B. Beckert. A completion-based method for mixed universal and rigid E-unification. In A. Bundy, editor, Automated Deduction --- CADE12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994. 106 107  Home/Search   Document Not in Database   Summary   Related Articles   Check

....There are cases where this restriction can be lifted in tableau calculi, i.e. where it is sound to instantiate di#erent occurrences of a free variable di#erently. If that is the case, one calls the free variable universal for the formula in which it occurs, see [12] As has been recognized in [5], using universal variables is crucial for e#cient equality handling in tableaux. The calculi presented in this paper do not use universal variables, but we expect our results to be easily adaptable to calculi that use them. 3 The Calculus We shall describe a clausal free variable tableau ....

.... Or, vice versa, if it is su#cient to just put pending ext expansions in a FIFO queue, why should it not be good enough to use the same queue for superposition steps In an e#cient tableau prover, particularly in the presence of equality, one needs to take universal variables into account, see [5]. The superposition rules with universal variables correspond essentially to unfailing Knuth Bendix completion [2] which does not terminate in general. UKBA behaves very well in practice, so it is probably not sensible to artificially introduce additional conditions to enforce termination. As ....

Bernhard Beckert. A completion-based method for mixed universal and rigid E-unification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, LNAI 814, pages 678--692. Springer-Verlag, 1994.

.... one uses simple sucient criteria to detect universality of free variables, the most common one being to ag all free variables introduced in a extension as universal, and to preserve universality through all non splitting 4 They have been used to solve a similar problem in equality handling in [3]. 5 rule applications. After a rule application, those free variables which occur on more than one of the subformulae become rigid. The bene t of universal variables is that they may be instantiated independently for all formulae and may also be renamed as needed, whereas rigid variables have ....

Bernhard Beckert. A completion-based method for mixed universal and rigid e- unication. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, LNAI 814, pages 678-692. Springer-Verlag, 1994.

....equalities, this method will prove the theorem using the equalities. In tableau based methods, some of these techniques do not work the same way because variables are treated differently in tableaux. Beckert, Gallier, and others have worked on efficient methods for handling equality in tableaux [18, 55]. Set theory and higher order logic. Reasoning in most mathematical theories (or in most other realms of reasoning) involves reasoning about sets of objects. Reasoning about sets of objects in first order logic is usually done using a well known yet complex list of axioms called the axioms of ....

....the current implementation of IPR. Section 5.1.1 goes into more detail on the topic of breadth first unification. 3.2.1 Unification Strategies For each of these strategies, there are certainly refinements that make it more efficient. For example, Beckert s mixed universal and rigid unification [18] or Oppacher and Suen s condense algorithm [94] can be incorporated into any of these strategies with some additional bookkeeping. Depth First Backtracking Unification. In this strategy, we use a limit, q, on the number of times the g rule may be applied to a single formula. An implementation of ....

[Article contains additional citation context not shown here]

Bernhard Beckert. A completion-based method for mixed universal and rigid E-unification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692. Springer-Verlag, 1994.

..... p(X) q(Y ) # X # a r(Y ) # X # a where Y is a new free variable. Another use for constrained formulae is equality handling: In [NR01] Sect. 5, Nieuwenhuis and Rubio point out the importance of using ordering constraints to reduce the search space in automated equality reasoning, and [Bec94] presents a constraint based method for equality handling in tableaux that can be neatly integrated with the incremental closure approach. To use ordering constraints, one simply has to extend the constraint language to contain an ordering predicate # in addition to the syntactic equality #. ....

Bernhard Beckert. A completion-based method for mixed universal and rigid E-unification. In Alan Bundy, editor, Proc. 12th Conf. on Automated Deduction CADE, Nancy/France, LNAI 814, pages 678--692. Springer, 1994.

