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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| Vydáno v: | Theoretical computer science Ročník 155; číslo 2; s. 291 - 319 |
|---|---|
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| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Elsevier B.V
11.03.1996
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| ISSN: | 0304-3975, 1879-2294 |
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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 |
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| Cites_doi | 10.1016/0304-3975(95)80024-4 10.1016/0304-3975(92)90302-V 10.1145/321386.321399 10.1070/RM1961v016n05ABEH004112 10.1109/TEC.1960.5221603 10.1145/363347.363387 10.1016/0304-3975(80)90069-9 10.5109/13383 10.1147/rd.32.0114 10.1142/S0218196792000062 10.1016/0304-3975(86)90088-5 10.1145/321239.321249 10.1016/0304-3975(93)90287-4 10.1093/comjnl/36.7.623 10.1016/S0747-7171(87)80027-5 |
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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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