W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, 2nd edition, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....we have given a high level description of skeptical query answering that abstracts from an underlying credulous reasoner. In this paper, we make the aforementioned meta algorithm precise and employ it to extend an existing approach to credulous query answering [21] based on the Connection Method [1]. On leave from FG Intellektik, TH Darmstadt The reader may wonder why we have chosen Constrained Default Logic [19, 3] rather than Reiter s original approach. In fact, Constrained Default Logic serves us as an exemplary Default Logic enjoying the property of semi monotonicity, which ....

....Rational Default Logic on the large fragment of so called semi normal default theories [ All these interrelations render our exemplar, Constrained Default Logic, a prime candidate for exposing our approach. Of course, a similar question may arise concerning the choice of the Connection Method [1]. Unlike resolution based methods that decompose formulas in order to derive a contradiction, the Connection Method analyses the structure of formulas for proving their unsatisfiability. In fact, we will see that skeptical query answering requires numerous variants of similar subproofs. In such a ....

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W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, second edition, 1987.

....general and adequate proof methods and techniques for the logics under consideration. It is comparatively easy to invent a general proof method, but it is much more difficult to develop a general and adequate proof technique. For example, the resolution principle [Rob65] and the connection method [Bib87] are general proof methods for first order logic. But are they adequate What is the meaning of adequateness in the first place Roughly speaking, we will consider a technique as being adequate if it solves simpler problems faster than more difficult ones. We illustrate the notion of adequateness ....

....unification helps us to transform recursive programs to iterative ones. The iterative structure can be compiled such that a proof might be detected faster than with the depth first search of Prolog. One of the important applications is datalogic, i.e. the field between logic and databases (cf. [Bib87]) It has been shown by e.g. Smith [SGG86] that cycles are the source of non terminating queries. Consequently, insights from cycle unification may be and have already been used to determine non terminating queries to deductive systems ( DVB89] DVB90] Cycle unification might also contribute to ....

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W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, 2 edition, 1987.

....the connection graph is almost fully connected, primarily to ensure that tautologies may always be deleted. But he did not prove confluence; i.e. his proof admits the possibility that a sequence of link activations will yield a connection graph for which a proof is no longer available. Bibel [5] established completeness for the first order case, but only if the copy rule is available. The graphs are again almost fully connected, and derivations are constructed so that tautologies are removable. Several of Siekmann s students attacked the problem. Smolka [24] established first order ....

....failure. This is in contrast to some calculi the tableau method and path dissolution, for example that also delete links (either explicitly or implicitly) but are easily seen to be strongly complete. Eisinger s example is a bit complicated: It cycles after sixty six steps. In 1987, Bibel [5] found a simpler example. He noted that a fairness condition would eliminate his example but not Eisinger s. Resolution with free selection function is not complete, and any example that demonstrates its incompleteness is easily seen also to be a counterexample to strong completeness of ....

Bibel, W., Automated Theorem Proving, 2 nd ed, Vieweg, Wiesbaden, 1987.

....very important fact because it means that a model of the original clause set or rather, a finite representation of it, L n is readily available at the end of the derivation; it does not have to be computed from the branch, as in other model generation calculi. The calculus is proof confluent [5]: any derivation of an unsatisfiable clause set extends to a refutation. In fact, because of the strong completeness result in Theorem 4.8, the calculus satisfies an even stronger property, as illustrated by the corollary below. In practical terms, the corollary implies that as long as a ....

W. Bibel. Automated Theorem Proving. Vieweg, 1982.

....problems 1 45, Folderol failed on 34, 38, and 41 44, while the Prolog version failed on 34, 37, 38, 43, and 45. HARP is much more powerful than Folderol by virtue of its heuristics and connection methods. But a collection of heuristics is hard to analyze scientifically. The matrix methods of [Bibel 1987] define a notion of path in a formula. A formula is a theorem if all paths within it contain a connection. The number of paths is exponential but each connection rules out a whole class of paths. Matrix methods can determine that a proof can be constructed without constructing one, avoiding ....

Wolfgang Bibel. Automated Theorem Proving. Friedr. Vieweg & Sohn.

....very important fact because it means that a model of the original clause set or rather, a finite representation of it, L n is readily available at the end of the derivation; it does not have to be computed from the branch, as in other model generation calculi. The calculus is proof confluent [5]: any derivation of an unsatisfiable clause set extends to a refutation. In fact, because of the strong completeness result in Theorem 4.8, the calculus satisfies an even stronger property, as illustrated by the corollary below. In practical terms, the corollary implies that as long as a ....

