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 strings obtainable from a string in S by cutting off a prefixing u, formally: u−1S = { v ∈ Σ*: uv ∈ S }, cf. picture.[1] 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,[2] 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.[3] It may be computed recursively as follows.[4]

(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:[5]

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:[6]

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)= εif 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.[7]

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.[8] 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.

See also

References

  1. ↑ Janusz A. Brzozowski (1964). "Derivatives of Regular Expressions". J ACM. 11: 481–494. doi:10.1145/321239.321249.
  2. ↑ Brzozowski (1964), p.481, required A to consist of the 2n combinations of n bits, for some n.
  3. ↑ Brzozowski (1964), p.483, Theorem 4.1
  4. ↑ Brzozowski (1964), p.483, Theorem 3.2
  5. ↑ Brzozowski (1964), p.483, Theorem 3.1
  6. ↑ Brzozowski (1964), p.482, Definition 3.2
  7. ↑ Brzozowski (1964), p.483, Theorem 4.2
  8. ↑ Brzozowski (1964), p.484, Theorem 4.3
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