## F.4.1 Mathematical Logic (F.1.1, I.2.2-4)

We study the extension of techniques from Inductive Logic Programming (ILP) to temporal logic programming languages. Therefore we present two temporal logic programming languages and analyse the learnability of programs from these languages from finite sets of examples. In first order temporal logic the following topics are analysed: - How can we characterize the denotational semantics of programs? - Which proof techniques are best suited? - How complex is the learning task? In propositional temporal logic we analyse the following topics: - How can we use well known techniques from model checking in order to refine programs? - How complex is the learning task? In both cases we present estimations for the VC-dimension of selected classes of programs.

This technical report is the Emerging Trends proceedings of the 20th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2007), which was held during 10-13 September in Kaiserslautern, Germany. TPHOLs covers all aspects of theorem proving in higher order logics as well as related topics in theorem proving and veriﬁcation.

Automated theorem proving is a search problem and, by its undecidability, a very difficult one. The challenge in the development of a practically successful prover is the mapping of the extensively developed theory into a program that runs efficiently on a computer. Starting from a level-based system model for automated theorem provers, in this work we present different techniques that are important for the development of powerful equational theorem provers. The contributions can be divided into three areas: Architecture. We present a novel prover architecture that is based on a set-based compression scheme. With moderate additional computational costs we achieve a substantial reduction of the memory requirements. Further wins are architectural clarity, the easy provision of proof objects, and a new way to parallelize a prover which shows respectable speed-ups in practice. The compact representation paves the way to new applications of automated equational provers in the area of verification systems. Algorithms. To improve the speed of a prover we need efficient solutions for the most time-consuming sub-tasks. We demonstrate improvements of several orders of magnitude for two of the most widely used term orderings, LPO and KBO. Other important contributions are a novel generic unsatisfiability test for ordering constraints and, based on that, a sufficient ground reducibility criterion with an excellent cost-benefit ratio. Redundancy avoidance. The notion of redundancy is of central importance to justify simplifying inferences which are used to prune the search space. In our experience with unfailing completion, the usual notion of redundancy is not strong enough. In the presence of associativity and commutativity, the provers often get stuck enumerating equations that are permutations of each other. By extending and refining the proof ordering, many more equations can be shown redundant. Furthermore, our refinement of the unfailing completion approach allows us to use redundant equations for simplification without the need to consider them for generating inferences. We describe the efficient implementation of several redundancy criteria and experimentally investigate their influence on the proof search. The combination of these techniques results in a considerable improvement of the practical performance of a prover, which we demonstrate with extensive experiments for the automated theorem prover Waldmeister. The progress achieved allows the prover to solve problems that were previously out of reach. This considerably enhances the potential of the prover and opens up the way for new applications.