.... Y ]# # D simp c2u # [ X]# # C (D # # #) Y ]# # D [ X # Y ] #) #) # (C D # # #) where # is a simplifiable 6 subformula of #, is a mgu of # and #, and C D # # # is satisfiable 5 They have been used to solve a similar problem in equality handling in [3]. 6 The simplifiable subformula condition could be relaxed to permit, e.g. the simplification of #y.p(y) with [X] p(X) but this becomes rather technical, so we won t do it in this paper. This rule is sound and complete for the free variable tableau calculus with universal variables, but ....

Bernhard Beckert. A completion-based method for mixed universal and rigid e-unification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, LNAI 814, pages 678--692. Springer-Verlag, 1994.

....If in ordinary tableaux we can at least search for a closing substitution on one branch of a tableau, then in [BH 92, Bec 94a] it is proposed to close all branches of a tableau simultaneously. Now search for a substitution should be made by the inspection of the whole tableau. Although in [Bec 94b] it is stated that methods have to be developed for composing a substitution that closes all branches of a tableau simultaneously from a great number of substitutions closing single branches , it is not clear that such incremental methods exist at all 5 . Even if one proves that there are ....

B. Beckert. A completion-based method for mixed universal and rigid E-unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994.

....use is that which is used by Smullyan and Fitting [24, 10] We use the free variable semantic tableau method for theorem proving. The rule for applying knowledge from the knowledge base is consistent with any of the liberalized ffi rules [12, 8, 2] and with mixed rigid and universal variables [7]. Section 2 describes the rules which are used in forming the sequents in the knowledge base and includes soundness and completeness results. Section 3 describes the rule which allows sequents from the knowledge base to be used in a tableau proof. Soundness and completeness of the combination of ....

Bernhard Beckert. A completion-based method for mixed universal and rigid Eunification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, LNAI 814, pages 678--692. Springer-Verlag, 1994.

....symbol forall is the universal quantifier and the symbol for some is the existential quantifier. The symbol = is the identity predicate. 3 2. 2 Sequent Calculus The proof procedure used by IPR is the sequent calculus [6, 12] IPR uses free variables [4, 7] and mixed rigid and universal variables [1]. A sequent is an ordered pair h Gamma; Deltai where Gamma and Delta are finite sets of formulas. We denote this sequent by Gamma Delta. Gamma is the set of hypotheses and Delta is the set of conclusions or goals in the sequent. If one of the sets of formulas is empty then it may be ....

....was introduced by the ffi rule is called a skolem term. 5 Inference Rule 9 (Tree Substitution) If G is a proof tree then for any substitution oe of free variables in G we may apply the substitution across the entire tree simultaneously observing the rules regarding mixed and rigid variables [1]. Soundness and completeness of these rules for first order logic is proved elsewhere [6, 12] The particular ffi rule used was proved correct by Hahnle [7] In order to keep from running forever, a limit is set on the number of times the fl rule can be applied to any formula. This limit is ....

[Article contains additional citation context not shown here]

Bernhard Beckert. A completion-based method for mixed universal and rigid e-unification. In Alan Bundy, editor, Proc. 12th Conference on 17 Automated Deduction CADE, Nancy/France, LNAI 814, pages 678--692. Springer Verlag, 1994.

....For example, suppose we have the formulas x = fy : Ag and a 2 x on a branch. Then the comprehension schema cannot be applied unless the substitution of fy : Ag for x is made in the second formula. However, implementing a complete procedure that makes substitutions can be very inefficient [2]. Therefore, we introduce restricted equality substitution and extensionality rules here that are not terribly expensive to apply in realistic examples and that give more strength to a prover. The idea is to have a set of rules that apply equalities on the fly just when they are needed. This ....

Bernhard Beckert. A completion-based method for mixed universal and rigid E-unification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692. Springer-Verlag, 1994.

.... logic with equality can be designed based on incomplete, but terminating, procedures for rigid E uni cation [8] A simpler version of the problem is known to be decidable and also NP complete, and several corresponding algorithms have been proposed in the literature (not all of them correct) [9, 10, 5, 8, 11, 6]. In the current paper, we consider this standard, non simultaneous version of the problem. Most of the known algorithms for nding a complete set of (standard) rigid uni ers employ techniques familiar from syntactic uni cation, completion and paramodulation. Practical algorithms also usually ....

