Elena Di Lavore

Pietro Barbiero, Stefano Fioravanti, Francesco Giannini et al. · ibm-research

Explainable AI (XAI) aims to address the human need for safe and reliable AI systems. However, numerous surveys emphasize the absence of a sound mathematical formalization of key XAI notions -- remarkably including the term "explanation" which still lacks a precise definition. To bridge this gap, this paper presents the first mathematically rigorous definitions of key XAI notions and processes, using the well-funded formalism of Category theory. We show that our categorical framework allows to: (i) model existing learning schemes and architectures, (ii) formally define the term "explanation", (iii) establish a theoretical basis for XAI taxonomies, and (iv) analyze commonly overlooked aspects of explaining methods. As a consequence, our categorical framework promotes the ethical and secure deployment of AI technologies as it represents a significant step towards a sound theoretical foundation of explainable AI.

Filippo Bonchi, Alessandro Di Giorgio, Elena Di Lavore

Tape diagrams provide a convenient graphical notation for arrows of rig categories, i.e., categories equipped with two monoidal products, $\oplus$ and $\otimes$. In this work, we introduce Kleene-Cartesian rig categories, namely rig categories where $\otimes$ provides a Cartesian bicategory, while $\oplus$ a Kleene bicategory. We show that the associated tape diagrams can conveniently deal with imperative programs and various program logic.

Elena Di Lavore, Jonas Forster, Mario Román

Functor coalgebras capture a wide range of transition systems that must however evolve in discrete steps. We introduce graded coalgebras of graded monads and propose them to model continuous-time transition systems. We develop the theory of graded coalgebras-including graded distributive laws between graded monads-and we give conditions for the existence of terminal coalgebras. We define both branching-time and trace semantics, linking them to recent work on Feller-Dynkin processes. Finally, we develop coalgebraic modal logics for both process semantics and state criteria for invariance and expressivity.

Giovanni de Felice, Elena Di Lavore, Mario Román et al.

We present some categorical investigations into Wittgenstein's language-games, with applications to game-theoretic pragmatics and question-answering in natural language processing.