# Brzozowski derivative

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In theoretical computer science, in particular in formal language theory, the Brzozowski derivative u−1S of a set S of strings and a string u is defined as the set of all strings obtainable from a string in S by cutting off a prefixing u, formally: u−1S = { v ∈ Σ*: uvS }, cf. picture. It is named after the computer scientist Janusz Brzozowski who investigated their properties and gave an algorithm to compute the derivative of a generalized regular expression.

## Derivative of a regular expression

Given a finite alphabet A of symbols, a generalized regular expression denotes a possibly infinite set of finite-length strings of symbols from A. It may be built of:

• ∅ (denoting the empty set of strings),
• ε (denoting the singleton set containing just the empty string),
• a symbol a from A (denoting the singleton set containing the single-symbol string a),
• RS (where R and S are, in turn, generalized regular expressions; denoting their set's union),
• RS (denoting the intersection of R 's and S 's set),
• ¬R (denoting the complement of R 's set with respect to the set of all strings of symbols from A),
• RS (denoting the set of all possible concatenations of strings from R 's and S 's set),
• R* (denoting the set of n-fold repetitions of strings from R 's set, for any n≥0, including the empty string).

In an ordinary regular expression, neither ∧ nor ¬ is allowed. The string set denoted by a generalized regular expression R is called its language, denoted as L(R).

## Computation

For any given generalized regular expression R and any string u, the derivative u−1R is again a generalized regular expression. It may be computed recursively as follows.

 (ua)−1R = a−1(u−1R) for a symbol a and a string u ε−1R = R

Using the previous two rules, the derivative with respect to an arbitrary string is explained by the derivative with respect to a single-symbol string a. The latter can be computed as follows:

 a−1a = ε a−1b = ∅ for each symbol b≠a a−1ε = ∅ a−1∅ = ∅ a−1(R*) = a−1RR* a−1(RS) = (a−1R)S ∨ ν(R)a−1S a−1(R∧S) = (a−1R) ∧ (a−1S) a−1(R∨S) = (a−1R) ∨ (a−1S) a−1(¬R) = ¬(a−1R)

Here, ν(R) is an auxiliary function yielding a generalized regular expression that evaluates to the empty string ε if R 's language contains ε, and otherwise evaluates to ∅. This function can be computed by the following rules:

 ν(a) = ∅ for any symbol a ν(ε) = ε ν(∅) = ∅ ν(R*) = ε ν(RS) = ν(R) ∧ ν(S) ν(R ∧ S) = ν(R) ∧ ν(S) ν(R ∨ S) = ν(R) ∨ ν(S) ν(¬R) = ε if ν(R) = ∅ ν(¬R) = ∅ if ν(R) = ε

## Properties

A string u is a member of the string set denoted by a generalized regular expression R if and only if ε is a member of the string set denoted by the derivative u−1R.

Considering all the derivatives of a fixed generalized regular expression R results in only finitely many different languages. If their number is denoted by dR, all these languages can be obtained as derivatives of R with respect to string of length below dR. Furthermore, there is a complete deterministic finite automaton with dR states which recognises the regular language given by R, as laid out by the Myhill–Nerode theorem.