L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. Journal of Logic and Computation, 4:217{ 247, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... most successfull methods for automated deduction with equality [BG98,NR01] For many years all known completeness results for Knuth Bendix completion and ordered paramodulation required the term ordering to be well founded, monotonic and total (or extendable to a total ordering) on ground terms [HR91,BDH86,BD94,BG94], until in [BGNR99] the monotonicity requirement was dropped and well foundedness and the subterm property were shown to be sucient (note that any such ordering can be totalized without loosing these two properties) After this, the fundamental question arises whether more requirements can be ....

....by all rules in E, and we de ne B = Right(R) as the set of B terms. We call a B position of t to any position in PB (t) and a non B position to any position of t not in PB (t) We denote by B s the set of terms t in B s.t. s . t. The way we de ne R is quite similar to the de nition in [BG94] and especially the one in [BGNR99] The main di erence is that only non B positions are required to be irreducible. As a consequence R can be non terminating and overlapping. However, if the ordering is a west ordering, i.e. it ful ls the subterm property as well, taking as , the generated ....

Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. Journal of Logic and Computation, 4(3):217-247, 1994.

.... most successful methods for automated deduction with equality [BG98,NR01] For many years all known completeness results for Knuth Bendix completion and ordered paramodulation required the term ordering to be well founded, monotonic and total (or extendable to a total ordering) on ground terms [HR91,BDH86,BD94,BG94], until in [BGNR99] the monotonicity requirement was dropped and well foundedness and the subterm property were shown to be sufficient (note that any such ordering can be totalized without loosing these two properties) After this, the fundamental question arises whether more requirements can be ....

....(which includes the subterm relation) for comparing the left hand sides of the rules is crucial in order to ensure that a rule in E is included in R if and only if it is irreducible at non B positions by the other rules in R. The way we define R is quite similar to the definition in [BG94] and especially the one in [BGNR99] The main difference is that only non B positions are required to be irreducible. As a consequence R can be non terminating and overlapping. However, if the ordering is a west ordering, i.e. it fulfils the subterm property as well, taking as , the generated R ....

Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

....is easily seen also to be a counterexample to strong completeness of unrestricted cg resolution. Ordered resolution predates cg resolution and is attributed to Reynolds [18] Variations have been proven complete by Kowalski and Hayes [13] by Joyner [12] and by Bachmair and Ganzinger [2]. Bachmair and Ganzinger also proved strong completeness. Ordered resolution is the basis of modern saturation based calculi that are used in many successful automated deduction systems. Ordered resolution is related to cg resolution insofar as it was shown to be a special case of cg resolution in ....

....q is not the head of A. Thus, before the subsumption deletions were made, q was fully linked in both A and C. Thus, the deletions would have made both occurrences of q pure. Hence, A would also be deleted by the pure rule. The proof method for the next theorem is due to Bachmair and Ganzinger [2]; the proof presented below is a refinement of an elegant exposition by Paliath Narendran. An interpretation is described as a set of atoms, indicating that atoms in the set are assigned true and atoms not in the set are assigned false. be an unsatisfiable semi full connection graph that is ....

Bachmair, L. and Ganzinger, H., Rewrite-based equational theorem proving with selection and simplification, Journal of Logic and Computation 4(3), 217--247, 1994.

....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. Rewrite-based equational theorem proving with selection and simpli cation. Journal of Logic and Computation, 4(3):217-247, June 1994.

....organization of the search process. In the next section we describe our approach to learning and cover knowledge acquisition, representation, and application. Section 4 contains experimental results, and we conclude in section 5. 2 Superposition Based Theorem Proving The superposition calculus [BG94,BG98] and its variants are generally recognized as the most powerful calculi currently available to tackle theorem proving problems with equality. The well known prover SPASS [WAB 99] and our own system E [Sch99,Sch01] are based on particular instances of this calculus, and all other leading ....

L. Bachmair and H. Ganzinger. Rewrite-Based Equational Theorem Proving with Selection and Simpli cation. Journal of Logic and Computation, 3(4):217-247, 1994.

