Journal of Symbolic Logic

On Adding $(\xi)$ to Weak Equality in Combinatory Logic

Martin W. Bunder, J. Roger Hindley, and Jonathan P. Seldin

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Because the main difference between combinatory weak equality and $\lambda\beta$-equality is that the rule \begin{equation*}\tag{\xi} X = Y \vdash \lambda x.X = \lambda x.Y\end{equation*} is valid for the latter but not the former, it is easy to assume that another way of defining combinatory $\beta$-equality is to add rule $(\xi)$ to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in $(\xi)$ very carefully. If one tries to use one of the more common abstraction algorithms, the result will be an equality, $=_\xi$, that is either equivalent to $\beta\eta$-equality (and so strictly stronger than $\beta$-equality) or else strictly weaker than $\beta$-equality. This paper will study the relations $=_\xi$ for several commonly used abstraction algorithms, distinguish between them, and axiomatize them.

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J. Symbolic Logic, Volume 54, Issue 2 (1989), 590-607.

First available in Project Euclid: 6 July 2007

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Bunder, Martin W.; Hindley, J. Roger; Seldin, Jonathan P. On Adding $(\xi)$ to Weak Equality in Combinatory Logic. J. Symbolic Logic 54 (1989), no. 2, 590--607.

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