B. Beckert. A completion-based method for mixed universal and rigid Euni cation. In A. Bundy, editor, 12th Intl Conf on Automated Deduction, CADE12, pages 678-692, 1994. LNAI 814.

....length proofs of rigid E unifiability can be constructed. Others have also studied systems for dealing with rigid E unification. We are not sure of the relationships of these methods to our own. In [17] a complete method is given for building in equality reasoning in the connection calculus. In [5], a method is given to combine classical and rigid E unification. In [13] a rule based method is given for finding complete sets of rigid E unifiers. This uses congruence closure. In [4] a method is given to add rigid E unification to an ordered theory resolution calculus. An algorithm for ....

B. Beckert. A completion-based method for mixed universal and rigid E- unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., pages 678--692, Nancy, France, June/July 1994. Volume 814 of Lecture Notes in Artificial Intelligence.

....equalities, this method will prove the theorem using the equalities. In tableau based methods, some of these techniques do not work the same way because variables are treated differently in tableaux. Beckert, Gallier, and others have worked on efficient methods for handling equality in tableaux [18, 55]. 14 Set theory and higher order logic. Reasoning in most mathematical theories (or in most other realms of reasoning) involves reasoning about sets of objects. Reasoning about sets of objects in first order logic is usually done using a well known yet complex list of axioms called the axioms of ....

....by the current implementation of IPR. Section 5.1.1 goes into more detail on the topic of breadthfirst unification. 3.2.1 Unification Strategies For each of these strategies, there are certainly refinements that make it more efficient. For example, Beckert s mixed universal and rigid unification [18] or Oppacher and Suen s condense algorithm [94] can be incorporated into any of these strategies with some additional bookkeeping. Depth First Backtracking Unification. In this strategy, we use a limit, q, on the number of times the fl rule may be applied to a single formula. An implementation of ....

[Article contains additional citation context not shown here]

Bernhard Beckert. A completion-based method for mixed universal and rigid Eunification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, volume 814 of Lecture Notes in Artificial Intelligence, pages 678-- 692. Springer-Verlag, 1994.

....used to efficiently handle equality in universal formula semantic tableaux, that are an extension of free variable tableaux. 1 Introduction One of the main goals of Automated Deduction is to efficiently handle first order logic with equality. In this paper we describe how mixed E unification [2], a combination of the classical universal E unification and rigid E unificati on [8] can be used to efficiently handle equality in universal formula semantic tableaux [4] that are an extension of free variable tableaux [7] Constructing a tableau for a first order formula OE can be ....

....described in [8, 9] these, however, are non deterministic and unsuited for implementation, since the guess that is part of the algorithm is highly complex. Recently, deterministic completion based methods have been introduced, both for purely rigid E unification [1] and mixed E unification [2]. 7 7 The method described in [2] has been implemented as part of the tableau based theorem prover 3 T A P [3] The implementation is written in Quintus Prolog. Besides the possibility to prove theorems from predicate logic with equality, the E unification module can be used stand alone to ....

[Article contains additional citation context not shown here]

B. Beckert. A completion basedmethod for mixed universal and rigid E-unification. In Proceedings, 12th International Conference on Automated Deduction (CADE), Nancy, LNCS. Springer, 1994.

....the classical universal E unification and rigid E unification, called mixed E unification, can be used to efficiently handle equality in universal formula semantic tableaux, that are an extension of free variable tableaux. 1 Introduction In this paper we describe how mixed E unification [2], a combination of the classical universal E unification and rigid E unification [8] can be used to efficiently handle equality in universal formula semantic tableaux [4] that are an extension of free variable tableaux [7] There are two techniques for handling equality in semantic ....

