Finite domain and symbolic inference methods for extensions of first-order logic

One of the long term goals of the research field Knowledge Representation and Reasoning is to build a knowledge base system (KBS). In such a system, knowledge about a domain of discourse is stored and different tasks are solved by applying various inference methods on that knowledge. An example is a KBS storing knowledge about course scheduling at a university. By applying suitable forms of inference, schedules can be generated automatically, hand-made schedules can be checked, existing schedules can be revised, etc., all using the same background knowledge. In our work, we contribute to the goal of building a KBS by investigating various forms of inference that are useful to solve many practical tasks. Specifically, we study the following forms of inference in the context of finite structures: constraint propagation, grounding, model revision, and debugging. Also, we investigate symbolic, i.e., structure-independent, constraint propagation. As underlying logic, we use FO(·) [1]. This logic extends order-sorted first-order logic with inductive definitions, aggregates, integer arithmetic, and partial functions. While deductive query answering is well-known to be undecidable for FO and, a fortiori, for the extension FO(·), the tasks we study are in NP. In the rest of this abstract, we summarize our results for the various forms of inference. We will use the following terminology. An n-ary three-valued relation over a domain D is a function mapping each tuple in D to one of the truth values true, false, or unknown. Such a three-valued relation R̃ is said to approximate a relation S if {d | R̃(d) = true} ⊆ S ⊆ {d | R̃(d) ∈ {true, unknown}}. A partial or three-valued finite structure over a vocabulary Σ is a structure with a finite domain D that assigns three-valued relations over D to each symbol in Σ. Such a partial structure Ĩ approximates a structure J if for every symbol P ∈ Σ, the three-valued relation assigned to P by Ĩ approximates the relation assigned to P by J .