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A Mechanization of Strong Kleene Logic for Partial Functions
 PROCEEDINGS OF THE 12TH CADE
, 1994
"... Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
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Cited by 28 (11 self)
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Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a resolution calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi.
Exploiting Data Dependencies in ManyValued Logics
 Journal of Applied NonClassical Logics
, 1996
"... . The purpose of this paper is to make some practically relevant results in automated theorem proving available to manyvalued logics with suitable modifications. We are working with a notion of manyvalued firstorder clauses which any finitelyvalued logic formula can be translated into and that h ..."
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Cited by 21 (7 self)
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. The purpose of this paper is to make some practically relevant results in automated theorem proving available to manyvalued logics with suitable modifications. We are working with a notion of manyvalued firstorder clauses which any finitelyvalued logic formula can be translated into and that has been used several times in the literature, but in an ad hoc way. We give a manyvalued version of polarity which in turn leads to natural manyvalued counterparts of Horn formulas, hyperresolution, and a DavisPutnam procedure. We show that the manyvalued generalizations share many of the desirable properties of the classical versions. Our results justify and generalize several earlier results on theorem proving in manyvalued logics. KEYWORDS: manyvalued logic, polarity, Horn formula, direct products of structures, resolution, DavisPutnam procedure Introduction The purpose of this paper is to make some practically relevant results in automated theorem proving available to manyvalue...
Elimination of Cuts in Firstorder Finitevalued Logics
 J. Inform. Process. Cybernet. (EIK
, 1994
"... A uniform construction for sequent calculi for finitevalued firstorder logics with distribution quantifiers is exhibited. Completeness, cutelimination and midsequent theorems are established. As an application, an analog of Herbrand's theorem for the fourvalued knowledgerepresentation logic of B ..."
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Cited by 15 (5 self)
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A uniform construction for sequent calculi for finitevalued firstorder logics with distribution quantifiers is exhibited. Completeness, cutelimination and midsequent theorems are established. As an application, an analog of Herbrand's theorem for the fourvalued knowledgerepresentation logic of Belnap and Ginsberg is presented. It is indicated how this theorem can be used for reasoning about knowledge bases with incomplete and inconsistent information.
Automated Theorem Proving by Resolution for FinitelyValued Logics Based on Distributive Lattices with Operators
 An International Journal of MultipleValued Logic
, 1999
"... In this paper we present a method for automated theorem proving in manyvalued logics whose algebra of truth values is a nite distributive lattice with operators. This class of manyvalued logics includes many logics that occur in a natural way in applications. The method uses the Priestley dual of t ..."
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Cited by 11 (2 self)
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In this paper we present a method for automated theorem proving in manyvalued logics whose algebra of truth values is a nite distributive lattice with operators. This class of manyvalued logics includes many logics that occur in a natural way in applications. The method uses the Priestley dual of the algebra of truth values instead of the algebra itself; this dual is used as a finite set of possible worlds. We first present a procedure that constructs, for every formula in the language of such a logic, a set of signed clauses such that is a theorem if and only if is unsatisfiable. Compared to related approaches, the method presented here leads in many cases to a reduction of the number of clauses that are generated, especially when the set of truth values is not linearly ordered. We then discuss several possibilities for checking the unsatisfiability of , among which a version of signed hyperresolution, and give several examples.
Systematic Construction of Natural Deduction Systems for Manyvalued Logics
 In Proc. 23rd International Symposium on Multiplevalued Logic
, 1993
"... A construction principle for natural deduction systems for arbitrary nitelymanyvalued rst order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and ..."
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Cited by 10 (3 self)
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A construction principle for natural deduction systems for arbitrary nitelymanyvalued rst order logics is exhibited. These systems are systematically obtained from sequent calculi, which in turn can be automatically extracted from the truth tables of the logics under consideration. Soundness and cutfree completeness of these sequent calculi translate into soundness, completeness and normal form theorems for the natural deduction systems.
