Geometry of Interaction for MALL via Hughes--Van Glabbeek Proof-Nets

Abstract

This article presents, for the first time, a Geometry of Interaction (GoI) interpretation inspired from Hughes--Van Glabbeek (HvG) proof-nets for multiplicative additive linear logic (MALL). Our GoI dynamically captures HvG’s geometric correctness criterion—the toggling cycle condition—in terms of algebraic operators. Our new ingredient is a scalar extension of the *-algebra in Girard’s *-ring of partial isometries over a Boolean polynomial ring with literals of eigenweights as indeterminates. To capture feedback arising from cuts, we construct a finer-grained execution formula. The expansion of this execution formula is longer than that for collections of slices for multiplicative GoI, hence it is harder to prove termination. Our GoI gives a dynamical, semantical account of Boolean valuations (in particular, pruning sub-proofs), conversion of weights (in particular, α-conversion), and additive (co)contraction, peculiar to additive proof-theory. Termination of our execution formula is shown to correspond to HvG’s toggling criterion. The slice-wise restriction of our execution formula (by collapsing the Boolean structure) yields the well-known correspondence, explicit or implicit in previous works on multiplicative GoI, between the convergence of execution formulas and acyclicity of proof-nets. Feedback arising from the execution formula by restricting to the Boolean polynomial structure yields autonomous definability of eigenweights among cuts from the rest of the eigenweights.