We study the connections between graph models and "wavestyle " Geometry of Interaction (GoI) #-models. The latters arise when Abramsky's GoI construction, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tensor, using the countable power as traced strong monoidal functor !. Abramsky hinted that the category of sets and relations is the basic setting for traditional "static semantics". Here we support this view by showing that a large class of graph-like models can be viewed as arising from a suitable generalization of the GoI construction. Furthermore, we show that the class of untyped #-theories induced by wave-style GoI models is richer than that induced by game models.

We study a class of \wave-style" Geometry of Interaction (GoI) -models based on the category Rel of sets and relations. Wave GoI models arise when Abramsky's GoI axiomatization, which generalizes Girard's original GoI, is applied to a traced monoidal category with the categorical product as tensor, using \countable power" as the traced strong monoidal functor !. Abramsky hinted that the category Rel is the basic setting for traditional denotational \static semantics". However, Rel, together with the cartesian product, apparently escapes Abramsky's original GoI construction. Here we show that Rel can be axiomatized as a strict GoI situation, i.e. a strict variant of Abramsky's GoI situation, which gives rise to a rich class of strict graph models. These are models of restricted -calculi in the sense of [HL99], such as Church's -I-calculus and the KN -calculus.