We will present several results on two types of continuous models of -calculus, namely graph models and extensional models. By introducing a variant of Engeler's model construction, we are able to generalize the results of [7] and to give invariants that determine a large family of graph models up to applicative isomorphism. This covers all graph models considered in the litterature so far. We indicate briefly how these invariants may be modified in order to determine extensional models as well.Furthermore, we (...) use our construction to exhibit graph models that are not equationally equivalent. We indicate once again how the construction passes on to extensional models. (shrink)

Our goal in this paper is to analyze the interpretation of arbitrary unsolvable λ-terms in a given model of λ-calculus. We focus on graph models and (a special type of) stable models. We introduce the syntactical notion of a decoration and the semantical notion of a critical sequence. We conjecture that any unsolvable term β-reduces to a term admitting a decoration. The main result of this paper concerns the interconnection between those two notions: given a graph model or stable model (...) D, we show that any unsolvable term admitting a decoration and having a non-empty interpretation in D generates a critical sequence in the model. In the last section, we examine three classical graph models, namely the model B(ω) of Plotkin and Scott, Engeler's model D A and Park's model D P . We show that B(ω) and D A do not contain critical sequences whereas D P does. (shrink)

Our goal in this paper is to analyze the interpretation of arbitrary unsolvable $\lambda$-terms in a given model of $\lambda$-calculus. We focus on graph models and (a special type of) stable models. We introduce the syntactical notion of a decoration and the semantical notion of a critical sequence. We conjecture that any unsolvable term $\beta$-reduces to a term admitting a decoration. The main result of this paper concerns the interconnection between those two notions: given a graph model or stable model (...) $\mathfrak{D}$, we show that any unsolvable term admitting a decoration and having a non-empty interpretation in $\mathfrak{D}$ generates a critical sequence in the model. In the last section, we examine three classical graph models, namely the model $\mathfrak{B}(\omega)$ of Plotkin and Scott, Engeler's model $\mathfrak{D}_A$ and Park's model $\mathfrak{D}_P$. We show that $\mathfrak{B}(\omega)$ and $\mathfrak{D}_A$ do not contain critical sequences whereas $\mathfrak{D}_P$ does. (shrink)