Journal of Symbolic Logic

The Interpretation of Unsolvable $\lambda$-Terms in Models of Untyped $\lambda$-Calculus

Rainer Kerth

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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.

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J. Symbolic Logic, Volume 63, Issue 4 (1998), 1529-1548.

First available in Project Euclid: 6 July 2007

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Kerth, Rainer. The Interpretation of Unsolvable $\lambda$-Terms in Models of Untyped $\lambda$-Calculus. J. Symbolic Logic 63 (1998), no. 4, 1529--1548.

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