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Representing Paraconsistent Reasoning via Quantified Propositional Logic
 In Inconsistency Tolerance, volume 3300 of LNCS
, 2005
"... Abstract. Quantified propositional logic is an extension of classical propositional logic where quantifications over atomic formulas are permitted. As such, quantified propositional logic is a fragment of secondorder logic, and its sentences are usually referred to as quantified Boolean formulas (QB ..."
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Abstract. Quantified propositional logic is an extension of classical propositional logic where quantifications over atomic formulas are permitted. As such, quantified propositional logic is a fragment of secondorder logic, and its sentences are usually referred to as quantified Boolean formulas (QBFs). The motivation to study quantified propositional logic for paraconsistent reasoning is based on two fundamental observations. Firstly, in recent years, practicably efficient solvers for quantified propositional logic have been presented. Secondly, complexity results imply that there is a wide range of paraconsistent reasoning problems which can be efficiently represented in terms of QBFs. Hence, solvers for QBFs can be used as a core engine in systems prototypically implementing several of such reasoning tasks, most of them lacking concrete realisations. To this end, we show how certain paraconsistent reasoning principles can be naturally formulated or reformulated by means of quantified Boolean formulas. More precisely, we describe polynomialtime constructible encodings providing axiomatisations of the given reasoning tasks. In this way, a whole variety of a priori distinct approaches to paraconsistent reasoning become comparable in a uniform setting. 1
A Logic of Limited Belief for Reasoning with Disjunctive Information
 Proceedings of the 9th International Conference on Principles of Knowledge Representation and Reasoning (KR04
, 2004
"... The goal of producing a general purpose, semantically motivated, and computationally tractable deductive reasoning service remains surprisingly elusive. By and large, approaches that come equipped with a perspicuous model theory either result in reasoners that are too limited from a practical point ..."
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The goal of producing a general purpose, semantically motivated, and computationally tractable deductive reasoning service remains surprisingly elusive. By and large, approaches that come equipped with a perspicuous model theory either result in reasoners that are too limited from a practical point of view or fall off the computational cliff. In this paper, we propose a new logic of belief called SL which lies between the two extremes. We show that query evaluation based on SL for a certain form of knowledge bases with disjunctive information is tractable in the propositional case and decidable in the firstorder case. Also, we present a sound and complete axiomatization for propositional SL.
Threevalued Logics for Inconsistency Handling
"... While threevalued paraconsistent logic is a valuable framework for reasoning under inconsistency, the corresponding basic inference relation is too cautious and fails in discriminating in a finegrained way the set of expected consequences of belief bases. To address both issues, we point out more ..."
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While threevalued paraconsistent logic is a valuable framework for reasoning under inconsistency, the corresponding basic inference relation is too cautious and fails in discriminating in a finegrained way the set of expected consequences of belief bases. To address both issues, we point out more refined inference relations. We analyze them from the logical and computational points of view and we compare them with respect to their relative cautiousness.
When data dependencies over SQL tables meet the Logics of Paradox and S3
 In Proceedings of the 29th ACM SIGMODSIGARTSIGACT Symposium on Principles of Database Systems (PoDS). ACM
"... We study functional and multivalued dependencies over SQL tables with NOT NULL constraints. Under a noinformation interpretation of null values we develop tools for reasoning. We further show that in the absence of NOT NULL constraints the associated implication problem is equivalent to that in pr ..."
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We study functional and multivalued dependencies over SQL tables with NOT NULL constraints. Under a noinformation interpretation of null values we develop tools for reasoning. We further show that in the absence of NOT NULL constraints the associated implication problem is equivalent to that in propositional fragments of Priest’s paraconsistent Logic of Paradox. Subsequently, we extend the equivalence to Boolean dependencies and to the presence of NOT NULL constraints using Schaerf and Cadoli’s S3 logics where S corresponds to the set of attributes declared NOT NULL. The findings also apply to Codd’s interpretation “value at present unknown”utilizing a weak possible world semantics. Our results establish NOT NULL constraints as an effective mechanism to balance the expressiveness and tractability of consequence relations, and to control the degree by which the existing classical theory of data dependencies can be soundly approximated in practice.
Paraconsistent preferential reasoning by signed quantified Boolean formulae
 In Proc. 16th European Conference on Artificial Intelligence (ECAI’04), R. López de Mántaras and
"... Abstract. We introduce a uniform approach of representing a variety of paraconsistent nonmonotonic formalisms by quantified Boolean formulae (QBFs) in the context of fourvalued semantics. This framework provides a useful platform for capturing, in a simple and natural way, a wide range of methods ..."
