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Thomas Lukasiewicz - One of the best experts on this subject based on the ideXlab platform.

  • expressive probabilistic description logics
    Artificial Intelligence, 2008
    Co-Authors: Thomas Lukasiewicz
    Abstract:

    The work in this paper is directed towards sophisticated formalisms for reasoning under probabilistic uncertainty in ontologies in the Semantic Web. Ontologies play a central role in the development of the Semantic Web, since they provide a precise definition of shared terms in web resources. They are expressed in the standardized web ontology language OWL, which consists of the three increasingly expressive sublanguages OWL Lite, OWL DL, and OWL Full. The sublanguages OWL Lite and OWL DL have a formal semantics and a reasoning support through a mapping to the expressive description logics SHIF(D) and SHOIN(D), respectively. In this paper, we present the expressive probabilistic description logics P-SHIF(D) and P-SHOIN(D), which are probabilistic extensions of these description logics. They allow for expressing rich terminological probabilistic Knowledge about concepts and roles as well as assertional probabilistic Knowledge about instances of concepts and roles. They are semantically based on the notion of probabilistic lexicographic entailment from probabilistic default reasoning, which naturally interprets this terminological and assertional probabilistic Knowledge as Knowledge about random and concrete instances, respectively. As an important additional feature, they also allow for expressing terminological default Knowledge, which is semantically interpreted as in Lehmann's lexicographic entailment in default reasoning from Conditional Knowledge bases. Another important feature of this extension of SHIF(D) and SHOIN(D) by probabilistic uncertainty is that it can be applied to other classical description logics as well. We then present sound and complete algorithms for the main reasoning problems in the new probabilistic description logics, which are based on reductions to reasoning in their classical counterparts, and to solving linear optimization problems. In particular, this shows the important result that reasoning in the new probabilistic description logics is decidable/computable. Furthermore, we also analyze the computational complexity of the main reasoning problems in the new probabilistic description logics in the general as well as restricted cases.

  • Probabilistic Default Reasoning with Conditional Constraints
    Annals of Mathematics and Artificial Intelligence, 2002
    Co-Authors: Thomas Lukasiewicz
    Abstract:

    We present an approach to reasoning from statistical and subjective Knowledge, which is based on a combination of probabilistic reasoning from Conditional constraints with approaches to default reasoning from Conditional Knowledge bases. More precisely, we introduce the notions of z -, lexicographic, and Conditional entailment for Conditional constraints, which are probabilistic generalizations of Pearl's entailment in system Z , Lehmann's lexicographic entailment, and Geffner's Conditional entailment, respectively. We show that the new formalisms have nice properties. In particular, they show a similar behavior as reference-class reasoning in a number of uncontroversial examples. The new formalisms, however, also avoid many drawbacks of reference-class reasoning. More precisely, they can handle complex scenarios and even purely probabilistic subjective Knowledge as input. Moreover, conclusions are drawn in a global way from all the available Knowledge as a whole. We then show that the new formalisms also have nice general nonmonotonic properties. In detail, the new notions of z -, lexicographic, and Conditional entailment have similar properties as their classical counterparts. In particular, they all satisfy the rationality postulates proposed by Kraus, Lehmann, and Magidor, and they have some general irrelevance and direct inference properties. Moreover, the new notions of z - and lexicographic entailment satisfy the property of rational monotonicity. Furthermore, the new notions of z -, lexicographic, and Conditional entailment are proper generalizations of both their classical counterparts and the classical notion of logical entailment for Conditional constraints. Finally, we provide algorithms for reasoning under the new formalisms, and we analyze its computational complexity.

  • default reasoning from Conditional Knowledge bases complexity and tractable cases
    Artificial Intelligence, 2000
    Co-Authors: Thomas Eiter, Thomas Lukasiewicz
    Abstract:

    Abstract Conditional Knowledge bases have been proposed as belief bases that include defeasible rules (also called defaults) of the form “ φ→ψ ”, which informally read as “generally, if φ then ψ ”. Such rules may have exceptions, which can be handled in different ways. A number of entailment semantics for Conditional Knowledge bases have been proposed in the literature. However, while the semantic properties and interrelationships of these formalisms are quite well understood, about their computational properties only partial results are known so far. In this paper, we fill these gaps and first draw a precise picture of the complexity of default reasoning from Conditional Knowledge bases: Given a Conditional Knowledge base KB and a default φ→ψ , does KB entail φ→ψ ? We classify the complexity of this problem for a number of well-known approaches (including Goldszmidt et al.'s maximum entropy approach and Geffner's Conditional entailment), where we consider the general propositional case as well as natural syntactic restrictions (in particular, to Horn and literal-Horn Conditional Knowledge bases). As we show, the more sophisticated semantics for Conditional Knowledge bases are plagued with intractability in all these fragments. We thus explore cases in which these semantics are tractable, and find that most of them enjoy this property on feedback-free Horn Conditional Knowledge bases, which constitute a new, meaningful class of Conditional Knowledge bases. Furthermore, we generalize previous tractability results from Horn to q-Horn Conditional Knowledge bases, which allow for a limited use of disjunction. Our results complement and extend previous results, and contribute in refining the tractability/intractability frontier of default reasoning from Conditional Knowledge bases. They provide useful insight for developing efficient implementations.