.... techniques used in work on automated theorem proving such as Kowalski s Clausal Graphs [14] which formed the basis of the Markgraf Karl Refutation Procedure [19] the implication graph can be viewed as a special case of this graph) and some rewrite based simplification methods [11] [3]. However, the methods used there are much more sophisticated and focus on aspects of first order theories that do not come up in the propositional setting in which we work. Acknowledgments: I am grateful to Yefim Dinitz and Avraham Melkman for their help and advice on graph algorithms and for ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

....paper is formed by sections 3 and 4, which analyze the static and dynamic aspects of the shared term representation, respectively. We conclude in section 5. 2 Unfailing Completion and Equational Theorem Proving E and Waldmeister are based on di erent, but related calculi. E is a superposition [BG94] prover for full clausal logic with optional selection of negative literals. Waldmeister is based on unfailing completion [BDP89] extended with narrowing to be able to deal with existentially quanti ed variables in the goal. Since non ground goals are handled di erently by both provers, we ....

L. Bachmair and H. Ganzinger. Rewrite-Based Equational Theorem Proving with Selection and Simpli cation. Journal of Logic and Computation, 3(4):217-247, 1994.

....and extensive experimental work. In this paper we will describe how E is constructed and which aspects of the prover are responsible for its power. There are four major elements that form the conceptual core of the prover: Calculus: E is based on a variant of the superposition calculus [4] with literal selection. Superposition is generally recognized as one of the most powerful calculi for proof problems with equality. The major reason for this is the compatibility with a wide variety of redundancy elimination criteria. E implements most known redundancy elimination techniques, as ....

....of the four major items mentioned above. We then discuss some aspects of the performance characteristics of the prover. The paper concludes with some sentences about possible future improvements. 2. Calculus E implements the calculus SP, a variant of the superposition calculus E described in [4] with a slightly di erent notion of literal selection and explicit inference rules for simpli cation. Pure superposition (as used in E) is a refutation based saturating calculus operating on the equational representation of formulae in clause normal form . The proof state is represented by a ....

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L. Bachmair and H. Ganzinger. Rewrite-Based Equational Theorem Proving with Selection and Simpli cation. Journal of Logic and Computation, 3(4):217-247, 1994.

....a non deterministic transition from state p, where the tree root is labeled with f , to states q and r dealing with the left resp. right subtree. The saturation process always terminates on the logic fragment at hand. We will use the Spass theorem prover [Wei97] as an implementation of saturation [BG94]. ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. Journal of Logic and Computation, 4(3):217-247, 1994.

....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 ....

....require the use of backtracking. For any logic L in between K and K (m) #, #) the expansion rules are given by appropriate subsets. Unnecessary duplication and superfluous inferences can be kept to a minimum by adopting a notion of redundancy which is in the spirit of Bachmair and Ganzinger [2]. A labelled formula F is redundant in a node if the node contains labelled formulae F 1 , F n (for n 0) which are smaller than F and L (F 1 . F n ) F . In this context a formula # is smaller than a formula # if # is a subformula of #, but a more general definition based ....

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

....T # ) #, s (# # , #, s[t # ] p # # , # #)# . vars(s) 3. Equality resolution: #, s #) t) 4. Factoring t, s # (#, t t # , # s )# . The model generation proof method ( BG90] BG94] used in the completeness proof for the strict superposition calculus for general constrained clauses without sequence variables in [NR95] or [NR01] apply as well to the case with sequence variables. Completeness holds for so called well constrained sets of clauses, which can be defined in the ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

....clauses. In the connection graph procedure [Kow75, Eis91] a literal is pure if there are no links connected to this literal. On a more abstract level one could say that a literal L is pure if all the inferences possible with L are redundant, and shift the problem to the de nition of redundancy [BG94]. Independent of the actual de nition of purity, a strategy for eliminating a clause C from a clause set without changing the satis ability and unsatis ability of could be to choose a literal L in C, to generate all resolvents with L until L becomes pure, and then to delete C. This could ....

Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. Journal of Logic and Computation, 4(3):217-247, 1994.

....proofs of their completeness is different in each case, and sometimes feel ad hoc. Amongst these techniques, we find Kowalski and Hayes semantic tree technique [9] which works nicely for ordered resolution, hyperresolution and semantic resolution; Bachmair and Ganzinger s forcing technique [1], which is often used for ordered resolution with selection and provides an explicit model construction in case no refutation can be found. These semantic methods have several advantages, including the fact that it is easy to show that several deletion strategies (tautology elimination, ....

....function satisfying (3) 4) 5) is complete. 4 Applications We first show that Corollary 1 allows us to justify some standard refinements of resolution. As announced in the introduction, we won t deal with every known refinement of resolution. In particular, free selection functions [1], even in the propositional case that we are now considering, do not seem to fit well in this framework. 4.1 Ordered resolution. Let be a strict ordering on atoms. Ordered resolution is the case where (C) is the set of all literals A such that A is maximal in C: i.e. there is no B in C, ....

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

....the same heavy restrictions on the theory that had been imposed in the early eighties. Then, in 1992, Ganzinger and Stuber in [GS92] proposed a method for proving consistency for a set of first order clauses with equality using a completion method based on a preliminary version of the method in [BG94]. It was a refutationally complete linear system, which could also generate a counterexample when presented with a false conjecture. In [KR90] Kounalis and Rusinowitch proposed the first extension to conditional theories, laying the foundations for the method that was to be implemented in the ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994. Revised version of Reseaech Report MPI-I-91-208, 1991. 9

.... A 2 (C A 1 ) where is the most general uni er of A 1 and A 2 . Factoring is not required for the completeness of our decision procedure, but it helps avoiding applications of splitting to clauses containing duplicate literals. The calculus is compatible with a general notion of redundancy [4, 5]. The splitting rule is similar to disjunction elimination in semantic tableaux. If the clause set N contains a clause C it can be split into clauses C 1 and C 2 , provided that C 1 and C 2 are variable disjoint. The original clause becomes redundant and the resolution refutation is performed ....

....clause. A derivation T from N is called fair if for any path N = N 0 ; N 1 ; in the tree T , with limit N1 = S j T k j N k , it is the case that each clause C that can be deduced from non redundant premises in N1 is contained in some set N j . Theorem 4. 1: Bachmair, Ganzinger, Waldmann [4, 6]) Let T be a fair R hyp theorem proving derivation from N . If N; N 1 ; is a path with limit N1 , then N1 is saturated up to redundancy. Furthermore, N is satis able if and only if there exists a path in T with limit N1 such that N1 is satis able. Theorem 4.2: Bachmair, Ganzinger, ....

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplication. J. Logic Computat., 4(3):217-247, 1994.

....improvements that are achievable, see Chapter 3. It is known [Lan75] that saturating the set E of equations in order to approximate only ground con uence is sucient for equational theorem proving. Many improvements of theorem proving procedures can be formulated within the concept of redundancy [BG94]. For example, equations whose ground instances already have a smaller proof are redundant and can be deleted. We propose to keep such equations for simpli cation purposes and show in Chapter 4 that despite that, no critical pairs have to be built with them. Therefore, this approach combines the ....

....and show in Chapter 4 that despite that, no critical pairs have to be built with them. Therefore, this approach combines the advantage of pruning the search space with the bene ts of a stronger simpli cation relation. The completeness of our proof system can be shown within the framework of [BG94]. But the theory developed there is much stronger than needed here and the proof techniques are rather involved. So, to make the paper self contained, we sketch a direct proof here. As outlined above, our main interest in this paper is not to improve the theory of theorem proving, but to improve ....

L. Bachmair and H. Ganzinger. Rewrite-based Equational Theorem Proving with Selection and Simplication. J. Logic and Computation, 4, pages 217-247, 1994.

