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.

....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 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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