....been described in [8, 9] these, however, are non deterministic and unsuited for implementation, since the guess that is part of the algorithm is highly complex. Deterministic completion based methods have been introduced recently, both for purely rigid E unification [1] and mixed E unification [2]. 9 Besides being completion based, there are several reasons why these methods are well suited for adding equality to free variable semantic tableaux: Firstly, the terms to be unified do not become part of the completion (in contrary to the method in [8] this is important because the ....

[Article contains additional citation context not shown here]

B. Beckert. A completion based method for mixed universal and rigid E-unification. In Proceedings, 12th International Conferenceon Automated Deduction (CADE), Nancy, LNCS. Springer, 1994.

....set of rigid equations are the substitutions fa=x 1 ; b=x 2 ; g(h n (a) yg, for every n 0. Since simultaneous rigid E unification was introduced by Gallier, Raatz and Snyder [17] there have been a number of publications on simultaneous rigid E unification itself and its use in theorem proving [15, 18, 16, 3, 6, 14, 7, 4, 5, 21, 29]. Some of these articles were based on the conjecture that simultaneous rigid E unification is decidable. There were several faulty proofs of the decidability of this problem (e.g. 15, 16, 21] Results on simultaneous rigid E unification known so far are the following: 1. Non simultaneous (i.e. ....

B. Beckert. A completion-based method for mixed universal and rigid E-unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678-- 692, Nancy, France, June/July 1994.

.... and its use in theorem proving [GNPS 88, GRS 89, GNPS 90, Petermann 90, Baumgartner 92a, Baumgartner 92b, BaFuPe 92, BecHan 92, GNRS 92, Gallier 92, Petermann 92, Beckert 93, Bibel 93, Goubault 93a, Goubault 93b, Goubault 93c, Goubault 93d, Petermann 93a, Petermann 93b, BecPet 94, Beckert 94a, Beckert 94b, Goubault 94, Petermann 94, De Kogel 95] Some of these articles were based on the conjecture that simultaneous rigid E unification is decidable. There were several faulty proofs of the decidability of this problem. Simultaneous rigid E unification can be formulated as follows. Given equations s ....

B. Beckert. A completion-based method for mixed universal and rigid E- unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994.

....rigid E unification by Gallier, Raatz and Snyder [35] there were a number of publications on simultaneous rigid E uni fication itself and its use in theorem proving, for example Gallier et.al. 34, 36, 32, 33] Baumgartner [5] Beckert and Hahnle [11] Becher and Petermann [8] Beckert [9], Goubault [38] and Petermann [54] Some of these articles were based on the conjecture that simultaneous rigid E unification is decidable. There were several faulty proofs of the decidability of this problem (e.g. 34, 32, 38] The refutation procedure for first order logic with equality using ....

....of fl rule (i.e. for a particular tableau) unlike algorithms based on the finite complete sets of unifiers in the sense of Gallier et.al. 33] or based on minus normalization (Kanger [40] The implementation of the method of [54] uses a completion 16 JELIA 96 based procedure by Beckert [9] of generation of complete sets of rigid E unifiers. This procedure is developed with the aim of solving a more general problem so called mixed E unification and has been implemented as part of the tableaubased theorem prover 3 T A P . Complete sets of unifiers both in the sense of Gallier ....

B. Beckert. A completion-based method for mixed universal and rigid E- unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994.

....rigid E unification by Gallier, Raatz and Snyder [36] there were a number of publications on simultaneous rigid E unification itself and its use in theorem proving, for example Gallier et.al. 35, 37, 33, 34] Baumgartner [5] Beckert and Hahnle [11] Becher and Petermann [8] Beckert [9], Goubault [39] and Petermann [56] Some of these articles were based on the conjecture that simultaneous rigid E unification is decidable. There were several faulty proofs of the decidability of this problem (e.g. 35, 33, 39] The refutation procedure for first order logic with equality using ....

