Kleene equality

Kleene equality,[1] 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.