## Journal of Symbolic Logic

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

Rainer Kerth

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745646

Mathematical Reviews number (MathSciNet)
MR1665771

Zentralblatt MATH identifier
0930.03013

JSTOR
Kerth, Rainer. The Interpretation of Unsolvable $\lambda$-Terms in Models of Untyped $\lambda$-Calculus. J. Symbolic Logic 63 (1998), no. 4, 1529--1548. https://projecteuclid.org/euclid.jsl/1183745646