Signed Formula Logic Programming: Operational Semantics and Applications
, 1995
"... . Signed formula can be used to reason about a wide variety of multiplevalued logics. The formal theoretical foundation of logic programming based on signed formulas is developed in [26]. In this paper, the operational semantics of signed formula logic programming is investigated through constraint ..."
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Cited by 9 (0 self)
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. Signed formula can be used to reason about a wide variety of multiplevalued logics. The formal theoretical foundation of logic programming based on signed formulas is developed in [26]. In this paper, the operational semantics of signed formula logic programming is investigated through constraint logic programming. Applications to bilattice logic programming and truthmaintenance are considered. Keywords: Logic for Artificial Intelligence, Multiplevalued Logic, Signed Formula, Constraint Logic Programming, TruthMaintenance, Bilattices * Please address all correspondence to: James Lu Department of Computer Science Bucknell University Lewisburg, PA 17837 U.S.A. Email: jameslu@bucknell.edu Phone: +1 717 524 1394 Fax: +1 717 524 1822 ** Work supported in part by the NSF under Grant CCR9225037. SIGNED FORMULA LOGIC PROGRAMMING 1 1. Introduction The logic of signed formulas facilitates the examination of questions regarding multiplevalued logics through classical logic. As...
Deduction in ManyValued Logics: a Survey
 Mathware & Soft Computing, iv(2):6997
, 1997
"... this article, there is considerable activity in MVL deduction which is why we felt justified in writing this survey. Needless to say, we cannot give a general introduction to MVL in the present context. For this, we have to refer to general treatments such as [153, 53, 93]. 2 A classification of man ..."
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Cited by 8 (1 self)
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this article, there is considerable activity in MVL deduction which is why we felt justified in writing this survey. Needless to say, we cannot give a general introduction to MVL in the present context. For this, we have to refer to general treatments such as [153, 53, 93]. 2 A classification of manyvalued logics according to their intended application
A Tableau Calculus for Partial Functions
, 1996
"... . Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago, but there has not been a satisfactory mechanization. ..."
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Cited by 6 (5 self)
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. Even though it is not very often admitted, partial functions do play a significant role in many practical applications of deduction systems. Kleene has already given a semantic account of partial functions using a threevalued logic decades ago, but there has not been a satisfactory mechanization. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. However, strong Kleene logic, where quantification is restricted and therefore not truthfunctional, does not fit the framework directly. We solve this problem by applying recent methods from sorted logics. This paper presents a tableau calculus that combines the proper treatment of partial functions with the efficiency of sorted calculi. Keywords: Partial functions, manyvalued logic, sorted logic, tableau. 1 Introduction Many practical applications of deduction systems in mathematics and computer science rely on the correct and efficient treatment of partial functions. For this purpose...
A Resolution Calculus for Presuppositions
 Proceedings of the 12th ECAI
, 1996
"... . The semantics of everyday language and the semantics of its naive translation into classical firstorder language considerably differ. An important discrepancy that is addressed in this paper is about the implicit assumption what exists. For instance, in the case of universal quantification natura ..."
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Cited by 4 (3 self)
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. The semantics of everyday language and the semantics of its naive translation into classical firstorder language considerably differ. An important discrepancy that is addressed in this paper is about the implicit assumption what exists. For instance, in the case of universal quantification natural language uses restrictions and presupposes that these restrictions are nonempty, while in classical logic it is only assumed that the whole universe is nonempty. On the other hand, all constants mentioned in classical logic are presupposed to exist, while it makes no problems to speak about hypothetical objects in everyday language. These problems have been discussed in philosophical logic and some adequate manyvalued logics were developed to model these phenomena much better than classical firstorder logic can do. An adequate calculus, however, has not yet been given. Recent years have seen a thorough investigation of the framework of manyvalued truthfunctional logics. Unfortunately, restricted quantifications are not truthfunctional, hence they do not fit the framework directly. We solve this problem by applying recent methods from sorted logics.