....During the last two decades this field has importantly progressed through new Knuth Bendix like completion techniques and their extensions to ordered paramodulation for first order clauses. These techniques have lead to important results on theorem proving in first order logic with equality [HR91,BDH86,BD94,BG94] (that have been applied to stateof the art theorem provers like Spass [Wei97] results on logic based complexity and decidability analysis [BG96,Nie98] on deduction with constrained clauses [KKR90,NR95] and on many other applications like inductive theorem proving, symbolic constraint ....

....thus further restricting the search space and allowing arbitrary selection strategies. We have also generalised, in a sense, the result in [dN96] for Horn clauses without equality, about completeness of resolution with an arbitrary selection of one single literal in each clause. 10 In [BG94] standard methods for proving compatibility with redundancy elimination techniques are given, by which, roughly, a clause is redundant if it follows from smaller clauses. These notions are not applicable to our proof transformation technique. But this is not surprising, since by these standard ....

Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

....is splitting equalities into inequalities. We can therefore disregard it. Joins 3 and meets are associative, commutative, idempotent (x x = x = x x) and monotonic in the associated partial ordering. We will henceforth consider all inequalities modulo associativity and commutativity. See [1] for further information on ordered resolution and [2] on lattices. 3 Preparatory Considerations In [11] we directly use J , M and D for deriving resolution bases. Here we choose a technically simpler way that puts more emphasis on semantic background considerations. The main idea is to develop ....

....the subformula property. In particular, this allows us to consider the cut rule from an algebraic point of view. Our focused calculi for transitive relations and lattices are a central starting point to many interesting applications. Besides the common mechanization of 14 equational reasoning [7,1], we plan in particular a consideration of (fragments of) naive set theory and of certain extensions of semilattices and distributive lattices such as Kleene algebra [8] regular algebra [4] relation algebra [10] or allegories [5] Mechanization of these theories in the context of proof support ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplication. J. Logic and Computation, 4(3):217-247, 1994.

....) 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 multiset extension of . Multisets of ....

....set of clauses will derive the empty clause. Fairness means that if some inference is possible it will be performed at some step unless one of the parent clauses gets simpli ed, subsumed, or deleted. The calculus SP is known to be refutationally complete for general rst order equational logic [BG94,NR01]. Note that for Horn clauses Equality Factoring is useless [KR91] In Table 1 the substitution is the most general uni er of u and u 0 , and u 0 is not a variable in Superposition and Paramodulation. We shall write Factoring instead of Equality Factoring for conciseness. In this Name Rule ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplication. J. of Logic and Comp., 4(3):217-247, 1994.

....During the last two decades this field has importantly progressed through new Knuth Bendix like completion techniques and their extensions to ordered paramodulation for first order clauses. These techniques have lead to important results on deduction in firstorder logic with equality, like [HR91,BDH86,BD94,BG94,BG98,NR01], results that have been applied to state of the art theorem provers like Spass [Wei97] and Vampire [RV01] These techniques have also led to results on logic based complexity and decidability analysis [BG01,Nie98] on deduction with constrained clauses [KKR90,NR95] on inductive theorem proving ....

....in many ways. For example, its refutation completeness is preserved when applied only if Dj p is not a variable, and if, for some ordering on terms and equations, the paramodulation steps involve only maximal terms of maximal equations of both premises. In this case it is called superposition [BG94]. Ordered paramodulation is the slightly less restricted version of superposition where inferences also take place on non maximal sides of equations [HR91] 2.1 Paramodulation with constrained clauses By expressing the ordering and unification restrictions as inherited constraints, ....

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Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

....only nitely many ordered links can ever exist, so only nitely many ordered cg resolvents are possible. We emphasize that the order in which links are activated has no impact on this analysis, and the proof is complete. ut The proof method for the next theorem 6 is due to Bachmair and Ganzinger [1]; the proof presented below is a re nement of an elegant exposition by Paliath Narendran. Interpretations in the proof will be described as a set of atoms, indicating that atoms in the set are assigned true and atoms not in the set are assigned false. Theorem 4. Let S be an unsatis able set of ....

