Partial derivatives of regular expressions and finite automaton constructions
We introduce a notion of partial derivative of a regular expression and apply it to finite automaton constructions. The notion is a generalization of the known notion of word derivative due to Brzozowski: partial derivatives are related to non-deterministic finite automata (NFA's) in the same n...
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| Published in: | Theoretical computer science Vol. 155; no. 2; pp. 291 - 319 |
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| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier B.V
11.03.1996
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| ISSN: | 0304-3975, 1879-2294 |
| Online Access: | Get full text |
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| Abstract | We introduce a notion of
partial derivative of a regular expression and apply it to finite automaton constructions. The notion is a generalization of the known notion of
word derivative due to Brzozowski: partial derivatives are related to non-deterministic finite automata (NFA's) in the same natural way as derivatives are related to deterministic ones (DFA's). We give a constructive definition of partial derivatives and prove several facts, in particular:
1.
(1) any derivative of a regular expression
r can be represented by a finite set of partial derivatives of
r;
2.
(2) the set of all partial derivatives of
r is finite and its cardinality is less than or equal to one plus the number of occurrences of letters from A appearing in
r;
3.
(3) any partial derivative of
r is either a regular unit, or a subterm of
r, or a concatenation of several such subterms.
These theoretical results lead us to a new algorithm for turning regular expressions into relatively small NFA's and allow us to provide certain improvements to Brzozowski's algorithm for constructing DFA's. We also report on a prototype implementation of our NFA construction and present several examples. |
|---|---|
| AbstractList | We introduce a notion of
partial derivative of a regular expression and apply it to finite automaton constructions. The notion is a generalization of the known notion of
word derivative due to Brzozowski: partial derivatives are related to non-deterministic finite automata (NFA's) in the same natural way as derivatives are related to deterministic ones (DFA's). We give a constructive definition of partial derivatives and prove several facts, in particular:
1.
(1) any derivative of a regular expression
r can be represented by a finite set of partial derivatives of
r;
2.
(2) the set of all partial derivatives of
r is finite and its cardinality is less than or equal to one plus the number of occurrences of letters from A appearing in
r;
3.
(3) any partial derivative of
r is either a regular unit, or a subterm of
r, or a concatenation of several such subterms.
These theoretical results lead us to a new algorithm for turning regular expressions into relatively small NFA's and allow us to provide certain improvements to Brzozowski's algorithm for constructing DFA's. We also report on a prototype implementation of our NFA construction and present several examples. |
| Author | Antimirov, Valentin |
| Author_xml | – sequence: 1 givenname: Valentin surname: Antimirov fullname: Antimirov, Valentin email: kucherov@loria.fr organization: CRIN (CNRS) & INRIA-Lorraine Campus Scientifique, BP 239 F-54506, Vandæuvre-lès-Nancy Cedex, France |
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partial derivative of a regular expression and apply it to finite automaton constructions. The notion is a generalization of the known... |
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