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78
Logics of Formal Inconsistency
 Handbook of Philosophical Logic
"... 1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory ..."
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Cited by 45 (19 self)
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1.1 Contradictoriness and inconsistency, consistency and noncontradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory
Logic Programs and Connectionist Networks
 Journal of Applied Logic
, 2004
"... One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic p ..."
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Cited by 44 (15 self)
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One facet of the question of integration of Logic and Connectionist Systems, and how these can complement each other, concerns the points of contact, in terms of semantics, between neural networks and logic programs. In this paper, we show that certain semantic operators for propositional logic programs can be computed by feedforward connectionist networks, and that the same semantic operators for firstorder normal logic programs can be approximated by feedforward connectionist networks. Turning the networks into recurrent ones allows one also to approximate the models associated with the semantic operators. Our methods depend on a wellknown theorem of Funahashi, and necessitate the study of when Funahasi's theorem can be applied, and also the study of what means of approximation are appropriate and significant.
Uniform Semantic Treatment of Default and Autoepistemic Logics
 ARTIFICIAL INTELLIGENCE
, 2000
"... We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the latti ..."
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Cited by 41 (23 self)
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We revisit the issue of epistemological and semantic foundations for autoepistemic and default logics, two leading formalisms in nonmonotonic reasoning. We develop a general semantic approach to autoepistemic and default logics that is based on the notion of a belief pair and that exploits the lattice structure of the collection of all belief pairs. For each logic, we introduce a monotone operator on the lattice of belief pairs. We then show that a whole family of semantics can be defined in a systematic and principled way in terms of fixpoints of this operator (or as fixpoints of certain closely related operators). Our approach elucidates fundamental constructive principles in which agents form their belief sets, and leads to approximation semantics for autoepistemic and default logics. It also allows us to establish a precise onetoone correspondence between the family of semantics for default logic and the family of semantics for autoepistemic logic. The correspondence exploits the modal interpretation of a default proposed by Konolige. Our results establish conclusively that default logic can be viewed as a fragment of autoepistemic logic, a result that has been long anticipated. At the same time, they explain the source of the difficulty to formally relate the semantics of default extensions by Reiter and autoepistemic expansions by Moore. These two semantics occupy different locations in the corresponding families of semantics for default and autoepistemic logics.
Logic programming revisited: logic programs as inductive definitions
 ACM Transactions on Computational Logic
, 2001
"... Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iffdefinitions. A second approach was to ..."
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Cited by 34 (21 self)
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Logic programming has been introduced as programming in the Horn clause subset of first order logic. This view breaks down for the negation as failure inference rule. To overcome the problem, one line of research has been to view a logic program as a set of iffdefinitions. A second approach was to identify a unique canonical, preferred or intended model among the models of the program and to appeal to common sense to validate the choice of such model. Another line of research developed the view of logic programming as a nonmonotonic reasoning formalism strongly related to Default Logic and Autoepistemic Logic. These competing approaches have resulted in some confusion about the declarative meaning of logic programming. This paper investigates the problem and proposes an alternative epistemological foundation for the canonical model approach, which is not based on common sense but on a solid mathematical information principle. The thesis is developed that logic programming can be understood as a natural and general logic of inductive definitions. In particular, logic programs with negation represent nonmonotone inductive definitions. It is argued that this thesis results in an alternative justification of the wellfounded model as the unique intended model of the logic program. In addition, it equips logic programs with an easy to comprehend meaning
A Logic of NonMonotone Inductive Definitions and its Modularity Properties
, 2004
"... Wellknown principles of induction include monotone induction and dierent sorts of nonmonotone induction such as inationary induction, induction over wellordered sets and iterated induction. In this work, we de ne a logic formalizing induction over wellordered sets and monotone and iterated ..."
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Cited by 29 (20 self)
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Wellknown principles of induction include monotone induction and dierent sorts of nonmonotone induction such as inationary induction, induction over wellordered sets and iterated induction. In this work, we de ne a logic formalizing induction over wellordered sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for NonMonotone Inductive De nitions (NMIDlogic). The semantics of the logic is strongly inuenced by the wellfounded semantics of logic programming.
A logic of nonmonotone inductive definitions
 ACM transactions on computational logic
, 2007
"... Wellknown principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over wellfounded sets and iterated induction. In this work, we define a logic formalizing induction over wellfounded sets and monotone and iterated i ..."
