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Controlled Integrations of the Cut Rule into Connection Tableau Calculi
"... In this paper techniques are developed and compared which increase the inferential power of tableau systems for classical firstorder logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a genera ..."
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Cited by 61 (3 self)
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In this paper techniques are developed and compared which increase the inferential power of tableau systems for classical firstorder logic. The mechanisms are formulated in the framework of connection tableaux, which is an amalgamation of the connection method and the tableau calculus, and a generalization of model elimination. Since connection tableau calculi are among the weakest proof systems with respect to proof compactness, and the (backward) cut rule is not suitable for the firstorder case, we study alternative methods for shortening proofs. The techniques we investigate are the folding up and the folding down operation. Folding up represents an efficient way of supporting the basic calculus, which is topdown oriented, with lemmata derived in a bottomup manner. It is shown that both techniques can also be viewed as controlled integrations of the cut rule. In order to remedy the additional redundancy imported into tableau proof procedures by the new inference rules, we develop and apply an extension of the regularity condition on tableaux and the mechanism of antilemmata which realizes a subsumption concept on tableaux. Using the framework of the theorem prover SETHEO, we have implemented three new proof procedures which overcome the deductive weakness of cutfree tableau systems. Experimental results demonstrate the superiority of the systems with folding up over the cutfree variant and the one with folding down.
A Treatise on ManyValued Logics
 Studies in Logic and Computation
, 2001
"... The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with som ..."
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Cited by 53 (3 self)
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The paper considers the fundamental notions of many valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous tnorms, leftcontinuous tnorms, Pavelkastyle fuzzy logic, fuzzy set theory, nonmonotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to manyvalued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into
Reasoning with Descriptions of Trees
 In ACL92, 7280
, 1992
"... In this paper we introduce a logic for describing trees which allows us to reason about both the parent and domination relationships. The use of domination has found a number of applications, such as in deterministic parsers based on Description the ory (Marcus, Hindle & Fleck, 1983), in a com pac ..."
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Cited by 45 (5 self)
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In this paper we introduce a logic for describing trees which allows us to reason about both the parent and domination relationships. The use of domination has found a number of applications, such as in deterministic parsers based on Description the ory (Marcus, Hindle & Fleck, 1983), in a com pact organization of the basic structures of Tree Adjoining Grammars (VijayShanker & Schabes, 1992), and in a new characterization of the ad joining operation that allows a clean integration of TAGs into the unificationbased framework (VijayShankeL 1992) Our logic serves to formalize the reasoning on which these applications are based.
A connection based proof method for intuitionistic logic
 TH WORKSHOP ON THEOREM PROVING WITH ANALYTIC TABLEAUX AND RELATED METHODS, LNAI 918
, 1995
"... We present a proof method for intuitionistic logic based on Wallen’s matrix characterization. Our approach combines the connection calculus and the sequent calculus. The search technique is based on notions of paths and connections and thus avoids redundancies in the search space. During the proof s ..."
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Cited by 29 (19 self)
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We present a proof method for intuitionistic logic based on Wallen’s matrix characterization. Our approach combines the connection calculus and the sequent calculus. The search technique is based on notions of paths and connections and thus avoids redundancies in the search space. During the proof search the computed firstorder and intuitionistic substitutions are used to simultaneously construct a sequent proof which is more human oriented than the matrix proof. This allows to use our method within interactive proof environments. Furthermore we can consider local substitutions instead of global ones and treat substitutions occurring in different branches of the sequent proof independently. This reduces the number of extra copies of formulae to be considered.
Logicbased Knowledge Representation
 Artificial Intelligence Today, Recent Trends and Developments, number 1600 in Lecture Notes in Computer Science
, 1996
"... . After a short analysis of the requirements that a knowledge representation language must satisfy, we introduce Description Logics, Modal Logics, and Nonmonotonic Logics as formalisms for representing terminological knowledge, timedependent or subjective knowledge, and incomplete knowledge res ..."
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Cited by 27 (0 self)
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. After a short analysis of the requirements that a knowledge representation language must satisfy, we introduce Description Logics, Modal Logics, and Nonmonotonic Logics as formalisms for representing terminological knowledge, timedependent or subjective knowledge, and incomplete knowledge respectively. At the end of each section, we briefly comment on the connection to Logic Programming. 1 Introduction This section is concerned with the question under which conditions one may rightfully claim to have represented knowledge about an application domain, and not just stored data occurring in this domain. 1 In the early days of Artificial Intelligence and Knowledge Representation, there was a heated discussion on whether logic can at all be used as a formalism for Knowledge Representation (see e.g. [135, 91, 92]). One aspect of the requirements on knowledge representation formalisms that can be derived from the considerations in this section is very well satisfied by logical for...
Tstringunification: unifying prefixes in nonclassical proof methods
 5 TH TABLEAUX WORKSHOP, LNAI 1071
, 1996
"... For an efficient proof search in nonclassical logics, particular in intuitionistic and modal logics, two similar approaches have been established: Wallen’s matrix characterization and Ohlbach’s resolution calculus. Beside the usual termunification both methods require a specialized stringunificat ..."
