L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, pp. 19--99. Elsevier, 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....that point on, the field of automated reasoning never looked forward. Resolution provides a simple inference rule for refutational proofs for first order statements in skolemized, prenex form. It spawned a multitude of strategies, heuristics, and extensions. Nearly forty years later, resolution [BG01] remains extremely popular as a general purpose proof search method primarily because the basic method can be implemented and extended with surprising ease. Resolution based methods have had some success in proving open problems in certain domains where general purpose search can be productive. ....

Leo Bachmair and Harald Ganzinger. Resolution theorem proving. In Robinson and Voronkov [RV01], pages 19--99.

....assigned to i j. This permits an efficient search for opposed formulas during the application of the (RES) rule. Finally, we use a special kind of tries to optimize subsumption checking (see below) Inference Rules. HyLoRes actually implements a version of ordered resolution with selection [5], where the application of the (RES) and (BOX) rules are restricted to certain selected formulas in the clause. Ordered resolution with selection greatly diminish the size of the saturated set, forbidding the generation of certain clauses, without compromising the completeness of the calculi. ....

....selection greatly diminish the size of the saturated set, forbidding the generation of certain clauses, without compromising the completeness of the calculi. Interestingly, the (still unpublished) proof of completeness of ordered resolution with selection for H ( #) closely follows the proof in [5], based on a step by step construction of a Herbrand model for any consistent input clause set. Once more, hybrid logics seem to provide the appropriate framework to merge first order and modal ideas. We are currently investigating the effect of different orders and selection functions on ....

L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19--99. Elsevier Science, 2001.

....inductive theorems, and thus to detect new redundancies. Section 6 concludes the paper. 2 Preliminaries Let us rst introduce the main notations used in the paper. For full or missing de nitions about term rewriting, we refer to [14] and for theorem proving in automated reasoning, we refer to [6]. Let A A be a binary relation on a set A. We denote the inverse of by , the symmetric closure by , the transitive closure by , the re exive and transitive closure by , and the re exive, symmetric and transitive closure by . We say that is con uent if, for every a; b; c ....

....axiomatization A of IE is a nite recursive set of purely universal formulas such that IE j= A, IE is the only Herbrand model of E [ A up to isomorphism, and for all ground terms s; t representative of its congruence class of IE , s 6 t ) A j= s 6= t. The method relies on saturation techniques [5, 6] for performing the proof by consistency of C [ A [ E, thus any saturationbased general purpose rst order theorem prover can be used for inductive validity. The (in)consistency proofs are performed in two stages: rst deductions on C [ E are computed by saturation, yielding new consequences; ....

L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume 1, chapter 2, pages 19-99. Elsevier Science, 2001.

....exists a substitution s such that D 1 s D 2 and :C 1 s :C 2 . A step clause C ) D is a tautology if D is a tautology. Note that, since we do not have negative occurrences to the left hand side of step clauses, C cannot be false) Tautologies are deleted. We adopt the terminology from [2]. A (linear) proof by fine grained resolution of a clause C from a set of clauses S is a sequence of clauses C 1 ; Cm such that C =Cm and each clause C i is either an element of S or else the conclusion by a deduction rule from C 1 ; C i 1 . A proof of false is called a refutation. A ....

L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, chapter 2, pages 19--99. Elsevier, 2001.

....1. is a liftable ordering. 2.2 Decidability result Theorem 1 (decidability of C) Let S be a nite set of clauses such that S belongs to C. The satis ability of S is decidable. Proof sketch We use splitting (see e.g. 26] ordered factorization and ordered binary resolution (see e.g. [2]) w.r.t. the partial ordering de ned above, using a classical redundancy criterion [2] also called a posteriori criterion in e.g. 15] we apply resolution on two clauses C 1 and C 2 only if no atom of the resolvent is greater than the resolved atom. Such an ordered strategy is complete [2, ....

....C) Let S be a nite set of clauses such that S belongs to C. The satis ability of S is decidable. Proof sketch We use splitting (see e.g. 26] ordered factorization and ordered binary resolution (see e.g. 2] w.r.t. the partial ordering de ned above, using a classical redundancy criterion [2], also called a posteriori criterion in e.g. 15] we apply resolution on two clauses C 1 and C 2 only if no atom of the resolvent is greater than the resolved atom. Such an ordered strategy is complete [2, 15] It only remains to show termination. First, after splitting, we only generate clauses ....

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume 1, chapter 2. North Holland, 2001.

