Strong Equality

In mathematics, Kleene equality, or strong equality, ($\simeq$) 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 $f$ and $g$, $f \simeq g$ means that for every $x$:
* $f(x)$ and $g(x)$ are both defined and $f(x) = g(x)$
* or $f(x)$ and $g(x)$ are both undefined.

Some authors
are using "quasi-equality", which is defined like this:
$(y_1 \sim y_2):\Leftrightarrow((y_1\downarrow \lor y_2\downarrow)\longrightarrow y_1=y_2),$
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:
$(f \simeq g):\Leftrightarrow (\forall x. (f(x)\sim g(x))).$

References

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Computability theory
Equivalence (mathematics)

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