Brzozowski derivative

Brzozowski derivative (on red background) of a dictionary string set with respect to "con"

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 rest-strings obtainable from a string in S by cutting off its prefix u (if possible), formally: u⁻¹S = { v ∈ Σ^*: uv ∈ S }, 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),
• R∨S (where R and S are, in turn, generalized regular expressions; denoting their set's union),
• R∧S (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(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 d_R, all these languages can be obtained as derivatives of R with respect to string of length below d_R. Furthermore, there is a complete deterministic finite automaton with d_R states which recognises the regular language given by R, as laid out by the Myhill–Nerode theorem.