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Models with both finite and arbitrary constant domains are investigated. We deal with the possible world semantics under the constant domain assumption and show that the expanding and varying domain assumptions are reducible to it. The main aim is to develop methods of proving decidability of the satisfiability problem for this language and apply them to description logics with most important temporal and epistemic operators, thereby obtaining satisfiability checking. The paper considers the standard concept description language ALC augmented with various kinds of modal operators which can be applied to concepts and axioms. Furthermore, we extend the inconsistency graph concept with a labeling that extends the hierarchy to include some other types of inconsistency measures. Then we introduce abstractions of the inconsistency graph and use them to construct a hierarchy of syntactic inconsistency measures.
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We then show that various measures can be computed using the inconsistency graph. It is based on the notion of an inconsistency graph for each knowledgebase (a bipartite graph with a set of vertices representing formulas in the knowledgebase, a set of vertices representing minimal inconsistent subsets of the knowledgebase, and edges representing that a formula belongs to a minimal inconsistent subset). To address these problems, we introduce a general framework for comparing syntactic measures of inconsistency.
#Ways to use the modal logic playground how to
But to date, it is not clear how to delineate the space of options for measures, nor is it clear how we can classify measures systematically. A number of proposals have been made to define measures of inconsistency. The aim of measuring inconsistency is to obtain an evaluation of the imperfections in a set of formulas, and this evaluation may then be used to help decide on some course of action (such as rejecting some of the formulas, resolving the inconsistency, seeking better sources of information, etc). Finally, we define a new concept, weak inconsistency measure, and show how to compute it. We show how to extend propositional logic inconsistency measures to such sets of formulas. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them.