....number of applications of fl rule (i.e. for a particular tableau) unlike algorithms based on the finite complete sets of unifiers in the sense of Gallier et.al. 34] or based on minus normalization (Kanger [41] The implementation of the method of [56] uses a completionbased procedure by Beckert [9] of generation of complete sets of rigid E unifiers. This procedure is developed with the aim of solving a more general problem so called mixed E unification and has been implemented as part of the tableau based theorem prover 3 T A P . Complete sets of unifiers both in the sense of Gallier ....

B. Beckert. A completion-based method for mixed universal and rigid E-unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994.

....fl rule (i.e. for a particular tableau) unlike algorithms based on the finite complete sets of unifiers in the sense of Gallier et.al. 20] or based on minus normalization (Kanger [22] Matulis [30] The implementation of the method of Petermann [35] uses a completion based procedure (Beckert [3]) of generation of complete sets of rigid E unifiers. This procedure is developed with the aim of solving a more general problem so called mixed E unification and has been implemented as part of the tableau based theorem prover 3 T A P . Complete sets of unifiers both in the sense of Gallier ....

B. Beckert. A completion-based method for mixed universal and rigid E- unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994.

....matings. 3. A solution to the substitution problem could be an incremental rigid unification algorithm, i.e. an algorithm which would allow to construct a solution for the simultaneous rigid E unification problem from solutions to separate rigid E unification problems. As it is noted in [Bec 94b] A simultaneous E unification problem can be solved by searching for common specialization to its components 2 . It is far from obvious that such an incremental algorithm for rigid E unification can be constructed. In theorem proving without equality, unification perfectly serves as a basis ....

....atoms to name formulas we obtain simple data structures. Second, we can only restrict to a small subset of all subformulas of fl, and hence a small set of names. Third, formulas with no fl names allow for unrestricted quantifier duplication and in fact give the same effect as mixed E unification [Bec 94b] Lemma 4.1 Let be a subformula of fl. Then 1. If is the least disjunctive superformula of then j= 8( oe : 2. If there is no disjunctive superformula of then j= 8( oe :fl) Proof. Immediate from the definition of disjunctive superformulas. 2 This lemma partially explains the ....

[Article contains additional citation context not shown here]

B. Beckert. A completion-based method for mixed universal and rigid E-unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994.

....system of rigid equations are fa=x 1 ; b=x 2 ; g(h n (a) yg, for every n 0. Since simultaneous rigid E unification was introduced by Gallier, Raatz and Snyder [24] there have been a number of publications on simultaneous rigid E unification itself and its use in theorem proving, for example [22, 25, 23, 3, 7, 21, 10, 4, 5, 29, 41]. Some of these articles were based on the conjecture that simultaneous rigid E unification is decidable. There were several faulty proofs of the decidability of this problem (e.g. 22, 23, 29] The following results on simultaneous rigid E unification have been known before our first ....

B. Beckert. A completion-based method for mixed universal and rigid E- unification. In A. Bundy, editor, Automated Deduction --- CADE-12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994.

....only applied when its use will close a branch or allow a theorem to be applied. Soundness and completeness of these rules for first order logic is proved elsewhere [17, 19, 31] The particular ffi rule used was proved correct by Hahnle [19] IPR also uses a mixture of rigid and universal variables [4]. In order to keep from running forever, a limit is set on the number of times the fl rule can be applied to any formula. The user may reset this limit at any time. This limit is referred to as the Q limit. 3 The sequent calculus proof procedure is complete for first order logic. In order to ....

....For example, in the formula (a member of s X) the equality reasoner tries to rewrite X as a class expression if possible because this allows further reasoning. Because of the presence of rigid variables in the sequent calculus many of the well known methods for handling equality are not applicable [4]. For example, there are ways of extending the congruence closure method to make it complete for a larger class of sequents (e.g. McAllester s fast grammar rewriting [21] but most of these work with strictly universal variables. Beckert [4] Fitting [14] and Gallier [16] have developed complete ....

