Geometry of Interaction for MALL via Hughes-vanGlabbeek Proof-Nets

This paper presents, for the first time, a Geometry of Interaction (GoI) interpretation using Hughes-vanGlabbeek (HvG) proof-nets for multiplicative additive linear logic (MALL). Our GoI captures dynamically 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. In order 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 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 structure yields definability of eigenweights among cuts from the rest of the eigenweights.