Quantitativity and fixed points in the infinitary relational semantics of linear logic

Grellois and Melli`es introduced in 2015 an infinitary variant of the relational model of linear logic [GM15a], in which the exponential can build multisets with finite or countable multiplicities. Their aim was to use such models for higher-order model-checking, in which denotational models need to capture finite programs generating finite width trees with branches of at most countable length, generated by infinitary recursion. While the traditional, finitary relational semantics admits a unique, uniform fixed point operator, they showed that the (countably) infinitary variant has several distinct fixed points. In this talk, we explain that there was an error in the paper: the fixed point they give, which reflects the traditional quantitative interpretation of the relational semantics of linear logic, is not the greatest fixed point as they claimed. We also discuss consequences, and relations between the greatest fixed point and this intermediate, quantitative fixed point.