Intuitionistic Dual-intuitionistic Nets

The intuitionistic sequent calculus (at most one formula on the right-hand side of sequents) comes with a natural dual system: the dual-intuitionistic sequent calculus (at most one formula on the left-hand side). We explain how the duality between these two systems exactly corresponds to the intensively studied duality between call-by-value systems and call-by-name systems for classical logic. Relying on the uniqueness of the computational behaviour underlying these four logics (intuitionistic, dual-intuitionistic, call-by-value classical and call-by-name classical), we define a generic syntax of nets which can be used for any of these logics. Starting from Griffin’s work [Gri90], the question of finding logical foundations of control operators in functional programming languages has been intensively studied. We focus on one particular line of work dealing with classical logical systems with a deterministic cut elimination procedure, translations into intuitionistic logic, linear logic analysis, constructions of denotational models, ... [Gir91a, Par92, LRS93, DJS97, SR98, HS02, Sel01]. An important by-product of these developments is the understanding of a duality between the call-by-name and the call-by-value evaluation procedures for classical logic [Fil89, Sel01]. In the present work, we try to clarify the relations between Gentzen’s sequent calculus for classical logic, intuitionistic logic, call-by-name systems for classical logic (such as LKT [DJS95]), call-by-value systems for classical logic (such as LKQ [DJS95]), dual-intuitionistic logic [Cze77], ... The material presented here is not particularly new, but we think putting all the ingredients together (in particular for dual-intuitionistic logic which is usually not present in the picture) helps to understand the deep relations they have. The heart of this being the existence of only one computational behaviour corresponding to the cut elimination procedures of all the considered systems. Uniqueness of computation in this setting allows us to introduce a common graphical syntax for all the presented systems. Taking inspiration from the work on polarized proof-nets [Lau03] for linear logic, we introduce id-nets (i.e. intuitionistic/dual-intuitionistic nets) which have the nice properties of proof-nets (such as a simple confluent and strongly normalizing cut-elimination procedure, the quotient of the irrelevant permutations of rules of the sequent calculus, ...) and which can be read both as a syntax for intuitionistic logic and for dual-intuitionistic logic.