Brzozowski derivative

In theoretical computer science, in particular in formal language theory, the Brzozowski derivative ${\displaystyle u^{-1}S}$ of a set ${\displaystyle S}$ of strings and a string ${\displaystyle u}$ is the set of all strings obtainable from a string in ${\displaystyle S}$ by cutting off the prefix ${\displaystyle u}$. Formally:

${\displaystyle u^{-1}S=\{v\in \Sigma ^{*}\mid uv\in S\}}$.

For example,

${\displaystyle c^{-1}\{{\text{cat}},{\text{cow}},{\text{dog}}\}=\{{\text{at}},{\text{ow}}\}.}$

The Brzozowski derivative was introduced under various different names since the late 1950s. Today it is named after the computer scientist Janusz Brzozowski who investigated its properties and gave an algorithm to compute the derivative of a generalized regular expression.