Nonmonotonic Probabilistic Logics between Model-Theoretic Probabilistic Logic and Probabilistic Logic under Coherence

Recently, it has been shown that probabilistic entailment under coherence is weaker than model-theoretic probabilistic entailment. Moreover, probabilistic entailment under coherence is a generalization of default entailment in System P. In this paper, we continue this line of research by presenting probabilistic generalizations of more sophisticated notions of classical default entailment that lie between model-theoretic probabilistic entailment and probabilistic entailment under coherence. That is, the new formalisms properly generalize their counterparts in classical default reasoning, they are weaker than model-theoretic probabilistic entailment, and they are stronger than probabilistic entailment under coherence. The new formalisms are useful especially for handling probabilistic inconsistencies related to conditioning on zero events. They can also be applied for probabilistic belief revision. More generally, in the same spirit as a similar previous paper, this paper sheds light on exciting new formalisms for probabilistic reasoning beyond the well-known standard ones.Comment: 10 pages; in Proceedings of the 9th International Workshop on Non-Monotonic Reasoning (NMR-2002), Special Session on Uncertainty Frameworks in Nonmonotonic Reasoning, pages 265-274, Toulouse, France, April 200

Probabilistic Default Reasoning with Conditional Constraints

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.Comment: 8 pages; to appear in Proceedings of the Eighth International Workshop on Nonmonotonic Reasoning, Special Session on Uncertainty Frameworks in Nonmonotonic Reasoning, Breckenridge, Colorado, USA, 9-11 April 200

Merging the local and global approaches to probabilistic satisfiability

AbstractThe probabilistic satisfiability problem is to verify the consistency of a set of probability values or intervals for logical propositions. The (tight) probabilistic entailment problem is to find best bounds on the probability of an additional proposition. The local approach to these problems applies rules on small sets of logical sentences and probabilities to tighten given probability intervals. The global approach uses linear programming to find best bounds. We show that merging these approaches is profitable to both: local solutions can be used to find global solutions more quickly through stabilized column generation, and global solutions can be used to confirm or refute the optimality of the local solutions found. As a result, best bounds are found, together with their step-by-step justification

An anytime deduction heuristic for first order probabilistic logic

This thesis describes an anytime deduction heuristic to address the decision and optimization form of the First Order Probabilistic Logic problem which was revived by Nilsson in 1986. Reasoning under uncertainty is always an important issue for AI applications, e.g., expert systems, automated theorem-provers, etc. Among the proposed models and methods for dealing with uncertainty, some as, e.g., Nilsson's ones, are based on logic and probability. Nilsson revisited the early works of Boole (1854) and Hailperin (1976) and reformulated them in an AI framework. The decision form of the probabilistic logic problem, also known as PSAT, consists of finding, given a set of logical sentences together with their probability value to be true, whether the set of sentences and their probability value is consistent. In the optimization form, assuming that a system of probabilistic formulas is already consistent, the problem is: Given an additional sentence, find the tightest possible probability bounds such that the overall system remains consistent with that additional sentence. Solution schemes, both heuristic and exact, have been proposed within the propositional framework. Even though first order logic is more expressive than the propositional one, more works have been published in the propositional framework. The main objective of this thesis is to propose a solution scheme based on a heuristic approach, i.e., an anytime deduction technique, for the decision and optimization form of first order probabilistic logic problem. Jaumard et al. [33] proposed an anytime deduction algorithm for the propositional probabilistic logic which we extended to the first order context

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic

A general approach to reasoning with probabilities

We propose a general scheme for adding probabilistic reasoning capabilities to a wide variety of knowledge representation formalisms and we study its properties. Syntactically, we consider adding probabilities to the formulas of a given base logic. Semantically, we define a probability distribution over the subsets of a knowledge base by taking the probabilities of the formulas into account accordingly. This gives rise to a probabilistic entailment relation that can be used for uncertain reasoning. Our approach is a generalisation of many concrete probabilistic enrichments of existing approaches, such as ProbLog (an approach to probabilistic logic programming) and the constellation approach to abstract argumentation. We analyse general properties of our approach and provide some insights into novel instantiations that have not been investigated yet

Foundations of Fuzzy Logic and Semantic Web Languages

This book is the first to combine coverage of fuzzy logic and Semantic Web languages. It provides in-depth insight into fuzzy Semantic Web languages for non-fuzzy set theory and fuzzy logic experts. It also helps researchers of non-Semantic Web languages get a better understanding of the theoretical fundamentals of Semantic Web languages. The first part of the book covers all the theoretical and logical aspects of classical (two-valued) Semantic Web languages. The second part explains how to generalize these languages to cope with fuzzy set theory and fuzzy logic

A Default-Logic Paradigm for Legal Reasoning and Factfinding

Unlike research in linguistics and artificial intelligence, legal research has not used advances in logical theory very effectively. This article uses default logic to develop a paradigm for analyzing all aspects of legal reasoning, including factfinding. The article provides a formal model that integrates legal rules and policies with the evaluation of both expert and non-expert evidence – whether the reasoning occurs in courts or administrative agencies, and whether in domestic, foreign, or international legal systems. This paradigm can standardize the representation of legal reasoning, guide empirical research into the dynamics of such reasoning, and put the representations and research results to immediate use through artificial intelligence software. This new model therefore has the potential to transform legal practice and legal education, as well as legal theory