Bachmair, L. and Ganzinger, H. Rewrite-based equational theorem proving with selection and simplication. Journal of Logic and Computation, 4(3):217-247, 1994.

....function symbol. Otherwise it is called non functional. An expression is shallow if it does not contain a non constant functional term. The set of variables of an expression E will be denoted by var(E) Next, we brie y recall the de nition of ordered resolution from Bachmair and Ganzinger [1, 2]. Derivations are controlled through an admissible ordering . In the full calculus a second parameter, a selection function, may be used, but for the results of this paper it is not essential. By de nition, an ordering is admissible, if (i) it is a total well founded ordering on the set of ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplication. J. Logic Computat., 4(3):217-247, 1994.

....ACUKT . In other words, fC 1 ; C n g j= ACUKT C 0 if and only if ACUKT[ fC 1 ; C n g j= C 0 . 3 Cancellative Superposition The cancellative superposition calculus (Waldmann [14] is a refutationally complete variant of the standard superposition calculus (Bachmair and Ganzinger [2]) for sets of clauses that contain the axioms ACUK and (optionally) T. It requires neither extended clauses, nor explicit inferences with the axioms ACUKT, nor symmetrizations. Compared with standard superposition or AC superposition calculi, the ordering restrictions of its inference rules are ....

....C 0 . Lifting the inference rules to non ground clauses is relatively straightforward as long as we restrict to clauses without unshielded variables. For the inference rules equality resolution and standard superposition, we proceed as in the standard superposition calculus (Bachmair and Ganzinger [2]) For the inference rules cancellation, cancellative superposition, and cancellative equality factoring, we have to take into account that, in a clause C =C 0 A, the maximal literal A need no longer have the form mu s : s 0 , where u is the unique maximal atomic term. Rather, a ....

Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplication. Journal of Logic and Computation, 4(3):217-247, 1994. 15

.... process can be organized in time O(n log n) Kap97,BT00] Other syntactic restrictions on non equational saturated sets S that are quite easily shown to lead to decision procedures include reductive Horn clauses (also called conditional equations) or universally reductive general clauses [BG94a], even in cases where the reduction ordering is not isomorphic to . This applies to certain restricted kinds of queries only. We will expand further on the subject of deciding entailment problems in the section 1.8 for the case of non equational rst order theories. 1.6 Paramodulation with ....

Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplication. Journal of Logic and Computation, 4(3):217-247, 1994.

.... ) 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: 1994, `Rewrite-Based Equational Theorem Proving with Selection and Simplication'. Journal of Logic and Computation 4(3), 217-247. 32 Ullrich Hustadt and Renate A. Schmidt

....of Joyner. Another approach to decidability stems from the inverse method due to Maslov, and was further developed by Zamov [19] and Tammet [18] in a very similar way as resolution in the work of Joyner. Both approaches contribute to [7] Further, we mention the work of Bachmaier and Ganzinger [2]. They generalize completion procedures similar to the well known Knuth Bendix completion to refutational first order theorem proving with equality. A different approach to using tableaux as a decision procedure for certain first order problems is due to Caferra [5] It is not based on ordering ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Technical Report MPI--I-9-1--208, Max-PlanckInstitut fur Informatik, MPI at Saarbrucken, Germany, 1991.

....order in which these rules are applied in R hyp is (i) factoring, ii) splitting and (iii) hyperresolution. As usual we make a minimal assumption that at any stage in the derivation the clause set contains no duplicate clauses. For soundness and refutational completeness of R hyp see [23] or [4, 5]. For the classes of clause sets we consider in the present paper the positive premises will always be ground, in particular, because we use splitting, the positive premises will always be ground unit clauses, and the conclusions are always positive ground clauses. Crucial for termination is that ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplication. J. Logic Computat., 4(3):217-247, 1994.