[Article contains additional citation context not shown here]

Bernhard Beckert. A completion-based method for mixed universal and rigid E-unification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, LNAI 814, pages 678--692. Springer Verlag, 1994.

....real word problems make heavy use of equality. It is, therefore, essential for an automated deduction system that is integrated with an interactive prover to employ ecient equality reasoning techniques. Part of 3 T A P is a special equality background reasoner that uses a completion based method [5] for solving E uni cation problems extracted from tableau branches. This equality reasoner is much more ecient than just including the equality axioms. In addition to the mere eciency of the tableau based foreground reasoner and that of the equality reasoner, the interaction between them plays a ....

Bernhard Beckert. A completion-based method for mixed universal and rigid E- unication. In A. Bundy, editor, Proc. 12th CADE, Nancy, France, LNCS 814, pages 678-692. Springer-Verlag, 1994.

....s; ti : s t) 2 Kg is the set of rigid E unification problems in K. Theorem 20. For any key K the set of solutions to the rigid E unification problems in P (K) is a complete set of refuters for K (w.r.t. the equality theory E) Various methods for computing rigid E unifiers have been described [12, 9, 5], the most efficient of which are completion based. 15 Fortunately, completionbased methods for rigid E unification can easily be used for incremental background reasoning: Let ( Phi i ) i0 be a sequence of incremental keys, then the following equations hold for the sequence (P ( Phi i ) i0 of ....

....2. add the new rewrite rules and E unification problems to the old ones, and 3. remove the rewrite rules that are not valid for the substitution oe i (these rules constitute information that cannot be reused) 6 Implementation A completion based method for solving mixed E unification problems [5], which is an extension of rigid E unification, 16 has been implemented as part of the 15 We use the version of total theory reasoning in semantic tableaux where branches are closed one after the other. To close all branches simultaneously, a simultaneous rigid E unification problem has to be ....

B. Beckert. A completion-based method for mixed universal and rigid E- unification. In A. Bundy, editor, Proceedings, 12th International Conference on Automated Deduction (CADE), Nancy, France, LNCS 814, pages 678--692. Springer, 1994.

....2 Phig is the set of rigid E unification problems in Phi. Theorem20. For any key Phi the set of solutions to the rigid E unification problems in P ( Phi) is a complete set of refuters for Phi (w.r.t. the equality theory E) Various methods for computing rigid E unifiers have been described [14, 11, 6], the most efficient of which are completion based. 6 Fortunately, completionbased methods for rigid E unification can easily be used for incremental background reasoning: Let ( Phi i ) i0 be a sequence of incremental keys, then the following equations hold for the sequence (P ( Phi i ) i0 of ....

....problem has to be solved. This is much more difficult than the nonsimultaneous version: simultaneous rigid E unification is undecidable [12] whereas the non simultaneous problem is NP complete [14] 6 Implementation A completion based method for solving mixed E unification problems [6], which is an extension of rigid E unification, 7 has been implemented as part of the tableau based theorem prover 3 T A P [7, 8] The E unification problems extracted from a branch (resp. key) are transformed into (sets of) constrained terms and rewrite rules; the constraints describe the sets ....

B. Beckert. A completion-based method for mixed universal and rigid E- unification. In A. Bundy, editor, Proceedings, 12th International Conference on Automated Deduction (CADE), Nancy, France, LNCS 814, pages 678--692. Springer, 1994.

....t(name,formula) The theorem named name; formula is its internal representation. s(sortpath) sortpath is a list of sorts representing a sort path. 5. 12.5 Pre processing Formulae The module preproc exports predicates for pre processing formulae; it uses a method based on removing anti links (Beckert et al. 1994) In Version 3.0 of 3 T A P this is an undocumented feature ; it is only prototypically implemented, and only experienced users should set the switches flattenformulas and removeantilinks to on, that control pre processing. If, despite this warning, you want to use the module preproc, please consult the ....

