Kennisrepresentatie met partiële functies en intensie concepten: theorie en toepassingen

Most of the knowledge representation formalisms assume that functions are total. In fact, in many applications, the domain of discourse is heterogeneous consisting of multiple types of objects resulting in basically all functions being partial. Furthermore, logic is often limited to the extensions of concepts (their value), while sometimes it is needed to talk about concepts themself (intensions). For example, counting chief positions in a company; counting extensions of chief positions (i.e., people behind them) is wrong as different positions could be assigned to the same person. Different applications and experiments of Knowledge Base systems show the requirement for partial functions and intensional concepts. Due to the lack of support for such language constructs, knowledge engineers developed different ways of getting around them. This thesis focuses on the systematic treatment of partial functions and intensional concepts resulting in increased expressivity, compactness, and elegance of the language. Research of these topics led to the following three main scientific results: A Prudent Logic of Partial Functions - We develop a base extension of three-valued predicate logic, in which partial function terms are guarded by domain expressions ensuring the well-definedness property despite the three-valued nature of the underlying logic. To tackle the verbosity of this core language, we propose different ways to increase convenience by using disambiguating annotations and non-commutative connectives. Furthermore, we present an extension of the logic with definitions allowing the declaration of partial functions rather than only constraining them. This approach results in a system with a refined distinction between meaningfulness and undefinedness. Namely, undefined sentences are often rejected on the grounds of having no meaning. However, we consider potentially undefined sentences ambiguous, and not necessary meaningless (e.g., the present king of France is bald); while the lack of meaning is reserved for the categorically undefined sentences (e.g., the present king of France is a prime number). On the relationship between types and partial functions: a Knowledge Representation perspective - Inspired by the parallels between guarding of partial functions and typing techniques in typed logics, we investigate the fundamental connection between partial functions and typing -- exploring when and why they are introduced, and how they differ from unary predicates. Types serve as domains for functions and predicates, grouping entities into homogeneous classes. However, when partial functions are involved, this approach can lead to complex and convoluted type structures. As a solution, we take the reverse approach: treating partial functions as the most primitive entity and formalizing types as unary total predicates. In this setup, the guarding relation of partial functions serves as the well-typing relation. We show that both many-sorted and order-sorted logic can be embedded within the logic of partial functions and that its expressivity goes beyond. Further, combining the idea of guards with the intensional logic in the context of order-sorted logic, we design the logic suitable for expressing subtyping polymorphism. In this logic, it is possible to cast a variable to an appropriate type based on its application by using typing assertions and quantification over concepts. An Epistemic Logic for Modeling Decisions in the Context of Incomplete Knowledge - When modeling decision processes, decisions are often made with incomplete knowledge about the actual state of affairs. This is distinct from situations where certain objects lack a value altogether, as in the case of partial functions. To formally capture this distinction, we extended the Decision Modeling and Notation (DMN) standard with partial functions and provided a model-theoretic semantics. Despite this advancement, the language still lacked constructs to express whether certain information is known or unknown. To address this, we introduced eDMN, an extension of DMN with epistemic connectives, motivated by the observation that decision-making inherently occurs within an agent's epistemic state (i.e., what agent knows and does not know).