Frank's triangular norms in Piaget's logical proportions

Starting from the Boolean notion of logical proportion in Piaget's sense, which turns out to be equivalent to analogical proportion, this note proposes a definition of analogical proportion between numerical values based on triangular norms (and dual co-norms). Frank's family of triangular norms is particularly interesting from this perspective. The article concludes with a comparative discussion with another very recent proposal for defining analogical proportions between numerical values based on the family of generalized means.

Analogical Proportions and Creativity: A Preliminary Study

Analogical proportions are statements of the form "$a$ is to $b$ as $c$ is to $d$", which expresses that the comparisons of the elements in pair $(a, b)$ and in pair $(c, d)$ yield similar results. Analogical proportions are creative in the sense that given 3 distinct items, the representation of a 4th item $d$, distinct from the previous items, which forms an analogical proportion with them can be calculated, provided certain conditions are met. After providing an introduction to analogical proportions and their properties, the paper reports the results of an experiment made with a database of animal descriptions and their class, where we try to "create" new animals from existing ones, retrieving rare animals such as platypus. We perform a series of experiments using word embeddings as well as Boolean features in order to propose novel animals based on analogical proportions, showing that word embeddings obtain better results.

Belief revision and incongruity: is it a joke?

Incongruity often makes people laugh. You have to be smart to say stupid things. It requires to be even smarter for understanding them. This paper is a shameless attempt to formalize this intelligent behavior in the case of an agent listening to a joke. All this is a matter of revision of beliefs, surprise and violation of norms.

An elementary belief function logic

Non-additive uncertainty theories, typically possibility theory, belief functions and imprecise probabilities share a common feature with modal logic: the duality properties between possibility and necessity measures, belief and plausibility functions as well as between upper and lower probabilities extend the duality between possibility and necessity modalities to the graded environment. It has been shown that the all-or-nothing version of possibility theory can be exactly captured by a minimal epistemic logic (MEL) that uses a very small fragment of the KD modal logic, without resorting to relational semantics. Besides, the case of belief functions has been studied independently, and a belief function logic has been obtained by extending the modal logic S5 to graded modalities using {\L}ukasiewicz logic, albeit using relational semantics. This paper shows that a simpler belief function logic can be devised by adding {\L}ukasiewicz logic on top of MEL. It allows for a more natural semantics in terms of Shafer basic probability assignments.

Analogical Relevance Index

Focusing on the most significant features of a dataset is useful both in machine learning (ML) and data mining. In ML, it can lead to a higher accuracy, a faster learning process, and ultimately a simpler and more understandable model. In data mining, identifying significant features is essential not only for gaining a better understanding of the data but also for visualization. In this paper, we demonstrate a new way of identifying significant features inspired by analogical proportions. Such a proportion is of the form of "a is to b as c is to d", comparing two pairs of items (a, b) and (c, d) in terms of similarities and dissimilarities. In a classification context, if the similarities/dissimilarities between a and b correlate with the fact that a and b have different labels, this knowledge can be transferred to c and d, inferring that c and d also have different labels. From a feature selection perspective, observing a huge number of such pairs (a, b) where a and b have different labels provides a hint about the importance of the features where a and b differ. Following this idea, we introduce the Analogical Relevance Index (ARI), a new statistical test of the significance of a given feature with respect to the label. ARI is a filter-based method. Filter-based methods are ML-agnostic but generally unable to handle feature redundancy. However, ARI can detect feature redundancy. Our experiments show that ARI is effective and outperforms well-known methods on a variety of artificial and some real datasets.

Some recent advances in reasoning based on analogical proportions

Analogical proportions compare pairs of items (a, b) and (c, d) in terms of their differences and similarities. They play a key role in the formalization of analogical inference. The paper first discusses how to improve analogical inference in terms of accuracy and in terms of computational cost. Then it indicates the potential of analogical proportions for explanation. Finally, it highlights the close relationship between analogical proportions and multi-valued dependencies, which reveals an unsuspected aspect of the former.

From Shallow to Deep Interactions Between Knowledge Representation, Reasoning and Machine Learning (Kay R. Amel group)

This paper proposes a tentative and original survey of meeting points between Knowledge Representation and Reasoning (KRR) and Machine Learning (ML), two areas which have been developing quite separately in the last three decades. Some common concerns are identified and discussed such as the types of used representation, the roles of knowledge and data, the lack or the excess of information, or the need for explanations and causal understanding. Then some methodologies combining reasoning and learning are reviewed (such as inductive logic programming, neuro-symbolic reasoning, formal concept analysis, rule-based representations and ML, uncertainty in ML, or case-based reasoning and analogical reasoning), before discussing examples of synergies between KRR and ML (including topics such as belief functions on regression, EM algorithm versus revision, the semantic description of vector representations, the combination of deep learning with high level inference, knowledge graph completion, declarative frameworks for data mining, or preferences and recommendation). This paper is the first step of a work in progress aiming at a better mutual understanding of research in KRR and ML, and how they could cooperate.

Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence (1999)

This is the Proceedings of the Fifteenth Conference on Uncertainty in Artificial Intelligence, which was held in Stockholm Sweden, July 30 - August 1, 1999

Modeling uncertain and vague knowledge in possibility and evidence theories

This paper advocates the usefulness of new theories of uncertainty for the purpose of modeling some facets of uncertain knowledge, especially vagueness, in AI. It can be viewed as a partial reply to Cheeseman's (among others) defense of probability.

Positive and Negative Explanations of Uncertain Reasoning in the Framework of Possibility Theory

This paper presents an approach for developing the explanation capabilities of rule-based expert systems managing imprecise and uncertain knowledge. The treatment of uncertainty takes place in the framework of possibility theory where the available information concerning the value of a logical or numerical variable is represented by a possibility distribution which restricts its more or less possible values. We first discuss different kinds of queries asking for explanations before focusing on the two following types : i) how, a particular possibility distribution is obtained (emphasizing the main reasons only) ; ii) why in a computed possibility distribution, a particular value has received a possibility degree which is so high, so low or so contrary to the expectation. The approach is based on the exploitation of equations in max-min algebra. This formalism includes the limit case of certain and precise information.