To increase automation in proof assistants and other verification tools based on higher-order formalisms, we propose to generalize superposition to an extensional, polymorphic, clausal version of higher-order logic (also called simple type theory). Our ambition is to achieve a graceful extension, which coincides with standard superposition on first-order problems and smoothly scales up to arbitrary higher-order problems. Bentkamp, Blanchette, Cruanes, and Waldmann [12] designed a family of superpositionlike calculi for a λ-free clausal fragment of higher-order logic, with currying and applied variables. We adapt their extensional nonpurifying calculus to support λ-terms (Sect.

Signature-based abduction aims at building hypotheses over a specified set of names, the signature, that explain an observation relative to some background knowledge. This type of abduction is useful for tasks such as diagnosis, where the vocabulary used for observed symptoms differs from the vocabulary expected to explain those symptoms. We present the first complete method solving signature-based abduction for observations expressed in the expressive description logic ALC, which can include TBox and ABox axioms, thereby solving the knowledge base abduction problem. The method is guaranteed to compute a finite and complete set of hypotheses, and is evaluated on a set of realistic knowledge bases.

Many forms of inductive logic programming (ILP) use \emph{metarules}, second-order Horn clauses, to define the structure of learnable programs and thus the hypothesis space. Deciding which metarules to use for a given learning task is a major open problem and is a trade-off between efficiency and expressivity: the hypothesis space grows given more metarules, so we wish to use fewer metarules, but if we use too few metarules then we lose expressivity. In this paper, we study whether fragments of metarules can be logically reduced to minimal finite subsets. We consider two traditional forms of logical reduction: subsumption and entailment. We also consider a new reduction technique called \emph{derivation reduction}, which is based on SLD-resolution. We compute reduced sets of metarules for fragments relevant to ILP and theoretically show whether these reduced sets are reductions for more general infinite fragments. We experimentally compare learning with reduced sets of metarules on three domains: Michalski trains, string transformations, and game rules. In general, derivation reduced sets of metarules outperforms subsumption and entailment reduced sets, both in terms of predictive accuracies and learning times.

We present an algorithm for the generation of prime implicates in equational logic, that is, of the most general consequences of formulæ containing equations and disequations between first-order terms. This algorithm is defined by a calculus that is proved to be correct and complete. We then focus on the case where the considered clause set is ground, i.e., contains no variables, and devise a specialized tree data structure that is designed to efficiently detect and delete redundant implicates. The corresponding algorithms are presented along with their termination and correctness proofs. Finally, an experimental evaluation of this prime implicate generation method is conducted in the ground case, including a comparison with state-of-the-art propositional and first-order prime implicate generation tools.