W. Bibel. Automated Theorem Proving. Vieweg, 1982.

....particularly precise form of human problem solving that is used especially in scientific applications. Because of its precision it lends itself to automation more easily than reasoning in general. In classical predicate logic, theorem provers based on resolution [23, 31] and the connection method [5, 6, 14, 4] have demonstrated that logical deduction can be simulated su#ciently well on a computer. Recently the characterizations of logical validity, which underly these systems, have been extended to intuitionistic logic and the modal logics K,K4,D,D4,T,S4, and S5 [17, 28, 29, 27] On this basis the ....

....an e#cient proof search in classical and non classical logics. They avoid notational redundancies contained in mathematical languages or sequent calculi and allow a very compact representation of a formal proof. Originally developed as foundation of Bibel s connection method for classical logic [5, 6], they have later been extended to non classical logics by Wallen [29] and serve as a basis for a uniform proof method for a rich variety of logics [19] Since our starting point will be a given matrix proof we will present only the basic ideas and syntactical concepts and refer to [29] or [19] ....

W. Bibel. Automated Theorem Proving. Vieweg, 1987.

....is driven only by a top down decomposition of formulae. In the rest of this paper we will show how to integrate extended rippling into a complete, and more e#cient proof procedure for constructive first order logic. 3 Matrix based Constructive Theorem Proving Matrix based proof search procedures [4, 6] can be understood as compact representations of tableaux or sequent proof techniques. They avoid the usual redundancies contained in these calculi and are driven by complementary connections, i.e. pairs of atomic formulae that may become leaves in a sequent proof, instead of the logical ....

....unary connections, A=label(u) and A=label(v) u is # complementary with respect to a theory i# #(A) False. v is # complementary with respect to i# True #( A) In a similar way the concept of complementary with respect to a theory which resembles Bibel s theory connections [6], could also be extended to nary connections w.r.t some theory . It enables us to formulate an extended matrix characterization of logical validity exactly like Theorem 1 and to use the same general proof technique as for constructive first order logic. The only modification is the test for ....

W. Bibel. Automated Theorem Proving. Vieweg, 1987.

....clauses (such as the GC procedure or the model elimination procedure that is implemented in PTTP [15] with more complicated rules for counting costs to compensate for the absence of simple proof trees. Alternatively, an assumption mechanism can be added to the matings or connection method [1, 2]. These proof procedures do not require multiple occurrences of the same instances of axioms. This approach would reduce requirements on the syntactic form of the axioms (e.g. the need for clauses) so that a cost could be associated with an arbitrary axiom formula instead of a clause. 6 ....

Bibel, W., Automated Theorem Proving, Friedr. Vieweg & Sohn, Braunschweig, West Germany, 1982.

....of analytic tableau and connection calculi. The most elegant format for introducing a generalized form of model elimination is by using the framework of connection tableaux, which we elaborate in this paper. The framework results from integrating concepts of the connection method [Bibel, 1981, Bibel, 1987] into the tableau calculus [Beth, 1955, Beth, 1959] Smullyan, 1968] Connection tableau procedures are successful and promising because of their goal orientedness and the existence of powerful techniques for reducing the search space. Due to their cut freeness, however, connection tableau ....

....the subgoal tree structure. 2.5 Tableaux and Related Formalisms The connection tableau format is closely related with two other well known frameworks in automated deduction, namely, connection matrices and model elimination chains. 2.5. 1 Connection Matrices The connection calculus presented in [Bibel, 1987] Chapter III.6 can be viewed as a version of the connection tableau calculus restricted to depth first selection functions. Here we shall consider a refinement of this connection calculus, without factorization, which is studied below. The favourite notation for displaying connection proofs is by ....

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W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, second edition, 1987.

....significant improvements. But because of the use of sequent calculi some redundancies remain. Proof nets [7] on the other hand, can handle only a fragment of the logic. Matrix characterizations of logical validity, originally developed as foundation of the connection method for classical logic [2, 3, 5], avoid many kinds of redundancies contained in sequent calculi and yield a compact representation of the search space. They have been extended successfully to intuitionistic and modal logics [24] and serve as a basis for a uniform proof search method [20] and a method for translating matrix ....

....exponential prefix substitution is a mapping # : # ) # which maps elements from # to strings from (# ) # only. Substitutions are extended homomorphically to strings and are assumed to be computed by unification. Complementarity. Matrix characterizations for classical [5] and non classical [24] logics are based on a notion of complementarity . Essentially this means that every path through a matrix must contain a unifiable connection. These requirements also hold for linear logic but have to be extended by a few additional properties. We shall specify all these ....