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Cited by 28 (16 self)
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Wellknown principles of induction include monotone induction and different sorts of nonmonotone induction such as inflationary induction, induction over wellfounded sets and iterated induction. In this work, we define a logic formalizing induction over wellfounded sets and monotone and iterated induction. Just as the principle of positive induction has been formalized in FO(LFP), and the principle of inflationary induction has been formalized in FO(IFP), this paper formalizes the principle of iterated induction in a new logic for NonMonotone Inductive Definitions (IDlogic). The semantics of the logic is strongly influenced by the wellfounded semantics of logic programming. This paper discusses the formalisation of different forms of (non)monotone induction by the wellfounded semantics and illustrates the use of the logic for formalizing mathematical and commonsense knowledge. To model different types of induction found in mathematics, we define several subclasses of definitions, and show that they are correctly formalized by the wellfounded semantics. We also present translations into classical first or second order logic. We develop modularity and totality results and demonstrate their use to analyze and simplify complex definitions. We illustrate the use of the logic for temporal reasoning. The logic formally extends Logic Programming, Abductive Logic Programming and Datalog, and thus formalizes the view on these formalisms as logics of (generalized) inductive definitions. Categories and Subject Descriptors:... [...]:... 1.
Logic programs with abstract constraint atoms
 In Proceedings of the 19th National Conference on Artificial Intelligence (AAAI04
, 2004
"... We propose and study extensions of logic programming with constraints represented as generalized atoms of the form C(X), where X is a finite set of atoms and C is an abstract constraint (formally, a collection of sets of atoms). Atoms C(X) are satisfied by an interpretation (set of atoms) M, if M ∩ ..."
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Cited by 22 (5 self)
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We propose and study extensions of logic programming with constraints represented as generalized atoms of the form C(X), where X is a finite set of atoms and C is an abstract constraint (formally, a collection of sets of atoms). Atoms C(X) are satisfied by an interpretation (set of atoms) M, if M ∩ X ∈ C. We focus here on monotone constraints, that is, those collections C that are closed under the superset. They include, in particular, weight (or pseudoboolean) constraints studied both by the logic programming and SAT communities. We show that key concepts of the theory of normal logic programs such as the onestep provability operator, the semantics of supported and stable models, as well as several of their properties including complexity results, can be lifted to such case.
The nomore++ approach to answer set solving
 PROCEEDINGS OF THE TWELFTH INTERNATIONAL CONFERENCE ON LOGIC FOR PROGRAMMING, ARTIFICIAL INTELLIGENCE, AND REASONING
, 2005
"... We present a new answer set solver, called nomore++, along with its underlying theoretical foundations. A distinguishing feature is that it treats heads and bodies equitably as computational objects. Apart from its operational foundations, we show how it improves on previous work through its new lo ..."
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Cited by 21 (10 self)
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We present a new answer set solver, called nomore++, along with its underlying theoretical foundations. A distinguishing feature is that it treats heads and bodies equitably as computational objects. Apart from its operational foundations, we show how it improves on previous work through its new lookahead and its computational strategy of maintaining unfoundedfreeness. We underpin our claims by selected experimental results.
Approximations, Stable Operators, WellFounded Fixpoints And Applications In Nonmonotonic Reasoning
, 2000
"... In this paper we develop an algebraic framework for studying semantics of nonmonotonic logics. Our approach is formulated in the language of lattices, bilattices, operators and fixpoints. The goal is to describe fixpoints of an operator O defined on a lattice. The key intuition is that of an approxi ..."
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Cited by 18 (8 self)
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In this paper we develop an algebraic framework for studying semantics of nonmonotonic logics. Our approach is formulated in the language of lattices, bilattices, operators and fixpoints. The goal is to describe fixpoints of an operator O defined on a lattice. The key intuition is that of an approximation, a pair (x, y) of lattice elements which can be viewed as an approximation to each lattice element z such that x z y. The key notion is that of an approximating operator, a monotone operator on the bilattice of approximations whose fixpoints approximate the fixpoints of the operator O. The main contribution of the paper is an algebraic construction which assigns a certain operator, called the stable operator, to every approximating operator on a bilattice of approximations. This construction leads to an abstract version of the wellfounded semantics. In the paper we show that our theory offers a unified framework for semantic studies of logic programming, default logic and autoepistemic logic.
Tableau calculi for answer set programming
 PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON LOGIC PROGRAMMING (ICLP’06
, 2006
"... We introduce a formal proof system based on tableau methods for analyzing computations made in Answer Set Programming (ASP). Our approach furnishes declarative and finegrained instruments for characterizing operations as well as strategies of ASPsolvers. First, the granulation is detailed enough ..."
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Cited by 17 (6 self)
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We introduce a formal proof system based on tableau methods for analyzing computations made in Answer Set Programming (ASP). Our approach furnishes declarative and finegrained instruments for characterizing operations as well as strategies of ASPsolvers. First, the granulation is detailed enough to capture the variety of propagation and choice operations of algorithms used for ASP; this also includes SATbased approaches. Second, it is general enough to encompass the various strategies pursued by existing ASPsolvers. This provides us with a uniform framework for identifying and comparing fundamental properties of algorithms. Third, the approach allows us to investigate the proof complexity of algorithms for ASP, depending on choice operations. We show that exponentially different bestcase computations can be obtained for different ASPsolvers. Finally, our approach is flexible enough to integrate new inference patterns, so to study their relation to existing ones. As a result, we obtain a novel approach to unfounded set handling based on loops, being applicable to nonSATbased solvers. Furthermore, we identify backward propagation operations for unfounded sets.