Henning Urbat

Sergey Goncharov, Marco Peressotti, Stelios Tsampas et al.

The bialgebraic abstract GSOS framework by Turi and Plotkin provides an elegant categorical approach to modelling the operational and denotational semantics of programming and process languages. In abstract GSOS, bisimilarity is always a congruence, and it coincides with denotational equivalence. This saves the language designer from intricate, ad-hoc reasoning to establish these properties. The bialgebraic perspective on operational semantics in the style of abstract GSOS has recently been extended to higher-order languages, preserving compositionality of bisimilarity. However, a categorical understanding of bialgebraic denotational semantics according to Turi and Plotkin's original vision has so far been missing in the higher-order setting. In the present paper, we develop a theory of adequate denotational semantics in higher-order abstract GSOS. The denotational models are parametric in an appropriately chosen semantic domain in the form of a locally final coalgebra for a behaviour bifunctor, whose construction is fully decoupled from the syntax of the language. Our approach captures existing accounts of denotational semantics such as semantic domains built via general step-indexing, previously introduced on a per-language basis, and is shown to be applicable to a wide range of different higher-order languages, e.g. simply typed and untyped languages, or languages with computational effects such as probabilistic or non-deterministic branching.

Fabian Lenke, Stefan Milius, Henning Urbat

Presheaves and nominal sets provide alternative abstract models of sets of syntactic objects with free and bound variables, such as lambda-terms. One distinguishing feature of the presheaf-based perspective is its elegant syntax-free characterization of substitution using a closed monoidal structure. In this paper, we introduce a corresponding closed monoidal structure on nominal sets, modeling substitution in the spirit of Fiore et al.'s substitution tensor for presheaves over finite sets. To this end, we present a general method to derive a closed monoidal structure on a category from a given action of a monoidal category on that category. We demonstrate that this method not only uniformly recovers known substitution tensors for various kinds of presheaf categories, but also yields notions of substitution tensor for nominal sets and their relatives, such as renaming sets. In doing so, we shed new light on different incarnations of nominal sets and (pre-)sheaf categories and establish a number of known and new correspondences between them.