Around finite second-order coherence spaces
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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 is applied to implicit computational complexity, resulting in a characterization of regular languages in Elementary Linear Logic and in the solution of an open question raised by Baillot. The second-order coherence space model is also effective, in the sense that the denotations of formulas and proofs are computable. We prove this and also study cardinality and complexity bounds in this semantics. This lays the groundwork for a sequel paper (j.w.w. Pierre Pradic) proposing a new approach to logarithmic space computation in linear logic.
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