  • probabilistic default reasoning with Conditional constraints
    arXiv: Artificial Intelligence, 2000
    Co-Authors: Thomas Lukasiewicz
    Abstract:

    We propose a combination of probabilistic reasoning from Conditional constraints with approaches to default reasoning from Conditional Knowledge bases. In detail, we generalize the notions of Pearl's entailment in system Z, Lehmann's lexicographic entailment, and Geffner's Conditional entailment to Conditional constraints. We give some examples that show that the new notions of z-, lexicographic, and Conditional entailment have similar properties like their classical counterparts. Moreover, we show that the new notions of z-, lexicographic, and Conditional entailment are proper generalizations of both their classical counterparts and the classical notion of logical entailment for Conditional constraints.

Giuseppe Sanfilippo - One of the best experts on this subject based on the ideXlab platform.

  • probabilistic entailment in the setting of coherence the role of quasi conjunction and inclusion relation
    International Journal of Approximate Reasoning, 2013
    Co-Authors: Angelo Gilio, Giuseppe Sanfilippo
    Abstract:

    In this paper, by adopting a coherence-based probabilistic approach to default reasoning, we focus the study on the logical operation of quasi conjunction and the Goodman-Nguyen inclusion relation for Conditional events. We recall that quasi conjunction is a basic notion for defining consistency of Conditional Knowledge bases. By deepening some results given in a previous paper we show that, given any finite family of Conditional events F and any nonempty subset S of F, the family F p-entails the quasi conjunction C(S); then, given any Conditional event E|H, we analyze the equivalence between p-entailment of E|H from F and p-entailment of E|H from C(S), where S is some nonempty subset of F We also illustrate some alternative theorems related with p-consistency and p-entailment. Finally, we deepen the study of the connections between the notions of p-entailment and inclusion relation by introducing for a pair (F,E|H) the (possibly empty) class K of the subsets S of F such that CS implies E|H. We show that the class K satisfies many properties; in particular K is additive and has a greatest element which can be determined by applying a suitable algorithm.

  • quasi conjunction and inclusion relation in probabilistic default reasoning
    European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, 2011
    Co-Authors: Angelo Gilio, Giuseppe Sanfilippo
    Abstract:

    We study in the setting of probabilistic default reasoning under coherence the quasi conjunction, which is a basic notion for defining consistency of Conditional Knowledge bases, and the Goodman & Nguyen inclusion relation for Conditional events. We deepen two results given in a previous paper: the first result concerns p-entailment from a finite family F of Conditional events to the quasi conjunction C(S), for each nonempty subset S of F; the second result analyzes the equivalence between p-entailment from F and p-entailment from C(S), where S is some nonempty subset of F. We also characterize p-entailment by some alternative theorems. Finally, we deepen the connections between p-entailment and inclusion relation, by introducing for a pair (F,E|H) the class of the subsets S of F such that C(S) implies E|H. This class is additive and has a greatest element which can be determined by applying a suitable algorithm.

  • quasi conjunction and inclusion relation in probabilistic default reasoning
    European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, 2011
    Co-Authors: Angelo Gilio, Giuseppe Sanfilippo
    Abstract:

    We study in the setting of probabilistic default reasoning under coherence the quasi conjunction, which is a basic notion for defining consistency of Conditional Knowledge bases, and the Goodman & Nguyen inclusion relation for Conditional events. We deepen two results given in a previous paper: the first result concerns p-entailment from a finite family F of Conditional events to the quasi conjunction C(S), for each nonempty subset S of F; the second result analyzes the equivalence between p-entailment from F and p-entailment from C(S), where S is some nonempty subset of F. We also characterize p-entailment by some alternative theorems. Finally, we deepen the connections between p-entailment and inclusion relation, by introducing for a pair (F,E|H) the class of the subsets S of F such that C(S) implies E|H. This class is additive and has a greatest element which can be determined by applying a suitable algorithm.

Angelo Gilio - One of the best experts on this subject based on the ideXlab platform.

  • probabilistic entailment in the setting of coherence the role of quasi conjunction and inclusion relation
    International Journal of Approximate Reasoning, 2013
    Co-Authors: Angelo Gilio, Giuseppe Sanfilippo
    Abstract:

    In this paper, by adopting a coherence-based probabilistic approach to default reasoning, we focus the study on the logical operation of quasi conjunction and the Goodman-Nguyen inclusion relation for Conditional events. We recall that quasi conjunction is a basic notion for defining consistency of Conditional Knowledge bases. By deepening some results given in a previous paper we show that, given any finite family of Conditional events F and any nonempty subset S of F, the family F p-entails the quasi conjunction C(S); then, given any Conditional event E|H, we analyze the equivalence between p-entailment of E|H from F and p-entailment of E|H from C(S), where S is some nonempty subset of F We also illustrate some alternative theorems related with p-consistency and p-entailment. Finally, we deepen the study of the connections between the notions of p-entailment and inclusion relation by introducing for a pair (F,E|H) the (possibly empty) class K of the subsets S of F such that CS implies E|H. We show that the class K satisfies many properties; in particular K is additive and has a greatest element which can be determined by applying a suitable algorithm.

