A map of dependencies among three-valued logics

International audienceThree-valued logics arise in several fields of computer science, both inspired by concrete problems (such as in the management of the null value in databases) and theoretical considerations. Several three-valued logics have been defined. They differ by their choice of basic connectives, hence also from a syntactic and proof-theoretic point of view. Different interpretations of the third truth value have also been suggested. They often carry an epistemic flavor. In this work, relationships between logical connectives on three-valued functions are explored. Existing theorems of functional completeness have laid bare some of these links, based on specific connectives. However we try to draw a map of such relationships between conjunctions, negations and implications that extend Boolean ones. It turns out that all reasonable connectives can be defined from a few of them and so all known three-valued logics appear as a fragment of only one logic. These results can be instrumental when choosing, for each application context, the appropriate fragment where the basic connectives make full sense, based on the appropriate meaning of the third truth-value

Some characterizations of T-power based implications

Recently, the so-called family of T-power based implications was introduced. These operators involve the use of Zadeh’s quantifiers based on powers of t-norms in its definition. Due to the fact that Zadeh’s quantifiers constitute the usual method to modify fuzzy propositions, this family of fuzzy implication functions satisfies an important property in approximate reasoning such as the invariance of the truth value of the fuzzy conditional when both the antecedent and the consequent are modified using the same quantifier. In this paper, an in-depth analysis of this property is performed by characterizing all binary functions satisfying it. From this general result, a fully characterization of the family of T-power based implications is presented. Furthermore, a second characterization is also proved in which surprisingly the invariance property is not explicitly used.Peer ReviewedPostprint (author's final draft

On Some Functional Equations Related to Alpha Migrative t-conorms

In this contribution, we analyse in details the recently introduced definition of migrative tconorms [see Fuzzy implications: alpha migrativity and generalised laws of importation, M. Baczy´nski, B. Jayaram, R. Mesiar, 2020]. We also focus on some general functional equations, which might be obtained from such a notion. We concentrate on some particular well-known families of fuzzy implications and show solutions of those equations among this kind of fuzzy implication functions

Formal transformation methods for automated fault tree generation from UML diagrams

With a growing complexity in safety critical systems, engaging Systems Engineering with System Safety Engineering as early as possible in the system life cycle becomes ever more important to ensure system safety during system development. Assessing the safety and reliability of system architectural design at the early stage of the system life cycle can bring value to system design by identifying safety issues earlier and maintaining safety traceability throughout the design phase. However, this is not a trivial task and can require upfront investment. Automated transformation from system architecture models to system safety and reliability models offers a potential solution. However, existing methods lack of formal basis. This can potentially lead to unreliable results. Without a formal basis, Fault Tree Analysis of a system, for example, even if performed concurrently with system design may not ensure all safety critical aspects of the design. [Continues.]</div

Can indices of ecological evenness be used to measure consensus?

In the context of group decision making with fuzzy preferences, consensus measures are employed to provide feedback and help guide automatic or semi-automatic decision reaching processes. These measures attempt to capture the intuitive notion of how much inputs, individuals or groups agree with one another. Meanwhile, in ecological studies there has been an ongoing research effort to define measures of community evenness based on how evenly the proportional abundances of species are distributed. The question hence arises as to whether there can be any cross-fertilization from developments in these fields given their intuitive similarity. Here we investigate some of the models used in ecology toward their potential use in measuring consensus. We found that although many consensus characteristics are exhibited by evenness indices, lack of reciprocity and a tendency towards a minimum when a single input is non-zero would make them undesirable for inputs expressed on an interval scale. On the other hand, we note that some of the general frameworks could still be useful for other types of inputs like ranking profiles and that in the opposite direction consensus measures have the potential to provide new insights in ecology

From Degrees of Belief to Binary Beliefs: Lessons from Judgment-Aggregation Theory

International audienceWhat is the relationship between degrees of belief and binary beliefs? Can the latter be expressed as a function of the former – a so-called “belief-binarization rule” – without running into difficulties such as the lottery paradox? We show that this problem can be usefully analyzed from the perspective of judgment-aggregation theory. Although some formal similarities between belief binarization and judgment aggregation have been noted before, the connection between the two problems has not yet been studied in full generality. We seek to fill this gap. This paper is organized around a baseline impossibility theorem, which we use to map out the space of possible solutions to the belief-binarization problem. Our theorem shows that, except in limiting cases, there exists no belief-binarization rule satisfying four initially plausible desiderata. Surprisingly, this result is a direct corollary of the judgmentaggregation variant of Arrow’s classic impossibility theorem in social choice theory

Three-valued logics, uncertainty management and rough sets

This paper is a survey of the connections between three-valued logics and rough sets from the point of view of incomplete information management. Based on the fact that many three-valued logics can be put under a unique algebraic umbrella, we show how to translate three-valued conjunctions and implications into operations on ill-known sets such as rough sets. We then show that while such translations may provide mathematically elegant algebraic settings for rough sets, the interpretability of these connectives in terms of an original set approximated via an equivalence relation is very limited, thus casting doubts on the practical relevance of truth-functional logical renderings of rough sets