Complete reducibility candidates

Deduction modulo is an extension of first-order predicate logic where axioms are replaced by a congruence relation on propositions and where many theories, such as arithmetic, simple type theory and some variants of set theory, can be expressed. An important question in deduction modulo is to find a condition of the theories that have the strong normalization property. Dowek and Werner have given a semantic sufficient condition for a theory to have the strong normalization property: they have proved a ”soundness” theorem of the form: if a theory has a model (of a particular form) then it has the strong normalization property. In this paper, we refine their notion of model in a way allowing not only to prove soundness, but also completeness: if a theory has the strong normalization property, then it has a model of this form. The key idea of our model construction is a refinement of Girard’s notion of reducibility candidates. By providing a sound and complete semantics for theories having the strong normalization property, this paper contributes to explore the idea that strong normalization is not only a proof-theoretic notion, but also a model-theoretic one. In this paper, we define a sound and complete semantics for theories having the strong normalization property in minimal deduction modulo. Deduction modulo [5] is a logical framework, based on Natural Deduction where axioms are replaced by a congruence relation on propositions, allowing to express proofs of many theories like arithmetic [9], simple type theory [6], some variants of set theory [7], etc... The absence of axioms ensures the fact that all cut-free proofs end with an introduction rule, as in usual Natural Deduction. Hence the cut elimination property of a theory entails its consistency. In this framework, cuts in proofs are represented by β-redexes, and the elimination of a cut, by β-reduction. Therefore strong normalization of the β-reduction ensures the cut elimination property of the corresponding theory, and furthermore its consistency. The usual tool to prove strong normalization is called reducibility candidates. The notion of reducibility candidates was first introduced by J.Y. Girard [11], following the work of W.W. Tait [18]. We can see a posteriori their work as proofs of strong normalization, obtained by the existence of a C-valued model, where C is the algebra of reducibility candidates. This work has been extended to, at least, two non-trivial logical frameworks: Pure Type Systems by P.A. Melliès and B. Werner [16], and Deduction modulo by G. Dowek and B. Werner [8]. By nontrivial, we mean that these logical frameworks can express strongly normalizing in ria -0 04 33 15 9, v er si on 1 18 N ov 2 00 9 Author manuscript, published in "Proof Search in Type Theory (2009)"