Cut Admissibility by Saturation

Deduction modulo is a framework in which theories are integrated into proof systems such as natural deduction or sequent calculus by presenting them using rewriting rules. When only terms are rewritten, cut admissibility in those systems is equivalent to the confluence of the rewriting system, as shown by Dowek, RTA 2003, LNCS 2706. This is no longer true when considering rewriting rules involving propositions. In this paper, we show that, in the same way that it is possible to recover confluence using Knuth-Bendix completion, one can regain cut admissibility in the general case using standard saturation techniques. This work relies on a view of proposition rewriting rules as oriented clauses, like term rewriting rules can be seen as oriented equations. This also leads us to introduce an extension of deduction modulo with conditional term rewriting rules. Whatever their origin, proofs rarely need to be search for without context: Program verification requires arithmetic, theories of lists or arrays, etc. Mathematical theorems are in general not proved in pure predicate logic. Consequently, even if (automated and interactive) proof systems have achieved a high degree of maturity, they need to be able to deal with theories in a efficient way. This explains the particular interest focused on the SMT (Satisfiability Modulo Theory) provers in the latter years. However, one of the drawbacks of the SMT approach is that the way theories are integrated is not completely generic, in the sense that each theory needs a special treatment. A more generic approach to integrate theories into a proof system was proposed by Dowek, Hardin and Kirchner [15]. In Deduction Modulo, a theory is represented by a congruence over formulæ, and proofs are search for modulo this congruence. In practice, this congruence is most often described as a rewriting system. However, using only term rewriting rules would not be enough to capture interesting theories. For instance, Vorobyov [24] showed that even quantifier-free Presburger arithmetic cannot be presented as a convergent term rewriting system. To overcome this, Deduction Modulo also deals with proposition rewriting rules, that rewrite atomic formulæ into formulæ. Thanks to this, it was possible to present many theories in Deduction Modulo : simple type theory (also known as higher-order logic) [14], arithmetic [17], set theory [16], B set theory [21], any pure type system, including the calculus of construction which is the 1 Although it may sound rather strange, the absence of subsequent to the term “modulo” follows the original works about this field.