## Derivatives of regular expressions (1964)

Venue: | JOURNAL OF THE ACM |

Citations: | 210 - 3 self |

### BibTeX

@ARTICLE{Brzozowski64derivativesof,

author = {Janusz A. Brzozowski},

title = {Derivatives of regular expressions},

journal = {JOURNAL OF THE ACM},

year = {1964},

volume = {11},

pages = {481--494}

}

### Years of Citing Articles

### OpenURL

### Abstract

Abstract. Kleene's regular expressions, which can be used for describing sequential circuits, were defined using three operators (union, concatenation and iterate) on sets of sequences. Word descriptions of problems can be more easily put in the regular expression language if the language is enriched by the inclusion of other logical operations. However, in the problem of converting the regular expression description to a state diagram, the existing methods either cannot handle expressions with additional operators, or are made quite complicated by the presence of such operators. In this paper the notion of a derivative of a regular expression is introduced and the properties of derivatives are discussed. This leads, in a very natural way, to the construction of a state diagram from a regular expression containing any number of logical operators.

### Citations

148 |
Gedanken-experiments on sequential machines
- Moore
- 1956
(Show Context)
Citation Context ...ssions which can have any number of logical connectives, and describe methods for obtaining state diagrams from such regular expressions. We are concerned with the usual model of a finite automaton M =-=[4, 7, 10, 11, 12]-=-. The n binary inputs x~, x2, • • • , x~ of M are represented by a single 2 '~- valued input x, taking the values from A~ = {0, 1, ... , ]¢ - 11, where/c = 2 ~. The internal states of M are ql, q:, • ... |

9 |
Delayed logic and finite state machines
- ARDEN
- 1960
(Show Context)
Citation Context ...OREM 4.6. An equation of the form X = AX + B, where 6(A) = 4), has the solution X = A • B, which is unique (up to equality of regular expressions). The theorem is a modification of a theorem of Arden =-=[8]-=-, wire has shown that X = XA ~ B, ~(A ) = ¢p, has the solution X = BA,. The proof of Theorem 4.6 parallels that of Arden's theorem and will not be given here. THEOREM 4.7. The set of characteristic eq... |

6 |
A survey of regular expressions and their applications
- Brzozowski
- 1962
(Show Context)
Citation Context ...ssions which can have any number of logical connectives, and describe methods for obtaining state diagrams from such regular expressions. We are concerned with the usual model of a finite automaton M =-=[4, 7, 10, 11, 12]-=-. The n binary inputs x~, x2, • • • , x~ of M are represented by a single 2 '~- valued input x, taking the values from A~ = {0, 1, ... , ]¢ - 11, where/c = 2 ~. The internal states of M are ql, q:, • ... |

4 |
A method for synthesizing sequential circuits
- H
- 1955
(Show Context)
Citation Context ...ssions which can have any number of logical connectives, and describe methods for obtaining state diagrams from such regular expressions. We are concerned with the usual model of a finite automaton M =-=[4, 7, 10, 11, 12]-=-. The n binary inputs x~, x2, • • • , x~ of M are represented by a single 2 '~- valued input x, taking the values from A~ = {0, 1, ... , ]¢ - 11, where/c = 2 ~. The internal states of M are ql, q:, • ... |

1 |
b S.C. Representation of events in nerve nets and finite automat
- KI
- 1956
(Show Context)
Citation Context ... R2 + 1); D1~,1 = (R1 + X, R2 + X); The state diagram is shown in Figure 4. 7. Conclusion introduce qx, q0, z = (0, 0), q~, z = (0,0), to qo, z -- (0, 0), toql,z = (0, I), q~0, z = (0, 0), to qx, z = =-=(1, 0)-=-, to q~o, z = (0, 0), toqx,z = (1,1). Regular expressions can be obtained more easily from word description of problems if one is allowed to use any logical connective in the formation of the expressi... |

1 | EI~(;o% C, C.; aND Wmcwr, J.B. Realization of events by logical nets - Coet - 1958 |

1 |
Regular expressions a~M state graphs for automats
- McNAuGH'rON, YaADA
- 1960
(Show Context)
Citation Context ...f the transition under input a takes the graph from q~ to qk then the equation for Rj contains the term aRk ; and k is a term of R~. if and only if the output for qj is Z = 1. There are other methods =-=[3, 5, 8]-=- of obtaining the regular expressions, but this otto is most closely related to the derivative approach. 2. To obtain the minimal state diagram from a regular expression, find the cimracterisfic deriv... |

1 |
Signal flow graph techniques for sequential circuit state diagrams
- BltzozowsKl, usKEY
- 1963
(Show Context)
Citation Context ...f the transition under input a takes the graph from q~ to qk then the equation for Rj contains the term aRk ; and k is a term of R~. if and only if the output for qj is Z = 1. There are other methods =-=[3, 5, 8]-=- of obtaining the regular expressions, but this otto is most closely related to the derivative approach. 2. To obtain the minimal state diagram from a regular expression, find the cimracterisfic deriv... |

1 |
H..~ND FEtNSTmN, N.H. Design of sequential m~tehines from their regular expressions
- G
- 1961
(Show Context)
Citation Context ...show that this difficulty can be overcome, and a state graph can always be constructed, but not necessarily wi~h the minimum number of states. It should be pointed out that the other existing methods =-=[3, 6]-=- have the same difficulties and, moreover, are limited to regular expressions with (+), (.) and (,) only. Definition 5.2. Two regular expressions are similar if one can be transformeds488 JANUSZ A. BR... |

1 |
Automata and finite automat,'L
- LII, Y
- 1960
(Show Context)
Citation Context |

1 | Finite automata and representatAon of events - MVmLL |

1 |
Finite automata and their decision problems
- RmN, Seo, et al.
- 1959
(Show Context)
Citation Context |

1 | The synthesis of sequential switching circuits - HU'FMaN - 1954 |

1 |
SequentiM functions
- RNEV
- 1958
(Show Context)
Citation Context ...uences and a finite sequence s, the derivative of R with respect to s is denoted by D~R and is D~R = {t I st 5: RI. The notion of derivative of a set (under different names) was introduced previously =-=[10, 14, 15]-=-, but was not applied ~o regular expressions. We now present an Mgorithm for finding derivatives of regular expressions. We shall need to know when a regular expression contains )t. For this purpose w... |