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Abstract. We introduce a uniform approach of representing a variety of paraconsistent nonmonotonic formalisms by quantified Boolean formulae (QBFs) in the context of fourvalued semantics. This framework provides a useful platform for capturing, in a simple and natural way, a wide range of methods for preferential reasoning. Offtheshelf QBF solvers may therefore be incorporated for simulating the corresponding consequence relations. 1
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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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
Tutorial: Complexity of ManyValued Logics
 In Proc. 31st International Symposium on MultipleValued Logics, IEEE CS Press, Los Alamitos
, 2001
"... this article selfcontained. ..."
The implication problem of data dependencies over SQL table definitions: axiomatic, algorithmic and logical characterizations
"... We investigate the implication problem for classes of data dependencies over SQL table definitions. Under Zaniolo’s “no information ” interpretation of null markers we establish an axiomatization and algorithms to decide the implication problem for the combined class of functional and multivalued de ..."
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Cited by 7 (4 self)
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We investigate the implication problem for classes of data dependencies over SQL table definitions. Under Zaniolo’s “no information ” interpretation of null markers we establish an axiomatization and algorithms to decide the implication problem for the combined class of functional and multivalued dependencies in the presence of NOT NULL constraints. The resulting theory subsumes three previously orthogonal frameworks. We further show that the implication problem of this class is equivalent to that in a propositional fragment of Cadoli and Schaerf ’s family of paraconsistent S3 logics. In particular, S is the set of variables that correspond to attributes declared NOT NULL. We also show how our equivalences for multivalued dependencies can be extended to Delobel’s class of full firstorder hierarchical decompositions, and the equivalences for functional dependencies can be extended to arbitrary Boolean dependencies. These dualities allow us to transfer several findings from the propositional fragments to the corresponding classes of data dependencies, and vice versa. We show that our results also apply to Codd’s null interpretation “value unknown at present”, but not to Imielinski’s orrelations utilizing Levene and Loizou’s weak possible world semantics. Our findings establish NOT NULL constraints as an effective mechanism to balance not only the certainty in database relations but also the expressiveness with the efficiency of entailment relations. They also control the degree by which the implication of data dependencies over total relations is soundly approximated
Computing inconsistency measure based on paraconsistent semantics
 J. Log. Comput
, 2011
"... Abstract. Measuring inconsistency in knowledge bases has been recognized as an important problem in several research areas. Many methods have been proposed to solve this problem and a main class of them is based on some kind of paraconsistent semantics. However, existing methods suffer from two li ..."
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Abstract. Measuring inconsistency in knowledge bases has been recognized as an important problem in several research areas. Many methods have been proposed to solve this problem and a main class of them is based on some kind of paraconsistent semantics. However, existing methods suffer from two limitations: 1) They are mostly restricted to propositional knowledge bases; 2) Very few of them discuss computational aspects of computing inconsistency measures. In this paper, we try to solve these two limitations by exploring algorithms for computing an inconsistency measure of firstorder knowledge bases. After introducing a fourvalued semantics for firstorder logic, we define an inconsistency measure of a firstorder knowledge base, which is a sequence of inconsistency degrees. We then propose a precise algorithm to compute our inconsistency measure. We show that this algorithm reduces the computation of the inconsistency measure to classical satisfiability checking. This is done by introducing a new semantics, named S[n]4 semantics, which can be calculated by invoking a classical SAT solver. Moreover, we show that this auxiliary semantics also gives a direct way to compute upper and lower bounds of inconsistency degrees. That is, it can be easily revised to compute approximating inconsistency measures. The approximating inconsistency measures converge to the precise values if enough resources are available. Finally, by some nice properties of the S[n]4 semantics, we show that some upper and lower bounds can be computed in Ptime, which says that the problem of computing these approximating inconsistency measures is tractable. 1
Paraconsistent reasoning and preferential entailments by signed quantified boolean formulae
 2007. Maier, Yue Ma, Pascal Hitzler / Paraconsistent OWL and Related Logics 33 hal00705876, version 1  8 Jun 2012
"... We introduce a uniform approach of representing a variety of paraconsistent nonmonotonic formalisms by quantified Boolean formulae (QBFs) in the context of multiplevalued logics. We show that this framework provides a useful platform for capturing, in a simple and natural way, a wide range of met ..."
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We introduce a uniform approach of representing a variety of paraconsistent nonmonotonic formalisms by quantified Boolean formulae (QBFs) in the context of multiplevalued logics. We show that this framework provides a useful platform for capturing, in a simple and natural way, a wide range of methods for preferential reasoning. The outcome is a subtle approach to represent the underlying formalisms, which induces a straightforward way to compute the corresponding entailments: by incorporating offtheshelf QBF solvers it is possible to simulate within our framework various kinds of preferential formalisms, among which are Priest’s logic LPm of reasoning with minimal inconsistency, Batens ’ adaptive logic ACLuNs2, Besnard and Schaub’s inference relation =n, a variety of formulapreferential systems, some bilatticebased preferential relations (e.g., =I1 and =I2), and consequence relations for reasoning with graded uncertainty (such as the fourvalued logic =4c).