Analogical proportions (AP) are statements of the form "a is to b ascis to d". They compare the pairs of items(a,b) and(c, d) in terms of their differences and similarities. The explicit use of APs in analogical reasoning has contributed to a renewal of its applications, leading to many developments, especially in the last decade; see [30] for a survey. However, even if much has been already done both at the theoretical and at the practical levels, the very nature of APs may not yet be fully understood and their full potential explored. In the following, we survey recent works on APs along three directions: their role in classification tasks [4]; their use for providing explanations [20]; their relation with multi-valued dependencies [21]. This just intends to be an introductory paper, and the reader is referred to the above references for more details on each issue.

Conditional independence structures describe independencies of one set of variables from another set of variables conditioned upon a third set of variables. These structures are invaluable means for compact representations of knowledge because independencies can be exploited for useful factorizations. Conditional independence structures appear in different disguise in various areas of knowledge representation, be it the conditional independence of sets of random variables in probabilistic graphical models such as Bayesian networks or as conditional functions related to belief revision, or as in- dependencies induced by (embedded) multivalued dependencies in data bases. This paper investigates conditional independencies for Boolean functions using Fourier analysis. We define three notions of independence based on the notion of influence of a variable on a function and draw connections to multivalued dependencies.

Conditional independence structures describe independencies of one set of variables from another set of variables conditioned upon a third set of variables. These structures are invaluable means for compact representations of knowledge because independencies can be exploited for useful factorizations. Conditional independence structures appear in different disguise in various areas of knowledge representation, be it the conditional independence of sets of random variables in probabilistic graphical models such as Bayesian networks or as conditional functions related to belief revision, or as in- dependencies induced by (embedded) multivalued dependencies in data bases. This paper investigates conditional independencies for Boolean functions using Fourier analysis. We define three notions of independence based on the notion of influence of a variable on a function and draw connections to multivalued dependencies.