W. Bibel. Automated Theorem Proving. Vieweg, 1987.

....but impractical to express axiomatically. In that case, a more viable option is to rely on an algorithmic representation of the theory, its decision procedure, and use it as a background reasoner. Although the main idea of theory reasoning can be found in several early works (such as [Bib82, Plo72] to name just a few) the first systematic treatment of it was given by Stickel in [Sti85] which describes a theory version of the resolution calculus and the matings calculus. After that work, nearly all existing calculi for automatic reasoning have been extended to theory reasoning (see ....

Wolfgang Bibel. Automated Theorem Proving. Friedr. Vieweg & Sohn, Braunschweig, Germany, 1982. 33

....of our action: The first one moves block b from the top of block c on the table, the second one moves block a from the table on block b , and the third one moves block c from the table on block a. The set of arcs represents the spanning set of connections which makes the matrix complementary (See [Bib87] for information on the Connection Method. Note that the columns of the matrix can be (partially) ordered and are written down in such an order in Fig. 1 such that for all connections the positive literal is right of the negative (overlined) one. Such an ordering defines an executable order ....

W. Bibel. Automated Theorem Proving. Vieweg, 1987. (second edition).

....and causality. A naive formalization, where the initial situation and the launder action are represented by the formulas q dd (1.1) dd nd; 1.2) respectively, does allow to show that qnd is a logical consequence of (1.1) and (1. 2) However, as illustrated by the connection proof (see e.g. [3]) depicted in Figure 1 it also allows to show that q dd nd is a logical consequence of (1.1) and (1.2) In other words, exchanging a dirty dollar bill will give us not only a new dollar bill but we will also retain the dirty bill. Technically, this unexpected result is due to the fact that the ....

....situation where Bert has a dirty dollar and a quarter can be represented by dd(s 0 ) q(s 0 ) 1.3) dd :nd nd Fig. 1: A connection proof for (1:1) 1:2) ddndq , where the formula has been transformed into its disjunctive normal form :dd:q (dd:nd) dd ndq) and written in matrix form [3]. Hence, each row of the matrix represents a conjunction of literals, and the formula is obtained by the disjunction of the rows. One should observe that the literal :dd is connected twice. where the constant s 0 denotes the initial situation. The situational fluent res(a; S) represents the ....

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W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, second edition, 1987.

....method and using l imited inferences [Holldobler, 1991; Holldobler, 1990a] It is the first connectionist inference system which can handle arbitrary first order terms. The system is built around a connectionist unification algorithm for first order terms [Holldobler, 1990b] We know from [Bibel, 1987] or [Stickel, 1987] that a proof for a formula is found if we can identify a spanning and complementary set of connections, where a connection consists of a positive and negative literal having the same predicate symbol. Informally, the spanning conditions ensure that the connections in the set ....

W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, second edition, 1987.

....value assigned to X in sv is changed to v , and the values of all other variables are left unchanged. Fe(sit :Fe(r(add(Sit :Fe(r(wait(Sit (r(add; Sit (r(wait(Sit (r(heat(Sit (sit Fig. 2. proof of (5) 6) 7) 8) using the connection method [3], where r are used as abbreviations result result(heat; result(wait; result(add; sulfur example may help to illustrate this idea. Initially, only iron is present. Bibel uses an additional literal state , whose role is to record the actions taken. state(st 0 ) Fe (11) The add ....

W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, second edition, 1987.

....claim is made that the underlying hypotheses have allowed to construct a highly structured, very efficient decider. 1 Introduction Most of the work in automated theorem proving has been oriented towards the definition of (semi )deciders for FOL with an uniform inference mechanism (for instance [Rob65, Bib82]) The main idea underlying the work described in this paper is that, instead of having only one general purpose decision procedure for all of FOL, it is better (with regard to timecomplexity) to have many special purpose particularly suited deciders for (some) known decidable fragments of FOL. ....

W. Bibel. Automated Theorem Proving. Vieweg, Braunschweig, 1982.

....the expressive power of interactive proof assistants with the automatic capabilities of rst order theorem proving, both for reasoning about mathematics and for reasoning about programs. It provides a theorem prover for rst order intuitionistic and classical logic based on the connection method [3,10], a tool for generating proof objects in the style of sequent proofs [11] and is coupled with mechanisms for integrating the prover into the Nuprl proof program development system [4,1] and the MetaPRL proof environment [8,9] These components enable a user to invoke the automatic prover on proof ....

W.Bibel. Automated Theorem Proving. Vieweg, 1987.

....Since the size of any DNF equivalent of the formula is 35 O(2 N ) the O(2 N ) running time inevitably results. 2 Some algorithms do not create such an intermediate DNF representation, but instead generate all implicates directly through inference techniques such as resolution (e.g. [2,3]) The implicates of the formulas above can be computed efficiently using these techniques. However, in [36] a class of formulas Phi = Phi; Phi; was introduced for which conversion to either CNF or to DNF is expensive. Each Phi is unsatisfiable, contains 2 Delta i 2 literals, and has a ....