Beckert, Bernhard. 1994b. A completion-based method for mixed universal and rigid E- unification. Pages 678--692 of: Bundy, Alan (ed), Proceedings, 12th International Conference on Automated Deduction (CADE), Nancy/France. LNAI 814. Springer Verlag.

....fair (deterministic) heuristics and strategies. 1 Equality Handling A special background reasoner is used for handling equality in 3 T A P : Mixed E unification problems are extracted from tableau branches and passed on to the background reasoner, that employs the completion based method from [1] to solve mixed E unification problems and, thus, to close tableau branches. The equality reasoner can be invoked multiply during the construction of a single tableau branch. After a futile try to find a unifier, the data computed by the background reasoner is reused for later calls, in particular ....

Bernhard Beckert. A completion-based method for mixed universal and rigid E-unification. In A. Bundy, editor, Proceedings, 12th International Conference on Automated Deduction (CADE), Nancy, France, LNCS 814, pages 678--692. Springer, 1994.

....5. 1 Universal, Rigid and Mixed E Unification There are different versions of E unification that are important for handling equality in semantic tableaux: the classical universal E unification [22] rigid E unification [11] and mixed E unification which is a combination of both [5]. The different versions of E unification allow equalities to be used differently in an equational proof: in the universal case the equalities can be applied several times with different instantiations for the variables they contain; in the rigid case they can be applied more than once but with ....

....version of semantic tableaux that equality is to be added to: Universal E unification can only be used in the ground case. For handling equality in free variable tableaux, rigid E unification problems have to be solved. 6 For tableaux with universal formulae both versions have to be combined [5]; then, equalities contain two types of variables, namely universal and rigid ones. To distinguish them syntactically, equalities (8x 1 ) Delta Delta Delta (8xn ) l r) are used that can be explicitly quantified w.r.t. variables they contain. Definition9. A mixed E unification problem hE; s; ....

[Article contains additional citation context not shown here]

B. Beckert. A completion based method for mixed universal and rigid E-unification. In Proceedings, 12th International Conference on Automated Deduction (CADE), Nancy, LNCS. Springer, 1994.

....For instance, the modified calculus still yields a decision procedure for the function free case (unlike hyper resolution) The second improvement is the extension with a dedicated inference rule for equality. We will use the completion based method for mixed universal and rigid E unification of [Bec94]. This procedure was developed to be used within free variable semantic tableaux, and its application within the modified hyper tableaux calculus is very natural. Its recent version [BP96] is in particular attractive because it improves the interaction between the foreground reasoner (i.e. hyper ....

B. Beckert. A completion-based method for mixed universal and rigid E-unification. In A. Bundy, editor, 12th International Conference on Automated Deduction (CADE 12), volume 814 of LNCS, pages 678--692. Springer, 1994.

....of an sufficiently efficient implementation that would be short enough to print it here and that, thus, would be in accordance with the philosophy of lean deduction. However, there is a complex implementation of the completion based method for solving rigid E unification problems described in (Beckert, 1994). The source code is available from the authors. Q 6 Can leanT A P be extended to non classical logics As long as the language underlying a logic is semi decidable and a complete and sound tableau calculus is known, it should be possible to adapt leanT A P for deduction in this calculus. More ....

Beckert, Bernhard. 1994. A Completion-Based Method for Mixed Universal and Rigid E-Unification.

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B. Beckert. A completion-based method for mixed universal and rigid E-unification. In A. Bundy, editor, Automated Deduction --- CADE12. 12th International Conference on Automated Deduction., volume 814 of Lecture Notes in Artificial Intelligence, pages 678--692, Nancy, France, June/July 1994. 106 107

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Bernhard Beckert. A completion-based method for mixed universal and rigid E-unification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, LNAI 814, pages 678--692. Springer-Verlag, 1994.

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Bernhard Beckert. A completion-based method for mixed universal and rigid E-unification. In Alan Bundy, editor, Proc. 12th Conference on Automated Deduction CADE, Nancy/France, LNAI 814, pages 678--692. Springer-Verlag, 1994.

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