Logic for computer science: foundations of automatic theorem proving
Logic for computer science: foundations of automatic theorem proving
Term rewriting and all that
Complete Sets of Reductions for Some Equational Theories
Journal of the ACM (JACM)
Autarkic Computations in Formal Proofs
Journal of Automated Reasoning
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Abstract Saturation-Based Inference
LICS '03 Proceedings of the 18th Annual IEEE Symposium on Logic in Computer Science
Journal of Automated Reasoning
HOL-λσ: an intentional first-order expression of higher-order logic
Mathematical Structures in Computer Science
Abstract canonical presentations
Theoretical Computer Science - Clifford lectures and the mathematical foundations of programming semantics
ACM Transactions on Computational Logic (TOCL)
Confluence as a cut elimination property
RTA'03 Proceedings of the 14th international conference on Rewriting techniques and applications
Semantic cut elimination in the intuitionistic sequent calculus
TLCA'05 Proceedings of the 7th international conference on Typed Lambda Calculi and Applications
RTA'05 Proceedings of the 16th international conference on Term Rewriting and Applications
IJCAR '08 Proceedings of the 4th international joint conference on Automated Reasoning
Types for Proofs and Programs
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Deduction Modulo implements Poincarés principle by identifying deduction and computation as different paradigms and making their interaction possible. This leads to logical systems like the sequent calculus or natural deduction modulo. Even if deduction modulo is logically equivalent to first-order logic, proofs in such systems are quite different and dramatically simpler with one cost: cut elimination may not hold anymore. We prove first that it is even undecidable to know, given a congruence over propositions, if cuts can be eliminated in the sequent calculus modulo this congruence.Second, to recover the cut admissibility, we show how computation rules can be added following the classical idea of completion a laKnuth and Bendix. Because in deduction modulo, rewriting acts on terms as well as on propositions, the objects are much more elaborated than for standard completion. Under appropriate hypothesis, we prove that the sequent calculus modulo is an instance of the powerful framework of abstract canonical systemsand that therefore, cuts correspond to critical proofs that abstract completion allows us to eliminate.In addition to an original and deep understanding of the interactions between deduction and computation and of the expressivity of abstract canonical systems, this provides a mechanical way to transform a sequent calculus modulo into an equivalent one admitting the cut rule, therefore extending in a significant way the applicability of mechanized proof search in deduction modulo.