....independently (see Section 3.4) He also shows that his implementation does as well as the clausal algorithm of deKleer s on clause formulas. Path Based Approaches In [28] Jackson and Pais give an algorithm (MM) to compute prime implicates implicants based on the connection method of Bibel [2]. Given a set of CNF clauses the algorithm enumerates the set of paths which are prime implicants (see theorem 4.4.2) using heuristics to eliminate subsumed paths as early as possible. The algorithm is shown in Figure 3.3 (at the end of the chapter) T is the input, i.e set of CNF) 43 clauses. ....

Wolfgang Bibel. Automated theorem proving. Artificial intelligence. Vieweg, Braunschweig, Germany, 1987.

....for ecient proof search methods. They yield a very compact representation of the search space and thus avoid many kinds of redundancies which usually occur in the sequent calculus and tableaux proof search methods. Originally developed as foundation of Bibel s connection method for classical logic [2,4] they have later been extended to nonclassical logics by Wallen [27] Wallen s formulation serves as a basis of a uniform proof method for a rich variety of logics [19,21] and also allows to transform matrix proofs into sequent style proofs by a uniform procedure [23,24] By Wallen s conjecture ....

....algorithm presented in the following is driven by connections instead of the logical connectives. Once a complementary connection has been identi ed all paths containing this connection are deleted. This is similar to Bibel s connection method for classical logic and formulas in clausal form [4]. The theoretical basis of the following algorithm is described in detail in [21] where it is used for proof search in classical, intuitionistic and modal logics. Only a few modi cations were necessary to adapt it to MLL. De nition 7 ( related, related) Two positions u and v are related ....

W. Bibel. Automated theorem proving. Vieweg, 1987.

....useful in speeding up the construction of proofs even if many of the key ideas must be supplied interactively. When searching for a proof of a theorem, Tps rst tries to nd an expansion proof [11] of which an important component is a mating [1] otherwise known as a spanning set of connections [4]) Various search procedures are implemented in Tps , most notably those described in [6] 5] 10] and [9] The method of dual instantiation of de nitions discussed in [7] is also implemented in Tps . Once an expansion proof has been found, it is translated into a natural deduction proof by the ....

Wolfgang Bibel. Automated Theorem Proving. Vieweg, Braunschweig, second edition, 1987.

....without building in axioms through theory resolution. 1 1 Introduction Incorporating a theory into derived inference rules so that its axioms are never resolved upon has enormous potential for reducing the size of the exponential search space commonly encountered in resolution theorem proving [9, 18, 4, 35]. Theory resolution is a method of incorporating specialized reasoning procedures in a resolution theorem prover so that the reasoning task will be effectively divided into two parts: special cases, such as reasoning about inequalities or about taxonomic information, are handled efficiently by ....

....(6) 16 that does not depend on performing resolution inference operations. Hence it also overcomes the difficulty in total narrow theory resolution of retention of tautologies. The theory matings procedure is an extension of Andrews s matings procedure [3] see also Bibel s connection method [4]) Definition 12 Let C 1 ; Cm (m 1) be a set of clauses. Then each set of literals K 1 ; Km such that each K i is a literal of C i is a path through C 1 ; Cm . A path consists of one literal from each clause; it can also be regarded as one row of the dual, disjunctive ....

Bibel, W. Automated Theorem Proving. Friedr. Vieweg & Sohn, Braunschweig, West Germany, 1982.

....on clausal tableaux and chapter I.2.6 on 243 W. Bibel, P. H. Schmitt (eds. Automated Deduction. A basis for applications. Vol. III c 1998 Kluwer Academic Publishers. Printed in the Netherlands 244 FRIEDER STOLZENBURG AND BERND THOMAS theory reasoning. This may be traced back to the work of Bibel (1982) and Stickel (1985) It is interesting to notice that the success of the application to be presented here is due to the combination of both approaches. 2.1. The PTTP Technique Constraint logic programming (CLP) is a field of active research. There, logic programming with Horn clauses is enhanced ....

Bibel, W.: 1982, Automated Theorem Proving. Braunschweig, Wiesbaden: Vieweg.

....classical type theory. VTEX(za al) PIPS No. 81315 JARSAB21.tex; 19 07 1996; 14:31; v.4; p.1 322 PETER B. ANDREWS et al. TPS is based on an approach to automated theorem proving called the mating method [5] which is essentially the same as the connection method developed independently by Bibel [13]. The mating method arose from reflections [3] on what a proof by resolution [50] reveals about the logical structure of the theorem being proved, but a distinguishing characteristic of the mating method is that it does not require reduction to clausal form. Matings provide significant insight ....

