Kleene equality explained

In mathematics, Kleene equality,^[1] 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^[2] 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

1. Web site: Kleene equality in nLab. ncatlab.org.
2. A Set Theory with Support for Partial Functions. Farmer, William M.. Guttman, Joshua D. . Studia Logica: An International Journal for Symbolic Logic. 66. 1. 2000. 59–78.

• Book: Cutland , Nigel . Computability, an introduction to recursive function theory. Cambridge University Press. 1980. 251. 978-0-521-29465-2.