability and Building Models 7 level such that for ....

....strict ordering on the ground Resolution for Testing Modal Satis ability and Building Models 7 level such that for literals: A n A n : A 1 A 1 . This is extended to the non ground level in a canonical manner. For the exact de nition the interested reader may refer to [5, 6, 7, 8]. A selection function assigns to each clause a possibly empty set of occurrences of negative literals. If C is a clause, then the literal occurrences in S(C) are selected. No restrictions are imposed on the selection function. The calculus consists of general expansion rules of the form: N N 1 ....

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Bachmair, L. and H. Ganzinger: 1997, `A Theory of Resolution'. Research Report MPI-I-97-2-005, Max-Planck-Institut fur Informatik, Saarbrucken, Germany.

....We then present several applications of our results. Finally, we briefly consider a number of refinements and extensions to other search strategies. Kernel resolution: Review Kernel resolution can be seen as the consequence finding generalization of ordered resolution (Fermuller et al. 1993; Bachmair and Ganzinger 1999). We assume a total ordering o = x 1 , x n of the propositional variables P, so called atom ordering or A ordering. We speak of x i being earlier or smaller in the ordering than x j just in case i j. We also use l i for either x i or x i ; the ordering is extended to literals in ....

....are L acceptable, and L kernel resolution finds all prime implicates of #. For L# , only resolvents whose skip is empty are acceptable; thus, we can only generate acceptable resolvents by resolving on the smallest literal of each clause. This is simply ordered resolution in the sense of (Bachmair and Ganzinger 1999), a satisfiability method which has been recently shown by (Dechter and Rish 1994) to be quite e#cient on problems with low induced width, on which it outperforms Davis Putnam backtracking by orders of magnitude. For LK , only resolvents whose skip has at most K literals are acceptable. In ....

L. Bachmair and H. Ganzinger. A theory of resolution. In J.A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier, 1999.

....programming. The complexity of most of these BE procedures can be analysed in terms of a single structural parameter, induced width. Second, DR is, specifically, the bucket elimination version of propositional ordered resolution (in the sense of atom ordering, see e.g. Ferm uller et al. 1993; Bachmair Ganzinger 1999)) itself a rich source of complexity results (Basin Ganzinger 1996; Ferm uller et al. 1993) Finally, del Val 1999) has shown how to extend ordered resolution (with or without BE) to consequence finding tasks, such as finding consequences that contain only certain literals, or with bounded ....

Bachmair, L., and Ganzinger, H. 1999. A theory of resolution.

....various algebraic and rewriting techniques may be conveniently applied, and we can also consider de ning more general notions of redundancy and introducing more powerful simpli cation rules to speed up the proof. This outline has been shown to be successful in classical rst order theorem proving[1, 7, 9, 10]. We believe that it is true for some non classical cases(e.g. annotated logics) too. This paper is an initial try on this. We haven t done the experimental work so far, which should be done after a further investigation, to indicate at least in which sense the method is wellsuited for ....

....did. We have to nd new ways to do that. Secondly, when an atom is assigned to a special value, some other atoms may disappear at the same time. We have to know how to track them especially in the case that a partial ordering between atoms is introduced. Like most of theorem proving systems[1, 3, 9, 10, 26], the following factor computation is necessary. Factoring: c, is an mgu of atoms p i (i 1) occurring in c, c ) We call (c ) a factor of c. We see that single overlaps are employed in superposition instead of multiple overlaps. Furthermore, we have Theorem 4.1.1 Superposition is ....

Bachmair, L., Ganzinger, H., A Theory of Resolution, in Robinson, J. and Voronkov, A., editors, Handbook of Automated Reasoning, Elsevier, Amsterdam (1997).

....are ordering refinements (see, for example, Fermuller et al. 1993, Joyner Jr. 1976, de Nivelle 1998, Hustadt and Schmidt 1998, 1999b) Notably, ordering refinements have been in the focus of research on automated deduction by resolution independent of the consideration of decidability issues (Bachmair and Ganzinger 1997). In the above example, the non terminating derivation involves unordered inferences which ordered resolution strategies do not generate (for any atom ordering which is compatible with the subterm ordering on terms) Transitivity did not come into the above derivation. In the presence of ....

Bachmair, L., and H. Ganzinger. 1997. A Theory of Resolution. Research Report MPI-I-97-2-005. Saarbrucken, Germany: Max-Planck-Institut fur Informatik.