Bibel, W.: Automated Theorem Proving, Vieweg, Braunschweig, 1987.

....particularly precise form of human problem solving which is used especially in scientific applications. Because of its precision it lends itself to automation more easily than reasoning in general. In classical predicate logic theorem provers based on resolution [21, 26] and the connection method [5, 6, 13, 4] have demonstrated that logical deduction can be simulated sufficiently well on a computer. Recently the characterizations of logical validity which underly these systems have been extended to intuitionistic logic and the modal logics K;K4;D;D4;T;S4, and S5 [16, 23, 24] On this basis the existing ....

....an efficient proof search in classical and non classical logics. They avoid notational redundancies contained in mathematical languages or sequent calculi and allow a very compact representation of a formal proof. Originally developed as foundation of Bibel s connection method for classical logic [5, 6] they have later been extended to nonclassical logics by Wallen [24] and serve as a basis for a uniform proof method for a rich variety of logics [18] Since our starting point will be a given matrix proof we will present only the basic ideas and syntactical concepts and refer to [24] or [18] for ....

W. Bibel. Automated Theorem Proving. Vieweg, 1987.

....of knowledge and belief, logics of programs, and for such tasks as the specification of distributed and concurrent systems. In many of these applications there is a need for automated proof search. For classical predicate logic theorem provers based on resolution [14, 17] the connection method [4, 5, 8, 3], or the tableaux calculus [2, 1] have demonstrated that formal reasoning can be automated sufficiently well but efficient proof procedures for non classical logics do not yet exist. Recently Wallen [15, 16] and Ohlbach [10] have extended the classical characterizations of logical validity on ....

....of non classical logics. In this paper we present an efficient proof procedure which allows a uniform treatment of classical, constructive, and modal logics. It is based on a unified representation of Wallen s matrix characterizations of logical validity and generalizes Bibel s connection method [4, 5] for classical predicate logic accordingly. In order to keep the general methodology of the connection method investigating paths by following connections unchanged we had to take into account a considerable extension of the notion of complementarity which strongly depends on the logic under ....

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W. Bibel. Automated Theorem Proving. Vieweg Verlag, 1987.

....i.e. the premise has to be somehow related to the conclusion. Although sequent style proofs are more comprehensible they turn out to be unsuited for automated proof search in general. Efficient proof methods based on the matrix characterizations for classical logic have been proposed by Bibel [1, 2]. Extensions of matrix characterizations have been proposed by Wallen [12] concerning modal logics and intuitionistic logic. It is desirable to relate logics and possible matrix representations a more generic way then it is practiced these days. As a promising starting point structural rules can ....

....While B induces just basic validity, BW allows weakening, i.e. literals may left unconnected, while BC supports contraction, i.e. literals may be part of several connections. If we add both contraction and weakening we obtain BCW which corresponds to the concept of validity as proposed in [2]. Finally by adding the inverse weakening rule we obtain the all relation, i.e. A relates every matrices. While B is the most constrained validity concept expressible by our approach, A represents the other extreme. It is worth to note that the more constraints are added to the validity ....

W. Bibel. Automated Theorem Proving. Vieweg Verlag, 1987.

....efficient proof search methods. They yield a very compact representation of the search space and thus avoid many kinds of redundancies which usually occur in the sequent calculus and tableaux proof search methods. Originally developed as foundation of Bibel s connection method for classical logic [2, 3] they have later been extended to nonclassical logics by Wallen [26] Wallen s formulation serves as a basis of a uniform proof method for a rich variety of logics [18, 20] and also allows to transform matrix proofs into sequent style proofs by a uniform procedure [22, 23] By Wallen s conjecture ....

....algorithm presented in the following is driven by connections instead of the logical connectives. Once a complementary connection has been identified all paths containing this connection are deleted. This is similar to Bibel s connection method for classical logic and formulas in clausal form [3]. The theoretical basis of the following algorithm is described in detail in [20] where it is used for proof search in classical, intuitionistic and modal logics. Only a few modifications were necessary to adapt it to MLL. Definition7 (ff related, fi related) Two positions u and v are ....

W. Bibel. Automated theorem proving. Vieweg, 1987.

....signi cant improvements. But because of the use of sequent calculi some redundancies remain. Proof nets [7] on the other hand, can handle only a fragment of the logic. Matrix characterizations of logical validity, originally developed as foundation of the connection method for classical logic [2,3,5], avoid many kinds of redundancies contained in sequent calculi and yield a compact representation of the search space. They have been extended successfully to intuitionistic and modal logics [24] and serve as a basis for a uniform proof search method [20] and a method for translating matrix ....

