# Around finite second-order coherence spaces

## TL;DR

This paper demonstrates finite and effective semantics for a polymorphic linear language using coherence spaces and hypercoherences, with implications for computational complexity and higher-order model checking.

## Contribution

It introduces a finite, effective semantics for second-order linear logic models, extending to hypercoherence spaces and analyzing complexity bounds.

## Key findings

- Denotations of formulas are finite in Girard's coherence space semantics.
- The semantics are effectively computable, enabling practical applications.
- Finiteness also holds for hypercoherence models, despite additional challenges.

## 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.