....t for all i with 1 i m, or f = g and there is a j with 1 j n such that (s 1 ; s j 1 ) t 1 ; t j 1 ) s j t j , and s k t for all k with j k n, or s t j for a j with 1 j n. 2.3. Resolution We brie y review some notions of the resolution framework in [3]. A selection function assigns to a clause C a possibly empty set of occurrences of negative literals in C. The resolution calculus is parameterized by an atom ordering and a selection function . A literal is called eligible in a clause C if either it is selected in C by (we will also say ....

....well founded atom ordering which is total on ground atoms, and let be a selection function. Then, for a clause set S, M (S) I ( S) where S is the closure of S under ordered resolution and factorisation with selection. If is a liftable ordering, then M (S) is a model for S [3]. Furthermore, we only need such clauses in S which do not contain any eligible negative literals, since ground instances of clauses containing negative eligible literals are not productive in I (S) So the rst step of our model building method is the saturation of a given set of guarded ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplication. Journal of Logic and Computation, 4(3):217-247, 1994.

....and hence of A [ fcg. Now it remains to design efficient procedures that are able to reduce any nonnormal counter example. Since this is essentially a well known problem in first order saturation based theorem proving, we can rely on a large amount of existing results from this field. We refer to [BG94] for more details and restate only the main results. Let us recall here only the ground versions of the following inference rules for Horn clauses in sequent notation, where s t; Gamma denotes that s t and s u for all terms u occurring in Gamma: 6 superposition right: Gamma 0 l = r ....

.... that if E is consistent and saturated under superposition and equality resolution then I is (isomorphic to) T (F) R , where R is a convergent ground term rewrite system (TRS) such that each rule l ) r in R is generated by a ground instance Gamma l = r of a clause in E with l r; Gamma [BG94, NR95]. In what follows, let R denote this TRS. Clearly, a term (or a clause) is normal if, and only if, it is in normal form with respect to R. Similarly, we will call a ground substitution oe normal (or irreducible by R) if xoe is normal (or irreducible by R) for all x 2 Dom(oe) Let us assume from ....

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Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

....N is called saturated if it is closed under condensation, the deletion of subsumed clauses and any clause generated by an ordered resolution inference from clauses from N is subsumed by some clause in N . Ordered resolution in general allows more powerful notions of simpli cation redundancy (Bachmair Ganzinger 1994), but for the purpose of this paper subsumption and condensation suces. Corollary 1 (Bachmair Ganzinger (1994) Let N be a set of Horn clauses saturated by ordered resolution. Then either N contains the empty clause or I N is a model for N . Lemma 1. Let N be a monadic Horn theory saturated ....

....generated by an ordered resolution inference from clauses from N is subsumed by some clause in N . Ordered resolution in general allows more powerful notions of simpli cation redundancy (Bachmair Ganzinger 1994) but for the purpose of this paper subsumption and condensation suces. Corollary 1 (Bachmair Ganzinger (1994)) Let N be a set of Horn clauses saturated by ordered resolution. Then either N contains the empty clause or I N is a model for N . Lemma 1. Let N be a monadic Horn theory saturated by sort resolution that does not contain the empty clause. Then I N = T N 0 where N 0 is the set of all ....

Bachmair, L. & Ganzinger, H. (1994), `Rewrite-based equational theorem proving with selection and simplication', Journal of Logic and Computation 4(3), 217-247.

....order in which these rules are applied in R hyp is (i) factoring, ii) splitting and (iii) hyperresolution. As usual we make a minimal assumption that at any stage in the derivation the clause set contains no duplicate clauses. For soundness and refutational completeness of R hyp see [23] or [4, 5]. For the classes of clause sets we consider in the present paper the positive premises will always be ground, in particular, because we use splitting, the positive premises will always be ground unit clauses, and the conclusions are always positive ground clauses. Crucial for termination is that ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. J. Logic Computat., 4(3):217--247, 1994.

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Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Technical Report MPI-I-91208, Max-Planck-Institut fur Informatik, Saarbrucken, August 1991. To appear in Journal of Logic and Computation.