.... in a clause are selected and, hence, all positive premises (called electrons ) and also the conclusion must be positive clauses (without negative literals) The refutational completeness of resolution with arbitrary selection functions is not difficult to show, cf. Bachmair and Ganzinger, 1997b; Bachmair and Ganzinger, 1990). The reflexivity axiom is the only congruence axiom that can be used as an electron in a hyper resolution inference. But if it is the only electron we obtain only instances of reflexivity that can be ignored, such as xx uy f ( u; f ( z; f ( x; f ( ....

....position of a variable x in C 0 . Let t = s[u[t] x] be a more reduced substitution than s. Then C 0 t is also a clause in g(M) but smaller than C, and fC 0 t; D s tg j= D C[t] 10 That is, the ground inference is redundant in g(M) Summarizing the above results we obtain: THEOREM 9. (Bachmair and Ganzinger, 1990). i) Let N be a set of ground clauses that is saturated up to redundancy with respect to S . Then N is unsatisfiable if and only if N contains the empty clause. ii) If M is the limit of a fair theorem proving derivation with respect to S then g(M) is saturated up to redundancy with respect to ....

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Bachmair, L. and H. Ganzinger: 1997b, `A Theory of Resolution'. Research Report MPI-I-97-2-005, Max-Planck-Institut fr Informatik, Saarbrcken. To appear in the Handbook of Automated Reasoning.

....L. Bachmair and H. Ganzinger Additional Key Words and Phrases: Chaining calculi, equational logic, reduction orderings, rewrite systems, term rewriting, transitive relations 1. INTRODUCTION Resolution, in its different variants, forms the core of many current automated reasoning systems, see Bachmair and Ganzinger [1997] for a recent exposition of the theory of resolution. By and large such resolution refinements as hyper resolution, ordered resolution, or the set of support strategy are useful in practice, but for theories with transitive relations, such as logics with equality or inequality relations, they are ....

....law, then the only inferences involving transitivity are of the form C ; u S s D ; t S v : x S y) y S z) x S z Coe ; Doe ; uoe S voe where oe is a most general unifier of s and t. The key observation here is that it is not necessary to resolve with the non selected, positive literals; see Bachmair and Ganzinger [1997] for details. In essence this amounts to an encoding of transitivity as an inference rule: Ordered Chaining Calculi Delta 5 Chaining. C ; u S s D ; t S v Coe ; Doe ; uoe S voe where oe is a most general unifier of s and t. This chaining rule, which was already proposed by Slagle [1972] is ....

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Bachmair, L. and Ganzinger, H. 1997. A theory of resolution. Research Report MPI-I97 -2-005, Max-Planck-Institut fur Informatik, Saarbrucken, Saarbrucken. To appear in the Handbook of Automated Reasoning.

....and a selection function S such that N is saturated with respect to 0 and S. Hyper resolution is refutationally complete for any atom ordering and selection function S. That is, a set of clauses N that is saturated up to redundancy is inconsistent if and only if it contains the empty clause [Bachmair and Ganzinger 1997]. To compare saturation by hyper resolution with order locality, we shall use particular kinds of selection functions. Let be a term ordering. S is said to be based on if, for any clause C, S(C) is the set of all negative atoms in C that contain a term that is maximal in C with respect to . ....

Bachmair, L. and Ganzinger, H. 1997. A theory of resolution. Research Report MPI-I97 -2-005, Max-Planck-Institut fur Informatik, Saarbrucken, Saarbrucken. To appear in the Handbook of Automated Reasoning.

....remains complete if, for any potential hyper inference from side premises C 1 , C k and main premise D, rather than deriving the full conclusion, we derive any don t care non deterministically chosen partial conclusion. A proof of this fact in the non equational case has been given in (Bachmair Ganzinger 1997), and the proof does not dependent on any properties that are critical when adding equality. The criterion for which partial conclusion to choose is simply that the conclusion should be a guarded clause. With this modification of the calculus, the class of guarded clauses is closed under its ....

Bachmair, L. & Ganzinger, H. (1997), A theory of resolution, Research Report MPI-I-97-2-005, Max-Planck-Institut fur Informatik, Saarbrucken, Saarbrucken. To appear in the Handbook of Automated Reasoning.

....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 ....

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L. Bachmair and H. Ganzinger, 1995b. The theory of resolution. In preparation.

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