Possibility Theory and Possibilistic Logic: Tools for Reasoning Under and About Incomplete Information

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.

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Lipski's approach to incomplete information data bases restated and generalized in the setting of Zadeh's possibility theory

Information Systems ◽

10.1016/0306-4379(84)90014-0 ◽

1984 ◽

Vol 9 (1) ◽

pp. 27-42 ◽

Cited By ~ 100

Author(s):

Henri Prade

Keyword(s):

Incomplete Information ◽

Possibility Theory ◽

Data Bases

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

Scientia Iranica ◽

10.1016/j.scient.2011.04.008 ◽

2011 ◽

Vol 18 (3) ◽

pp. 566-573 ◽

Cited By ~ 12

Author(s):

D. Dubois ◽

H. Prade ◽

S. Schockaert

Keyword(s):

Possibility Theory ◽

Possibilistic Logic

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PRODUCT-BASED CAUSAL NETWORKS AND QUANTITATIVE POSSIBILISTIC BASES

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488505003618 ◽

2005 ◽

Vol 13 (05) ◽

pp. 469-493 ◽

Cited By ~ 5

Author(s):

SALEM BENFERHAT ◽

FAIZA HANED KHELLAF ◽

AICHA MOKHTARI

Keyword(s):

Possibility Theory ◽

Knowledge Bases ◽

Possibilistic Logic ◽

Product Operator ◽

Causal Networks ◽

One Step

In possibility theory, there are two kinds of possibilistic causal networks depending if possibilistic conditioning is based on the minimum or on the product operator. Similarly there are also two kinds of possibilistic logic: standard (min-based) possibilistic logic and quantitative (product-based) possibilistic logic. Recently, several equivalent transformations between standard possibilistic logic and min-based causal networks have been proposed. This paper goes one step further and shows that product-based causal networks can be encoded into product-based knowledge bases. The converse transformation is also provided.

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Modelling incomplete information in Boolean games using possibilistic logic

International Journal of Approximate Reasoning ◽

10.1016/j.ijar.2017.10.017 ◽

2018 ◽

Vol 93 ◽

pp. 1-23 ◽

Cited By ~ 3

Author(s):

Sofie De Clercq ◽

Steven Schockaert ◽

Ann Nowé ◽

Martine De Cock

Keyword(s):

Incomplete Information ◽

Possibilistic Logic ◽

Boolean Games

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Timed Possibilistic Logic

Fundamenta Informaticae ◽

10.3233/fi-1991-153-403 ◽

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.

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POSSIBILISTIC EVALUATION OF SETS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488513500177 ◽

2013 ◽

Vol 21 (03) ◽

pp. 325-346 ◽

Cited By ~ 11

Author(s):

ANTOON BRONSELAER ◽

JOSÉ ENRIQUE PONS ◽

GUY DE TRÉ ◽

OLGA PONS

Keyword(s):

Probability Theory ◽

Incomplete Information ◽

Constraint Satisfaction ◽

Possibility Theory ◽

Temporal Databases ◽

The Past ◽

Possibility Distributions

In the past decades, the theory of possibility has been developed as a theory of uncertainty that is compatible with the theory of probability. Whereas probability theory tries to quantify uncertainty that is caused by variability (or equivalently randomness), possibility theory tries to quantify uncertainty that is caused by incomplete information. A specific case of incomplete information is that of ill-known sets, which is of particular interest in the study of temporal databases. However, the construction of possibility distributions in the case of ill-known sets is known to be overly complex. This paper contributes to the study of ill-known sets by investigating the inference of uncertainty when constraints are specified over ill-known values. More specific, in this paper it is investigated how the knowledge about constraint satisfaction can be inferred if the constraints themselves are defined by means of ill-known values. It is shown how such reasoning can contribute to the study of (fuzzy) temporal databases.

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Inconsistency Management from the Standpoint of Possibilistic Logic

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488515400024 ◽

2015 ◽

Vol 23 (Suppl. 1) ◽

pp. 15-30 ◽

Cited By ~ 1

Author(s):

Didier Dubois ◽

Henri Prade

Keyword(s):

Fuzzy Sets ◽

Lower Bounds ◽

Propositional Logic ◽

Information Sources ◽

Possibility Theory ◽

Human Knowledge ◽

Possibilistic Logic ◽

Inconsistency Management ◽

The Rich ◽

Theory Framework

Uncertainty and inconsistency pervade human knowledge. Possibilistic logic, where propositional logic formulas are associated with lower bounds of a necessity measure, handles uncertainty in the setting of possibility theory. Moreover, central in standard possibilistic logic is the notion of inconsistency level of a possibilistic logic base, closely related to the notion of consistency degree of two fuzzy sets introduced by L. A. Zadeh. Formulas whose weight is strictly above this inconsistency level constitute a sub-base free of any inconsistency. However, several extensions, allowing for a paraconsistent form of reasoning, or associating possibilistic logic formulas with information sources or subsets of agents, or extensions involving other possibility theory measures, provide other forms of inconsistency, while enlarging the representation capabilities of possibilistic logic. The paper offers a structured overview of the various forms of inconsistency that can be accommodated in possibilistic logic. This overview echoes the rich representation power of the possibility theory framework.

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