....a mapping : M [ E ) M [ M [ E [ E ) which maps elements from M to strings from ( M [ M ) only. Substitutions are extended homomorphically to strings and are assumed to be computed by uni cation. Complementarity. Matrix characterizations for classical [5] and non classical [24] logics are based on a notion of complementarity . Essentially this means that every path through a matrix must contain a uni able connection. These requirements also hold for linear logic but have to be extended by a few additional properties. We shall specify all these ....

W. Bibel. Automated Theorem Proving. Vieweg, 1987.

....we reassess our evaluation of the di#culty of parallelizing theorem proving, including also subgoal reduction strategies. 2 Sequential theorem proving strategies We begin by recalling basic concepts and terminology that will be referred to in the following. Surveys on the subject include [12, 47, 53, 85, 11, 49, 84, 59, 24, 6, 34, 19], where the interested reader may find extensive bibliographies. 1 AND parallelism may also be considered as parallelism at the term level, since the data accessed in parallel are the literals of a goal clause, hence subexpressions of formulae. 3 (synthetic) synthetic) instance based ....

Wolgang Bibel. Automated Theorem Proving. Friedr. Vieweg & Sohn, 2nd edition, 1987.

....and complementarity of respective matrices correspond must be ensured by a characterization theorem. A matrix characterization of a logic yields a representation of the search space which avoids many redundancies inherent to methods based on sequent style or tableau proofs. Starting with Bibel s [4, 6] connection method for classical logic matrix characterizations have later been extended to many non classical logics by Wallen [20] Recently, matrix characterizations for fragments of linear logic have been developed [11, 16, 15] Different approaches have been undertaken to represent matrix ....

....tree. pos op(p) pos op(p) pos op(p) 0 t 00 t 01 u 000; 001 3m 010 V x: 011 V y: 0000; 0010 W x: 0100 2m 0110 2m 00000; 00100 r(x) 01000 r(x) 01100 r(f(y; y) t u t V x: r(010) 2m 2m r(f(011; 011) W x: r(0010) 3m W x: r(0000) 3m V y: Remark. The reader familiar with [4, 6] might be astonished by our rather complicated definition of matrices. However, it is necessary in order to capture many non classical logics and to keep the approach extensible. 3.2 Complementarity Concept We first define a basic complementarity concept using basic positions, i.e. leaving ....

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W. Bibel. Automated Theorem Proving. Vieweg, 1987.

....and simplification. In addition, we present a new treatment of the connection method via named matrices which gives a number of advantages. 1 Introduction We propose an approach to adding equality to extension proof methods which are known under two different names connection method [Bi87] and the method of matings [An81, GNRS92] Both methods express the same idea going back to the fundamental Herbrand theorem. According to this idea, the proof search can be considered as the problem of verifying that each path through a matrix of the goal formula is complementary (in the ....

....instances of a quantifier free formula M(x) he proposed to search for a substitution oe for x 1 ; x n such that every path in (M(x 1 ) M(x n ) oe becomes complementary. A generalization of the ideas of Prawitz to the predicate calculus with equality has been proposed by Bibel [Bi87] and Gallier et al. GNRS92] In this generalization complementary literals are replaced by eq connections instantiated by eq unifiers [Bi87] or mated sets instantiated by rigid E unifiers [GNRS92] which is the same despite different terminology. For some time, there was a hope that rigid ....

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W. Bibel. Automated theorem proving. Vieweg Verlag, 1987. 2nd edition.

....[Rob65] who collapses the two steps into a unique and uniform inference rule (i.e. resolution) However resolution is inherently a local inference rule. The major drawback of this fact is that even trivial theorems require several applications of the rule. The latest generation of procedures [And81, Bib82, MR] have rediscovered the separation between the two steps of instance generation and propositional analysis (which adopting Andrews terminology are called amplification and search over the set of vertical paths respectively) The key characteristic of FOLTAUT is a neat separation between such ....

W. Bibel. Automated Theorem Proving. Vieweg, Braunschweig, 1982. 16

....extensions of a problem have to be handled. An important example are completion based methods for equality reasoning, that are inherently incremental. We focus on theory reasoning in semantic tableaux [21, 11] and related methods such as model elimination [16] and the connection method [8] , where a background theory reasoner is used to close tableau branches resp. to compute connections or links (total theory reasoning) The paper is organized as follows: In Section 2 we introduce notation and recall the basic definitions of theory reasoning; in Section 3 we define the ....

