Around finite second-order coherence spaces

Abstract

Many applications of denotational semantics, such as higher-order model checking or the complexity of normalization, rely on finite semantics for monomorphic type systems. We exhibit such a finite semantics for a polymorphic purely linear language: more precisely, we show that in Girard's semantics of second-order linear logic using coherence spaces and normal functors, the denotations of multiplicative-additive formulas are finite. This model is also effective, in the sense that the denotations of formulas and proofs are computable, as we show. We also establish analogous results for a second-order extension of Ehrhard's hypercoherences; while finiteness holds for the same reason as in coherence spaces, effectivity presents additional difficulties. Finally, we discuss the applications our our work to implicit computational complexity in linear (or affine) logic. In view of these applications, we study cardinality and complexity bounds in our finite semantics.