Kleene equality

In mathematics, Kleene equality,¹ 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 ${\displaystyle x}$:²

• ${\displaystyle f(x)}$ 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 authors³ 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: ${\displaystyle (f\simeq g):\Leftrightarrow (\forall x.(f(x)\sim g(x))).}$