W. Bibel. Automated Theorem Proving. Vieweg, Braunschweig, second revised edition, 1987.

....seldomly be found automatically, as there are no complete proof search procedures embedded into these systems. It is therefore desirable to extend the reasoning power of proof assistants by integrating well understood techniques from automated theorem proving. Matrix based proof search procedures [3, 4] can be understood as compact representations of tableaux or sequent proof techniques. They avoid the usual redundancies contained in these calculi and are driven by complementary connections , i.e. pairs of atomic formulae that may become leaves in a sequent proof, instead of the logical ....

....of matrix proofs into sequent proofs. In Section 5 we discuss the integration of rippling techniques into matrix methods. We conclude with a discussion of possible applications of our work in program synthesis and verification. 2 The Connection Method for Non normal Form The connection method [3, 4] was originally designed as proof search method for formulae in clause normal form. But as normalization of formulae is often costly and as many non classical logics do not have normal forms, it is necessary to develop matrix methods for formulae in non normal form. Bibel [4] already describes a ....

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W. Bibel. Automated Theorem Proving. Vieweg, 1987.

....of Predicate Logic. We merge the two levels of structural underspecification in order to obtain a matrix based proof system which reasons as directly as possible on the original UDRS s. 1 1 Introduction 1. 1 Structurally underspecified sequent proofs the matrix method Matrix methods ( 1] [2], 17] collect sets of constraints on possible (Gentzen) sequent proof trees 1 , instead of searching for an actual instance of a sequent proof tree for a formula. Conceptually, a matrix proof search consists of two steps. First, the leaves ( paths ) of a sequent derivation are enumerated ....

....not the embedding of subformulas in d itself. 8 Theorem 1 (Validity) A DRS d is valid iff (with respect to D 1 ) 1. there is some multiplicity (number of copies of universal formulas) for :d 2. and all paths for :d are axiom links, 3. and the reduction ordering is admissible. Proof. See [1] [2] for the corresponding proof for Predicate Logic. Due to the very tight relation of D 1 with PL deduction, its soundness and completeness is immediate, provided that the translation function 0 implements faithfully the intended formal semantics of DRS s, cf. 7] 2 It will be more convenient to ....

Wolfgang Bibel. Automated Theorem Proving. Vieweg, Braunschweig, 2nd edition, 1987.

....the other hand, the success of theorem provers for classical logic [Wos et al. 1990, Letz et al. 1992, Bibel et al. 1994, Beckert and Posegga, 1994] has shown that formal reasoning can be automated sufficiently well. Matrix based proof search procedures like the connection method [Bibel, 1981, Bibel, 1987] can be understood as compact representations of tableaux, natural deduction, or sequent proof techniques. They avoid the usual redundancies contained in these calculi and are driven by complementary connections, i.e. possible leaves in a sequent proof, instead of the logical connectives of a ....

....proof search procedure to MLL, the multiplicative fragment of linear logic. We conclude with brief discussion of possible further extensions and the integration of matrix methods into interactive proof systems. 2 The Matrix Characterization of Logical Validity The connection method [Bibel, 1981, Bibel, 1987] was originally designed as proof search method for formulas in clause normal form. But as normalization of formulas is often costly and as many non classical logics do not have normal forms, it is necessary to develop connection methods for formulas in non clausal form. Bibel [Bibel, 1987] ....

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W. Bibel. Automated Theorem Proving. Vieweg, 1987.

....correctness, completeness, minimality, and termination and investigate its complexity. 1 Introduction Unification is one of the key operations which are used to guide an efficient search for a proof of a theorem in classical predicate logic. Within theorem provers based on the connection method [5, 6, 8, 4], resolution [17, 20] the tableaux calculus [3, 2] and others unification is required for making certain atomic formulae complementary . Complementarity is a key concept in the characterization of logical validity. In classical logic two atomic formulae (or atoms) are called complementary if ....

W. Bibel. Automated Theorem Proving. Vieweg Verlag, 1987.

....permitting proofs that require more rounds of search. We also consider some complexity issues, as discussed in part by [Gou94] His approach is non clausal, but the same analysis applies to a clausal framework. He shows that the fundamental problem associated with the mating [And81] or connection [Bib87] approach is Sigma p 2 complete. These approaches first choose a number of copies of each of the input clauses, and then seek to find a single substitution that makes the given set of copies of the clauses propositionally unsatisfiable. For 3 a general set of clauses, an arbitrary number of ....

W. Bibel. Automated Theorem Proving. Vieweg, Braunschweig /Weisbaden, 1987. second edition.

....general and adequate proof methods and techniques for the logics under consideration. It is comparatively easy to invent a general proof method, but it is much more difficult to develop a general and adequate proof technique. For example, the resolution principle [15] and the connection method [1] are general proof methods for first order logic. But are they adequate What is the meaning of adequateness in the first place Roughly speaking, we will consider a technique as being adequate if it solves simpler problems faster than more difficult ones. We illustrate the notion of adequateness ....

