Directed Virtual Reductions

This note deenes a new graphical local calculus, directed virtual reductions. It is designed to compute Girard's execution formula EX, an invariant of closed functional evaluation obtained from the \geometry of interaction" interpretation of-calculus 5]. The calculus is obtained by synchronizing another graphical local calculus presented in \local and asynchronous beta-reduction": virtual reductions 4]. This synchronization makes it easier to mechanize than general virtual reductions. In undirected virtual reductions the consistency of the computation is insured by an algebraic mechanism called the bar. This mechanism in general induces correction terms of any order. The directed virtual reduction has been designed to keep those terms at order one. A further synchronization, the combustion strategy will even wipe out rst order correction terms. Applied to sharing graphs, the combustion strategy yields Lamping's optimal graphical calculus as presented in 1]. But more eecient optimal implementations of-calculus are expected. The paper is conceived as a follow-up of 4] and supposes a familiarity with virtual reduction. 1.1 Digs and dags A dig is a directed graph, a dag is a directed acyclic graph, that is a dig with no directed cycle or circuit. The valence of a node in a dig is the number of edges of which it is the source; the valence of a dig is the maximal valence of its nodes. A path in a dig is an arrow of the free involutive category generated by the dig; a directed path in a dig is an arrow of the free category generated by the dig. Beware that paths may take edges in the reverse direction, but directed ones may not. We adopt the following convention when denoting concrete paths: 2 1 is meant to denote 1 then 2 , that is the product of paths is more like composition of functions than like concatenation; this allows lighter notations in the proofs. The reversion operation, which is as said an involution is denoted