....form with respect to the given theory. In this paper, we use these main ingredients of Shostak s method to integrate a canonizable and solvable theory into a refutationally complete theorem proving calculus for equational rst order clauses: the superposition calculus of Bachmair and Ganzinger [BG94]. From the Shostak point of view, our calculus can be seen as an extension from a calculus for (dis )equations to a calculus working on arbitrary clauses over mixed terms. From the superposition point of view, the canonizer and the solver become simpli cation devices that allow us to replace ....

....we obtain by the superposition rule (4) f(b) a 2, which with (2) gives a 2 6 a 2 f(b) b 2. This clause and (4) lead to a 2 6 a 2 a 2 b 2. From this, equality factoring deduces a b 4. With (3) we obtain the empty clause. We now employ the proof technique of model generation of [BG94] to show that every saturated clause set free of the empty clause has a model. This model will be a congruence induced by a convergent ground rewrite system. The canonizer b is part of that system. In the sequel S denotes an arbitrary, xed set of ground clauses according to Conv. 18. Every ....

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. Journal of Logic and Computation, 4(3):217-247, 1994.

....superposition calculus. Unfortunately, the completeness proofs proposed for these calculi are technically involved and quite complicated. The calculus described in this paper is obtained by applying the technique of extended rules (Peterson and Stickel 1981) to a superposition calculus of Bachmair and Ganzinger (1994). A similar calculus has been discussed by Wertz (1992) and our completeness proof, like Wertz s proof, is based on the model construction techniques we originally proposed in Bachmair and Ganzinger (1990) The main difference with our current approach is that we use non equality partial models ....

....instance of the conclusion of the general inference. 3.3 Redundancy Simplification techniques, such as tautology deletion, subsumption, demodulation, contextual rewriting, etc. represent an essential component of automated theorem provers. These techniques are based on a concept of redundancy (Bachmair and Ganzinger 1994), which we shall now adapt to the AC case. Let RA denote the set consisting of the reflexivity axiom, F the set of all congruence axioms, and T the set consisting of the transitivity axiom, cf. Section 2. Furthermore, for any ground term s, let Ts be the set of all ground instances u 6 v v 6 w ....

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L. Bachmair and H. Ganzinger, 1994. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, Vol. 4, No. 3, pp. 1--31. Revised Version of MPI-I-91-208, to appear.

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. Journal of Logic and Computation, 4:217{ 247, 1994.

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. J. of Logic and Computation, 4(3):217-247, 1994.

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. Journal of Logic and Computation, 4(3):217{ 247, 1994.

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplication. J. of Logic and Computation, 4(3):217247, June 1994.

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. J. of Logic and Comp., 4(3):217-247, 1994.

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Bachmair L., Ganzinger H. Rewrite-based equational theorem proving with selection and simplification, Journal of Logic and Computation, vol.4, n.3, pp.217-247, (1994).

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L. Bachmair and H. Ganzinger, Rewrite-based Equational Theorem Proving with Selection and Simplification. J. of Logic and Computation 4:217--247, 1994.

No context found.

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

No context found.

L. Bachmair and H. Ganzinger, Rewrite-based Equational Theorem Proving with Selection and Simplification. J. of Logic and Computation 4:217--247, 1994.

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. J. Logic Computat., 4(3):217-247, 1994.

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L. Bachmair and H. Ganzinger. Rewrite-Based Equational Theorem Proving with Selection and Simplification. Journal of Logic and Computation, 3(4):217--247, 1994.

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Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3):217--247, 1994.

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. In Journal of Logic and Computation 4(3), 1-31, 1994.

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L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simpli cation. J. Logic and Comp utation, 4(3):217-247, 1994.

No context found.

L. Bachmair and H. Ganzinger. Rewrite-based equational theorem proving with selection and simplification. In Journal of Logic and Computation 4(3), 1-31, 1994.

No context found.

Leo Bachmair and Harald Ganzinger. Rewrite-based equational theorem proving with selection and simplification. Journal of Logic and Computation, 4(3): 217--247, 1994.

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