....the cycle, and X = t 1 ; r = t n g is the set of exit equations after k iterations through the cycle. A cycle unification problem should not be confused with a theory unification problem hG =C F i , i.e. the problem whether there exists a substitution oe such that oeG =C oeF [1, 19]. The following proposition is an immediate consequence of the completeness and soundness of the connection method [1] or SLD resolution, e.g. 9] Due to lack of space we had to omit the proof of this proposition and all further theorems. They can be found in detail in [4] Proposition 1 oe is a ....

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W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, 2 edition, 1987.

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W. Bibel. Automated Theorem Proving. Vieweg, second edition, 1987.

.... is a well known proof procedure for classical first order logic and has successfully been realized in theorem provers like Setheo [22] or KoMeT [6] It is based on a matrix characterization of logical validity: A formula F is (classically) valid iff the matrix of F is (classically) complementary [4, 5]. 6 In propositional classical logic the matrix of a formula F is complementary if there is a spanning set C of connections for F . A connection is a pair of atomic formulas with the same predicate symbol but different polarities. 3 A connection corresponds to an axiom in the sequent calculus. ....

....extract reconstruction knowledge for the conversion procedure. More precisely, the history of matrix proofs will be integrated into the conversion process rather than using only the spanning matings. This makes our procedure depend on a particular proof search strategy, i.e. an extension procedure [5, 30]. But a compact encoding of this proof knowledge into the conversion process (which can be done in polynomial time in the size of the matrix proof) allows us to derive the reconstruction knowledge in terms of a few elegant conditions. Finally, the resulting conversion strategy integrates these ....

W. Bibel. Automated Theorem Proving. Vieweg Verlag, 1987.

....right siblings on WAIT(N ) For instance in the deduction tree shown in Figure 2.2. 3, ACT(W 8 1 ) Q 1 ; Q; S 1 ; V 2 ; W 8 1 ) and WAIT(W 8 1 ) W 1 1 ; W 1 ; V 2 1 ) WAIT(V 2 ) The terminology here is taken from a path checking method (the connection method) of Bibel ([Bi87, Ho82, Jo93]) in which ACT is the current ACTive path, and WAIT denotes those subgoals waiting to be solved. Notice that some deduction trees (for P) will not correspond to derivations of P. The reason for this lies in the fact that we have (quite deliberately at this stage) not placed any restrictions on ....

....2 A, and m facts of the form A 1 A 2 . The (propositional) language from which the database was taken was of size m, and jPj = 3. Table 1 shows the figures for the algorithm discussed in Section 2.3.3. In order to gain some comparison, we ran similar experiments using Bibel s connection method [Bi87], and the corresponding figures are shown in Table 2. m 20 30 40 50 60 70 80 2.1 3.5 5.5 8.1 11.7 16.5 23.5 Table 1. m 15 20 25 30 35 40 45 1.6 3.3 7.3 16.7 38.4 92 192 Table 2. 51 ....

W. Bibel, Automated Theorem Proving (Vieweg, Wiesbaden, 1987).

....6 will resume the discussion of the semantics of TL while in the next section we focus on its deductive aspects. 4 Basic deductive machinery As the original name of our approach, linear connection method (LCM) suggests, the basic deductive machinery is based on the connection method [Bib93, Bib87] This deductive method is characterized by the fundamental theorem which in turn characterizes validity of a formula 4 by the so called spanning property explained shortly. Many different logical calculi can be based on this method. In order to explain the spanning property we need the ....

....compound instance F 0 of) F has a spanning and unifiable mating which satisfies the linearity restriction and has no (regular [Fro96] cycles within any of M s r parts. Since none of our examples will need compound instances we refer to the ATP literature for the respective details (e.g. see [Bib87] Nor has any of our examples cycles. Therefore we also ignore this technically intricate issue and refer the interested reader to [Fro96] for the details. Let us briefly discuss the rationale behind the definition of r compatibility in order to get a better understanding of the linearity ....

W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, 2. edition, 1987.

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W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, 2nd edition, 1987.

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W. Bibel. Automated Theorem Proving 2. Edition. Vieweg Verlag, 1987.

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Bibel, W., Automated Theorem Proving, Friedr. Vieweg & Sohn, Braunschweig, Wiesbaden, 1982.

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W. Bibel. Automated Theorem Proving. Vieweg Verlag, Braunschweig, 2nd edition, 1987.

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W. Bibel. Automated Theorem Proving. Vieweg, Braunschweig, second revised edition, 1987.

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Wolfgang Bibel (1987). Automated Theorem Proving. 2 nd rev.ed., Vieweg, Braunschweig.

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