  • quasi conjunction and inclusion relation in probabilistic default reasoning
    European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, 2011
    Co-Authors: Angelo Gilio, Giuseppe Sanfilippo
    Abstract:

    We study in the setting of probabilistic default reasoning under coherence the quasi conjunction, which is a basic notion for defining consistency of Conditional Knowledge bases, and the Goodman & Nguyen inclusion relation for Conditional events. We deepen two results given in a previous paper: the first result concerns p-entailment from a finite family F of Conditional events to the quasi conjunction C(S), for each nonempty subset S of F; the second result analyzes the equivalence between p-entailment from F and p-entailment from C(S), where S is some nonempty subset of F. We also characterize p-entailment by some alternative theorems. Finally, we deepen the connections between p-entailment and inclusion relation, by introducing for a pair (F,E|H) the class of the subsets S of F such that C(S) implies E|H. This class is additive and has a greatest element which can be determined by applying a suitable algorithm.

  • quasi conjunction and inclusion relation in probabilistic default reasoning
    European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty, 2011
    Co-Authors: Angelo Gilio, Giuseppe Sanfilippo
    Abstract:

    We study in the setting of probabilistic default reasoning under coherence the quasi conjunction, which is a basic notion for defining consistency of Conditional Knowledge bases, and the Goodman & Nguyen inclusion relation for Conditional events. We deepen two results given in a previous paper: the first result concerns p-entailment from a finite family F of Conditional events to the quasi conjunction C(S), for each nonempty subset S of F; the second result analyzes the equivalence between p-entailment from F and p-entailment from C(S), where S is some nonempty subset of F. We also characterize p-entailment by some alternative theorems. Finally, we deepen the connections between p-entailment and inclusion relation, by introducing for a pair (F,E|H) the class of the subsets S of F such that C(S) implies E|H. This class is additive and has a greatest element which can be determined by applying a suitable algorithm.

  • on csaszar s condition in nonmonotonic reasoning
    Non-Monotonic Reasoning, 2004
    Co-Authors: Angelo Gilio
    Abstract:

    Csaszar’s condition is a well-known property introduced about 50 years ago in the axiomatic theory of Conditional probability. In recent years such condition has been reconsidered by some authors, who have studied its role in the coherence-based approach to Conditional probability. In this paper we consider the probabilistic entailment of a Conditional Knowledge base by another one. We represent Loop rule in a generalized way and, using Csaszar’s condition, we give a simple probabilistic interpretation of it. Then, exploiting the rules Cautious Monotonicity and Cut, we obtain some related results on p-entailment by the Knowledge base associated with Loop rule. We also determine the best probability bounds for the quasi-conjunction of two Conditional events and we give a probabilistic semantics for the QAND rule. Finally, we reconsider our results in the setting of Conditional objects.

Henri Prade - One of the best experts on this subject based on the ideXlab platform.

  • nonmonotonic reasoning Conditional objects and possibility theory
    Artificial Intelligence, 1997
    Co-Authors: Salem Benferhat, Didier Dubois, Henri Prade
    Abstract:

    Abstract This short paper relates the Conditional object-based and possibility theory-based approaches for reasoning with Conditional statements pervaded with exceptions, to other methods in nonmonotonic reasoning which have been independently proposed: namely, Lehmann's preferential and rational closure entailments which obey normative postulates, the infinitesimal probability approach, and the Conditional (modal) logics-based approach. All these methods are shown to be equivalent with respect to their capabilities for reasoning with Conditional Knowledge although they are based on different modeling frameworks. It thus provides a unified understanding of nonmonotonic consequence relations. More particularly, Conditional objects, a purely qualitative counterpart to Conditional probabilities, offer a very simple semantics, based on a 3-valued calculus, for the preferential entailment, while in the purely ordinal setting of possibility theory both the preferential and the rational closure entailments can be represented.

Hykel Hosni - One of the best experts on this subject based on the ideXlab platform.

  • boolean algebras of Conditionals probability and logic
    Artificial Intelligence, 2020
    Co-Authors: Tommaso Flaminio, Lluis Godo, Hykel Hosni
    Abstract:

    Abstract This paper presents an investigation on the structure of Conditional events and on the probability measures which arise naturally in that context. In particular we introduce a construction which defines a (finite) Boolean algebra of Conditionals from any (finite) Boolean algebra of events. By doing so we distinguish the properties of Conditional events which depend on probability and those which are intrinsic to the logico-algebraic structure of Conditionals. Our main result provides a way to regard standard two-place Conditional probabilities as one-place probability functions on Conditional events. We also consider a logical counterpart of our Boolean algebras of Conditionals with links to preferential consequence relations for non-monotonic reasoning. The overall framework of this paper provides a novel perspective on the rich interplay between logic and probability in the representation of Conditional Knowledge.