Sophie Tourret

A string diagram is a two-dimensional graphical representation that can be described as a one-dimensional term generated from a set of primitives using sequential and parallel compositions. Since different syntactic terms may represent the same diagram, this syntax is quotiented by a collection of coherence equations expressing equivalence up to deformation. This work lays foundations for automated reasoning about diagrammatic equivalence, motivated primarily by the verification of quantum circuit equivalences. We consider two classes of diagrams, for which we introduce normalizing term rewriting systems that equate diagrammatically equivalent terms. In both cases, we prove termination and confluence with the help of the proof assistant Isabelle/HOL.

Resolution and superposition provers rely on the given clause procedure to saturate clause sets. Using Isabelle/HOL, we formally verify four variants of the procedure: the well-known Otter and DISCOUNT loops as well as the newer iProver and Zipperposition loops. For each of the variants, we show that the procedure guarantees saturation, given a fair data structure to store the formulas that wait to be selected. Our formalization of the Zipperposition loop clarifies some fine points previously misunderstood in the literature.

AVATAR is an elegant and effective way to split clauses in a saturation prover using a SAT solver. But is it refutationally complete? And how does it relate to other splitting architectures? To answer these questions, we present a unifying framework that extends a saturation calculus (e.g., superposition) with splitting and that embeds the result in a prover guided by a SAT solver. The framework also allows us to studylocking, a subsumption-like mechanism based on the current propositional model. Various architectures are instances of the framework, including AVATAR, labeled splitting, and SMT with quantifiers.

Abduction in description logics finds extensions of a knowledge base to make it entail an observation. As such, it can be used to explain why the observation does not follow, to repair incomplete knowledge bases, and to provide possible explanations for unexpected observations. We consider TBox abduction in the lightweight description logic EL, where the observation is a concept inclusion and the background knowledge is a TBox, i.e., a set of concept inclusions. To avoid useless answers, such problems usually come with further restrictions on the solution space and/or minimality criteria that help sort the chaff from the grain. We argue that existing minimality notions are insufficient, and introduce connection minimality. This criterion follows Occam's razor by rejecting hypotheses that use concept inclusions unrelated to the problem at hand. We show how to compute a special class of connection-minimal hypotheses in a sound and complete way. Our technique is based on a translation to first-order logic, and constructs hypotheses based on prime implicates. We evaluate a prototype implementation of our approach on ontologies from the medical domain.

Superposition is among the most successful calculi for first-order logic. Its extension to higher-order logic introduces new challenges such as infinitely branching inference rules, new possibilities such as reasoning about formulas, and the need to curb the explosion of specific higher-order rules. We describe techniques that address these issues and extensively evaluate their implementation in the Zipperposition theorem prover. Largely thanks to their use, Zipperposition won the higher-order division of the CASC-J10 competition.