Timed Possibilistic Logic

1991 ◽  
Vol 15 (3-4) ◽  
pp. 211-234
Author(s):  
Didier Dubois ◽  
Jérôme Lang ◽  
Henri Prade
Keyword(s):  
Temporal Logic ◽  
Fuzzy Set ◽  
Lattice Structure ◽  
Possibility Theory ◽  
Temporal Scale ◽  
Logical Framework ◽  
Classical Formula ◽  
Possibilistic Logic ◽  
Fuzzy Subsets

This paper is an attempt to cast both uncertainty and time in a logical framework. It generalizes possibilistic logic, previously developed by the authors, where each classical formula is associated with a weight which obeys the laws of possibility theory. In the possibilistic temporal logic we present here, each formula is associated with a time set (a fuzzy set in the more general case) which represents the set of instants where the formula is certainly true (more or less certainly true in the general case). When a particular instant is fixed we recover possibilistic logic. Timed possibilistic logic generalizes possibilistic logic also in the sense that we substitute the lattice structure of the set of the (fuzzy) subsets of the temporal scale to the lattice structure underlying the certainty weights in possibilistic logic. Thus many results from possibilistic logic can be straightforwardly generalized to timed possibilistic logic. Illustrative examples are given.

scholarly journals Characterizing and extending answer set semantics using possibility theory

2014 ◽  
Vol 15 (1) ◽  
pp. 79-116 ◽  
Author(s):  
KIM BAUTERS ◽  
STEVEN SCHOCKAERT ◽  
MARTINE DE COCK ◽  
DIRK VERMEIR
Keyword(s):  
Possibility Theory ◽  
Combinatorial Problems ◽  
The Body ◽  
Uncertain Information ◽  
Possibilistic Logic ◽  
Possibility Distributions ◽  
Answer Set Semantics ◽  
Answer Sets ◽  
Answer Set

AbstractAnswer Set Programming (ASP) is a popular framework for modelling combinatorial problems. However, ASP cannot be used easily for reasoning about uncertain information. Possibilistic ASP (PASP) is an extension of ASP that combines possibilistic logic and ASP. In PASP a weight is associated with each rule, whereas this weight is interpreted as the certainty with which the conclusion can be established when the body is known to hold. As such, it allows us to model and reason about uncertain information in an intuitive way. In this paper we present new semantics for PASP in which rules are interpreted as constraints on possibility distributions. Special models of these constraints are then identified as possibilistic answer sets. In addition, since ASP is a special case of PASP in which all the rules are entirely certain, we obtain a new characterization of ASP in terms of constraints on possibility distributions. This allows us to uncover a new form of disjunction, called weak disjunction, that has not been previously considered in the literature. In addition to introducing and motivating the semantics of weak disjunction, we also pinpoint its computational complexity. In particular, while the complexity of most reasoning tasks coincides with standard disjunctive ASP, we find that brave reasoning for programs with weak disjunctions is easier.

Fundamental Concepts

2019 ◽  
pp. 27-77
Keyword(s):  
Decision Making ◽  
Probability Theory ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Probability Distributions ◽  
Possibility Theory ◽  
Multi Objective ◽  
Basic Ideas ◽  
Fuzzy Stochastic Programming

In this chapter, the authors discuss some basic concepts of probability theory and possibility theory that are useful when reading the subsequent chapters of this book. The multi-objective fuzzy stochastic programming models developed in this book are based on the concepts of advanced topics in fuzzy set theory and fuzzy random variables (FRVs). Therefore, for better understanding of these advanced areas, the authors at first presented some basic ideas of probability theory and probability density functions of different continuous probability distributions. Afterwards, the necessity of the introduction of the concept of fuzzy set theory, some important terms related to fuzzy set theory are discussed. Different defuzzification methodologies of fuzzy numbers (FNs) that are useful in solving the mathematical models in imprecisely defined decision-making environments are explored. The concept of using FRVs in decision-making contexts is defined. Finally, the development of different forms of fuzzy goal programming (FGP) techniques for solving multi-objective decision-making (MODM) problems is underlined.

UNCERTAIN AND FUZZY OBJECT BASES: A DATA MODEL AND ALGEBRAIC OPERATIONS

2011 ◽  
Vol 19 (02) ◽  
pp. 275-305 ◽  
Author(s):  
TRU H. CAO ◽  
HOA NGUYEN
Keyword(s):  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Object Oriented ◽  
Possibility Theory ◽  
Object Properties ◽  
Common Framework ◽  
Formal Basis ◽  
Object Oriented Systems ◽  
Is Value

Fuzzy set theory and probability theory are complementary for soft computing, in particular object-oriented systems with imprecise and uncertain object properties. However, current fuzzy object-oriented data models are mainly based on fuzzy set theory or possibility theory, and lack of a rigorous algebra for querying and managing uncertain and fuzzy object bases. In this paper, we develop an object base model that incorporates both fuzzy set values and probability degrees to handle imprecision and uncertainty. A probabilistic interpretation of relations on fuzzy sets is introduced as a formal basis to coherently unify the two types of measures into a common framework. The model accommodates both class attributes, representing declarative object properties, and class methods, representing procedural object properties. Two levels of property uncertainty are taken into account, one of which is value uncertainty of a definite property and the other is applicability uncertainty of the property itself. The syntax and semantics of the selection and other main data operations on the proposed object base model are formally defined as a full-fledged algebra.

scholarly journals Rules and meta-rules in the framework of possibility theory and possibilistic logic

2011 ◽  
Vol 18 (3) ◽  
pp. 566-573 ◽  
Author(s):  
D. Dubois ◽  
H. Prade ◽  
S. Schockaert
Keyword(s):  
Possibility Theory ◽  
Possibilistic Logic

FUZZY SIMILARITY AND MEASURE OF SIMILARITY WITH LUKASIEWICZ IMPLICATOR

2008 ◽  
Vol 04 (02) ◽  
pp. 191-206 ◽  
Author(s):  
ISMAT BEG ◽  
SAMINA ASHRAF
Keyword(s):  
Fuzzy Sets ◽  
Fuzzy Set ◽  
Fuzzy Measure ◽  
Fuzzy Similarity ◽  
Measure Of Similarity ◽  
Universe Of Discourse ◽  
Fuzzy Subsets

Given a universe of discourse X. A fuzzy similarity mapping ST,Inc : F(X) × F(X) → F(X) is defined, where F(X) denotes the set of all fuzzy subsets of X. Mapping ST,Inc maps two fuzzy sets A and B to a fuzzy set ST,Inc(A,B) in X called their fuzzy set of similarity. A measure of similarity between A and B is then obtained by applying the composite of fuzzy measure and fuzzy similarity mapping on the pair (A,B). Several properties of fuzzy set of similarity and the measure of fuzzy similarity are obtained within the framework of Lukasiewicz fuzzy implicator and its respective t-norm and t-conorm. Many examples of measure of similarity are also constructed.

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