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Nondeterministic matrices and modular semantics of rules
 in Logica Universalis
, 2005
"... Abstract. We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, ..."
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Cited by 20 (9 self)
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Abstract. We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, which are multivalued structures in which the value assigned by a valuation to a complex formula can be chosen nondeterministically out of a certain nonempty set of options. The method is applied in the area of logics with a formal consistency operator (known as LFIs), allowing us to provide in a modular way effective, finite semantics for thousands of different LFIs.
Nondeterministic Semantics for Logics with a Consistency Operator
 IN THE INTERNATIONAL JOURNAL OF APPROXIMATE REASONING
, 2006
"... In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s appr ..."
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Cited by 19 (11 self)
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In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. Da Costa’s approach has led to the family of logics of formal (in)consistency (LFIs). In this paper we provide in a modular way simple nondeterministic semantics for 64 of the most important logics from this family. Our semantics is 3valued for some of the systems, and infinitevalued for the others. We prove that these results cannot be improved: neither of the systems with a threevalued nondeterministic semantics has either a finite characteristic ordinary matrix or a twovalued characteristic nondeterministic matrix, and neither of the other systems we investigate has a finite characteristic nondeterministic matrix. Still, our semantics provides decision procedures for all the systems investigated, as well as easy proofs of important prooftheoretical properties of them.
Logical Nondeterminism as a Tool for Logical Modularity: An Introduction
 in We Will Show Them: Essays in Honor of Dov Gabbay, Vol
, 2005
"... It is well known that every propositional logic which satisfies certain very ..."
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Cited by 13 (10 self)
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It is well known that every propositional logic which satisfies certain very
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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Cited by 7 (4 self)
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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.
The Enduring Scandal of Deduction. Is Propositional Logic really Uninformative
 University of Hertfordshire & University of Ferrara
, 2007
"... Abstract. Deductive inference is usually regarded as being “tautological ” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of firstorder logic and with the (likely) intractabilit ..."
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Cited by 3 (0 self)
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Abstract. Deductive inference is usually regarded as being “tautological ” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of firstorder logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the prooftheoretical point of view and propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth ” or “informativeness ” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic ” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed.
A logic for approximate firstorder reasoning
 In Proc. CSL’01
, 2001
"... Abstract. In classical approaches to knowledge representation, reasoners are assumed to derive all the logical consequences of their knowledge base. As a result, reasoning in the firstorder case is only semidecidable. Even in the restricted case of finite universes of discourse, reasoning remains ..."
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Cited by 1 (1 self)
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Abstract. In classical approaches to knowledge representation, reasoners are assumed to derive all the logical consequences of their knowledge base. As a result, reasoning in the firstorder case is only semidecidable. Even in the restricted case of finite universes of discourse, reasoning remains inherently intractable, as the reasoner has to deal with two independent sources of complexity: unbounded chaining and unbounded quantification. The purpose of this study is to handle these difficulties in a logicoriented framework based on the paradigm of approximate reasoning. The logic is semantically founded on the notion of resource, an accuracy measure, which controls at the same time the two barriers of complexity. Moreover, a stepwise technique is included for improving approximations. Finally, both sound approximations and complete ones are covered. Based on the logic, we develop an approximation algorithm with a simple modification of classical instancebased theorem provers. The procedure yields approximate proofs whose precision increases as the reasoner has more resources at her disposal. The algorithm is interruptible, improvable, dual, and can be exploited for anytime computation. Moreover, the algorithm is flexible enough to be used with a wide range of propositional satisfiability methods.