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Cited by 23 (12 self)
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For an efficient proof search in nonclassical logics, particular in intuitionistic and modal logics, two similar approaches have been established: Wallen’s matrix characterization and Ohlbach’s resolution calculus. Beside the usual termunification both methods require a specialized stringunification to unify the socalled prefixes of atomic formulae (in Wallen’s notation) or worldpaths (in Ohlbach’s notation). For this purpose we present an efficient algorithm, called TStringUnification, which computes a minimal set of most general unifiers. By transforming systems of equations we obtain an elegant unification procedure, which is applicable to the intuitionistic logic J and the modal logic S4. With some modifications we are able to treat the modal logics D, K, D4, K4, S5, and T. We explain our method by an intuitive graphical presentation, prove correctness, completeness, minimality, and termination and investigate its complexity.
A Uniform Proof Procedure for Classical and NonClassical Logics
 KI96: ADVANCES IN ARTIFICIAL INTELLIGENCE, LNAI 1137
, 1996
"... We present an efficient proof procedure for classical and nonclassical logics. The proof search is based on the matrixcharacterization of validity where the emphasis on paths and connections avoids redundancies occurring in sequent or tableaux calculi. Our uniform technique of pathchecking is a ..."
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Cited by 20 (16 self)
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We present an efficient proof procedure for classical and nonclassical logics. The proof search is based on the matrixcharacterization of validity where the emphasis on paths and connections avoids redundancies occurring in sequent or tableaux calculi. Our uniform technique of pathchecking is applicable to arbitrary formulae and a generalization of Bibel's connection method for classical logic and formulae in clauseform. This makes it suitable for intuitionistic logic and modal logics by simply modifying a component testing complementarity of two connected atoms. Since we avoid increasing the length of formulae by transforming them to any normal form, we reduce the search space even in comparison to the classical case. Besides a short and elegant algorithm for pathchecking we present details of a specialized stringunification algorithm which is necessary for dealing with the nonclassical logics under consideration.
A Deduction Method Complete for Refutation and Finite Satisfiability
 In Proc. 6th European Workshop on Logics in Artificial Intelligence, LNAI
, 1998
"... . Database and Artificial Intelligence applications are briefly discussed and it is argued that they need deduction methods that are not only refutation complete but also complete for finite satisfiability. A novel deduction method is introduced for such applications. Instead of relying on Skolemiza ..."
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Cited by 20 (4 self)
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. Database and Artificial Intelligence applications are briefly discussed and it is argued that they need deduction methods that are not only refutation complete but also complete for finite satisfiability. A novel deduction method is introduced for such applications. Instead of relying on Skolemization, as most refutation methods do, the proposed method processes existential quantifiers in a special manner which makes it complete not only for refutation, but also for finite satisfiability. A main contribution of this paper is the proof of these results. Keywords: Artificial Intelligence, Expert Systems, Databases, Automated Reasoning, Finite Satisfiability. 1 Introduction For many applications of automated reasoning, the tableaux methods [32, 16, 34, 18] have the following advantages: They not only detect unsatisfiability but also generate models; they are close to common sense reasoning, hence easy to enhance with an explanation tool; and they are quite easy to adapt to the special...
Guiding Program Development Systems by a Connection Based Proof Strategy
 5 TH INTERNATIONAL WORKSHOP ON LOGIC PROGRAM SYNTHESIS AND TRANSFORMATION, LECTURE NOTES IN COMPUTER SCIENCE 1048
, 1996
"... We present an automated proof method for constructive logic based on Wallen’s matrix characterization for intuitionistic validity. The proof search strategy extends Bibel’s connection method for classical predicate logic. It generates a matrix proof which will then be transformed into a proof within ..."
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Cited by 18 (13 self)
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We present an automated proof method for constructive logic based on Wallen’s matrix characterization for intuitionistic validity. The proof search strategy extends Bibel’s connection method for classical predicate logic. It generates a matrix proof which will then be transformed into a proof within a standard sequent calculus. Thus we can use an efficient proof method to guide the development of constructive proofs in interactive proof/program development systems.
Converting nonclassical matrix proofs into sequentstyle systems
 CADE13, LNAI 1104
, 1996
"... Abstract. We present a uniform algorithm for transforming matrix proofs in classical, constructive, and modal logics into sequent style proofs. Making use of a similarity between matrix methods and Fitting’s prefixed tableaus we first develop a procedure for extracting a prefixed sequent proof from ..."
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Cited by 14 (8 self)
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Abstract. We present a uniform algorithm for transforming matrix proofs in classical, constructive, and modal logics into sequent style proofs. Making use of a similarity between matrix methods and Fitting’s prefixed tableaus we first develop a procedure for extracting a prefixed sequent proof from a given matrix proof. By considering the additional restrictions on the order of rule applications we then extend this procedure into an algorithm which generates a conventional sequent proof. Our algorithm is based on unified representations of matrix characterizations for various logics as well as of prefixed and usual sequent calculi. The peculiarities of a logic are encoded by certain parameters which are summarized in tables to be consulted by the algorithm. 1