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The study of Description Logics have been historically mostly focused on features that can be translated to decidable fragments of first-order logic. In this paper, we leave this restriction behind and look for useful and decidable…
In many instances in first order logic or computable algebra, classical theorems show that many problems are undecidable for general structures, but become decidable if some rigidity is imposed on the structure. For example, the set of…
This paper proposes an alternative to standard first-order logic that seeks greater naturalness, generality, and semantic self-containment. The system removes the first-order restriction, avoids type hierarchies, and dispenses with external…
Explanation methods promise to make black-box classifiers more transparent. As a result, it is hoped that they can act as proof for a sensible, fair and trustworthy decision-making process of the algorithm and thereby increase its…
Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion.…
This article aims to provide a novel formalization of the concept of computational irreducibility in terms of the exactness of functorial correspondence between a category of data structures and elementary computations and a corresponding…
In functional programming languages, generalized algebraic data types (GADTs) are very useful as the unnecessary pattern matching over them can be ruled out by the failure of unification of type arguments. In dependent type systems, this is…
Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform…
We present a new type system with support for proofs of programs in a call-by-value language with control operators. The proof mechanism relies on observational equivalence of (untyped) programs. It appears in two type constructors, which…
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"…
The model of asynchronous programming arises in many contexts, from low-level systems software to high-level web programming. We take a language-theoretic perspective and show general decidability and undecidability results for asynchronous…
Completeness proofs in categorical semantics usually proceed by building a syntactic category whose composition is given by substitution. For untyped effectful Call-by-Value languages, this runs into a basic obstacle: there is no canonical…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
Concept induction, which is based on formal logical reasoning over description logics, has been used in ontology engineering in order to create ontology (TBox) axioms from the base data (ABox) graph. In this paper, we show that it can also…
We introduce OpSets, an executable framework for specifying and reasoning about the semantics of replicated datatypes that provide eventual consistency in a distributed system, and for mechanically verifying algorithms that implement these…
After surveying classical results, we introduce a generalized notion of inference system to support structural recursion on non-well-founded data types. Besides axioms and inference rules with the usual meaning, a generalized inference…
From an inconsistent database non-trivial arguments may be constructed both for a proposition, and for the contrary of that proposition. Therefore, inconsistency in a logical database causes uncertainty about which conclusions to accept.…
The lack of transparency in the decision-making processes of deep learning systems presents a significant challenge in modern artificial intelligence (AI), as it impairs users' ability to rely on and verify these systems. To address this…
We aim to explain a black-box classifier with the form: `data X is classified as class Y because X \textit{has} A, B and \textit{does not have} C' in which A, B, and C are high-level concepts. The challenge is that we have to discover in an…
We consider prescriptive type systems for logic programs (as in Goedel or Mercury). In such systems, the typing is static, but it guarantees an operational property: if a program is "well-typed", then all derivations starting in a…