Kleene equality

In mathematics, Kleene equality, named after Stephen Cole Kleene, or strong equality, (${\displaystyle \simeq }$) is an equality operator on partial functions, that states that on a given argument either both functions are undefined, or both are defined and their values on that arguments are equal.

For example, if we have partial functions ${\displaystyle f}$ and ${\displaystyle g}$, ${\displaystyle f\simeq g}$ means that for every ${\displaystyle x}$:

• ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are both defined and ${\displaystyle f(x)=g(x)}$
• or ${\displaystyle f(x)}$ and ${\displaystyle g(x)}$ are both undefined.

Some authors are using "quasi-equality", which is defined like this: ${\displaystyle (y_{1}\sim y_{2}):\Leftrightarrow ((y_{1}\downarrow \lor y_{2}\downarrow )\longrightarrow y_{1}=y_{2}),}$ where the down arrow means that the term on the left side of it is defined. Then it becomes possible to define the strong equality in the following way: ${\displaystyle (f\simeq g):\Leftrightarrow (\forall x.(f(x)\sim g(x))).}$

References

• Cutland, Nigel (1980). Computability, an introduction to recursive function theory. Cambridge University Press. p. 251. ISBN 978-0-521-29465-2.

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