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Section: New Results

Automated Theorem Proving

Guillaume Burel has shown that presenting theories by means of rewriting rules in Deduction modulo leads to more efficient proof search methods than using axioms, provided the rewriting system enjoys a proof theoretical property, namely cut admissibility.

He has been investigating which theories can be encoded as rewriting systems admitting cuts. Surprisingly, it turned out that any consistent theory in predicate logic can. This has been shown by studying the links between the set-of-support strategy of the Resolution method and the extension of the method based on Deduction modulo. He has also shown how to reduce the size of the corresponding rewriting systems [42] .

Guillaume Burel has also studied how to improve the confidence in iProver Modulo. When it finds a resolution proof, it is now able to produce a proof that can be checked by Dedukti. The encoding of Resolution proofs in the λΠ-calculus modulo that is used is shallow, making more plausible the long-term goal of interoperability of provers, both interactive and automated, through Dedukti.

Simon Cruanes has explored several ideas for combining the Superposition calculus—one of the most powerful calculi for automated reasoning within first-order logic with equality—with Deduction modulo. Combining the term rewriting system for a theory in Deduction modulo with the ordered rewriting on which Superposition is based on proved to be difficult, yielding incomplete calculi; in most cases it boils down to the fact that the combination of confluent terminating term rewriting systems is in general neither terminating nor confluent. In order to experiment quickly ideas by implementing them, he has written a Superposition-based prover in OCaml, with some special features—automatic ordering of rewrite rules in the input, non-clausal calculus to be able to use equivalence relations as rewrite rules. The prover is 8,000 lines of code and is designed to be flexible and modular, but still has decent performance and can prove some non-trivial theorems.

Together with Mélanie Jacquel (Cedric), David Delahaye and Catherine Dubois have investigated Zenon for verifying proof rules added to help the automation in the provers of Atelier B. They have augmented Zenon with specific rules for dealing with set operations and predicates, obtained by applying super deduction—a variant of Deduction modulo [33] .