....we allow arbitrary inferences between initial and universal clauses, the result is an initial clause. Universal clauses may subsume initial clauses but not vice versa. We define the notions of a derivation by fine grained resolution and a saturation procedure in a way similar to, e.g. [2]. This restriction justifies skolemisation: Skolem constants and functions do not sneak in the left hand side of step rules, and, hence, Skolem constants from different moments of time do not mix. We do not need to restrict the unifier s in this case since the restriction holds ....

L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, chapter 2, pages 19--99. Elsevier, 2001.

....the things that will be improved in the next versions of HyLoRes are the following. We are investigating both the theoretical and practical issues involved in performing direct ordered resolution for hybrid logics, where the resolution rules are restricted to the maximum literals in the clauses [8]. We want to make the prover much more aware of the characteristics of its input. At the moment, the prover simply check which formulas appear in the input (propositional, modal, basic hybrid, and binders) and uses the appropriate rules of the calculus. Particular rules heuristics for certain ....

L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19--99. Elsevier Science, 2001.

....clauses are assumed to be equal. We say an expression is functional if it contains a constant or a non nullary function symbol. Otherwise it is called non functional. Resolution. Now, we briefly recall the definition of ordered resolution extended with a selection function from Bachmair et al. [2, 3, 4]. Derivations are controlled by an admissible ordering and a selection function. Basically the idea is that inferences are restricted to literals maximal under the ordering while the selection function is used to override the ordering, and give preference to inferences with negative ....

....R forms a complete refutation system for clause sets. In general, the calculus R can be enhanced with standard simplification rules such as tautology deletion and subsumption deletion, in fact, it can be enhanced by any simplification rule which is compatible with a general notion of redundancy [3, 4]. Essentially, a ground clause is redundant in a set N with respect to the ordering if it follows from smaller instances of clauses in N , and a non ground clause is redundant in N if all its ground instances are redundant in N . A set N of clauses is saturated up to redundancy with respect to ....

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19--99. Elsevier, 2001.

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Leo Bachmair and Harald Ganzinger. Resolution theorem proving. In Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, volume 1, chapter 2, pages 19 100. North Holland, 2001.

....nitions more eciently, attempting to recover equivalences from clauses and treating them with speci c selection strategies for inferences. Admitting inferences on nonclausal formulas, the calculus described below is also related to the various calculi of nonclausal resolution and superposition [11, 12, 2, 4]. The main di erence here is that we also admit quanti ers, and that the local nature of classical superposition inferences is better preserved, facilitating the development of ecient implementations of our calculus in a theorem prover. In [1] the author argues, by discussing some examples, that ....

....Superposition, which leads to a syntactical variant of standard ordered resolution with selection. 6 Refutational Completeness We prove the refutational completeness of ES in the presence of strong redundancy criteria using the reduction of counterexamples framework of Bachmair and Ganzinger [4]. We assume that the reader is familiar with this method. We will construct a Herbrand model by well ordered recursion over on closed clauses. We construct a rewrite system R that contains rules of the form l ) r where either l and r are ground terms, or where l is a nonequational ground atom ....

Leo Bachmair and Harald Ganzinger. Resolution theorem proving. In Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, volume 1, chapter 2, pages 19-100. North Holland, 2001.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, pp. 19--99. Elsevier, 2001.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In J. A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier, 2000. To appear.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, pages 19--99. Elsevier, 2001.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19--99. Elsevier Science, 2001.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19--99. Elsevier Science, 2001.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19--99. Elsevier, 2001.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In Robinson and Voronkov [RV01], chapter 2, pages 19--99.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In Robinson and Voronkov [16], chapter 2, pages 19--99.

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L. Bachmair and H. Ganzinger. Resolution Theorem Proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19-99. Elsevier Science, 2001.

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L. Bachmair, H. Ganzinger, Resolution theorem proving, in: A. Robinson, A. Voronkov (Eds.), Handbook of Automated Reasoning, Elsevier, 2001, Ch. 2, pp. 19--99.

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Leo Bachmair and Haral Ganzinger. Resolution theorem proving. In Alan Robinson and Andrei Voronkov, editors, Handbook of Automated Reasoning, volume I, chapter 2, pages 19-99. Elsevier Science Publishers (North-Holland), Amsterdam, 2001.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, chapter 2, pages 19--99. Elsevier, 2001.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In J. A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning. Elsevier, 2000. To appear.

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L. Bachmair and H. Ganzinger. Resolution theorem proving. In A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume 1, chapter 2, pages 19-100. North Holland, 2001.

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Bachmair, L. and Ganzinger, H., Resolution Theorem Proving, in Robinson, A. and Voronkov, A., editors, The Handbook of Automated Reasoning, chapter 2, volume I, 19--99, Elsevier Science Publishers, 2001.

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