## Partial Derivatives of Regular Expressions and Finite Automata Constructions (1995)

Venue: | Theoretical Computer Science |

Citations: | 59 - 0 self |

### BibTeX

@ARTICLE{Antimirov95partialderivatives,

author = {Valentin Antimirov},

title = {Partial Derivatives of Regular Expressions and Finite Automata Constructions},

journal = {Theoretical Computer Science},

year = {1995},

volume = {155},

pages = {291--319}

}

### Years of Citing Articles

### OpenURL

### Abstract

. We introduce a notion of a partial derivative of a regular expression. It is a generalization to the non-deterministic case of the known notion of a derivative invented by Brzozowski. We give a constructive definition of partial derivatives, study their properties, and employ them to develop a new algorithm for turning regular expressions into relatively small NFA and to provide certain improvements to Brzozowski's algorithm constructing DFA. We report on a prototype implementation of our algorithm constructing NFA and present some examples. Introduction In 1964 Janusz Brzozowski introduced word derivatives of regular expressions and suggested an elegant algorithm turning a regular expression r into a deterministic finite automata (DFA); the main point of the algorithm is that the word derivatives of r serve as states of the resulting DFA [5]. In the following years derivatives were recognized as a quite useful and productive tool. Conway [8] uses derivatives to present various comp...

### Citations

275 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
Citation Context ...ain point of the algorithm is that the word derivatives of r serve as states of the resulting DFA [5]. In the following years derivatives were recognized as a quite useful and productive tool. Conway =-=[8]-=- uses derivatives to present various computational procedures in the algebra of regular expressions and to investigate some logical properties of this algebra. Krob [12] extends this differential calc... |

233 |
Finite automata and their decision problems
- Rabin, Scott
- 1959
(Show Context)
Citation Context ...ar to the well known relation between 4 Of course, one should bear in mind that the output of this procedure can have an exponential size. 10 NFA and DFA provided by the classical subset construction =-=[17]-=-. Really, suppose a regular term t is turned into an NFA as described in Theorem 12. This NFA can be transformed into an equivalent DFA by the subset construction; the states of the DFA will be repres... |

210 | Derivatives of regular expressions
- Brzozowski
(Show Context)
Citation Context ... an elegant algorithm turning a regular expression r into a deterministic finite automata (DFA); the main point of the algorithm is that the word derivatives of r serve as states of the resulting DFA =-=[5]-=-. In the following years derivatives were recognized as a quite useful and productive tool. Conway [8] uses derivatives to present various computational procedures in the algebra of regular expression... |

208 | Order-sorted algebra I: Equational deduction for multiple inheritance, overloading, exceptions and partial operations
- Goguen
- 1992
(Show Context)
Citation Context ...rasReg[A]. Actually, there exist different ways to choose the signature and to formalize the algebra. In this paper we follow the idea of [2] that Reg[A] should be regarded as an order-sorted algebra =-=[10]-=- having a sort A for an alphabet which is a subsort of a sort Reg for all the regular expressions. Here we also introduce two further subsorts of Reg, namely Reg0 and Reg1, to distinguish regular expr... |

117 |
Regular expression search algorithm
- Thompson
- 1968
(Show Context)
Citation Context ...rtial Derivatives In this section we apply partial derivatives to a classical problem of turning regular expressions into finite automata. There are several well-known algorithms performing this task =-=[14, 19, 4]-=-. Nevertheless, new algorithms, aimed at reducing sizes of resulting automata, improving their performance, etc., keep appearing (see e.g. a survey [20]). Using partial derivatives, we get yet another... |

106 |
Introducing OBJ3
- Goguen, Winkler
- 1988
(Show Context)
Citation Context ...s several improvements to Brzozowski's algorithm [5] due to the use of partial derivatives. We have implemented our algorithm turning regular expressions into NFA as an algebraic program in OBJ3 (see =-=[11]-=- for a description of the language). In Sect.5 we present several examples of NFA constructed by our program. In this version of the paper we omit all the proofs as well as some auxiliary propositions... |

100 |
From regular expressions to deterministic automata
- Berry, Sethi
- 1986
(Show Context)
Citation Context ...al algebra of K-rational expressions. Brzozowski and Leiss [6] employ the idea of derivatives to ascertain relations between regular expressions, finite automata and boolean networks. Berry and Sethi =-=[4]-=- give a solid theoretical background for McNaughton and Yamada's algorithm [14] through the notion of continuation which is a particular kind of derivative. Ginzburg [9] uses derivatives to develop a ... |

94 |
Regular Expressions and State Graphs for Automata
- McNaughton
- 1960
(Show Context)
Citation Context ... of derivatives to ascertain relations between regular expressions, finite automata and boolean networks. Berry and Sethi [4] give a solid theoretical background for McNaughton and Yamada's algorithm =-=[14]-=- through the notion of continuation which is a particular kind of derivative. Ginzburg [9] uses derivatives to develop a procedure for proving equivalence of regular expressions; further development o... |

92 | Reutenauer: Codes and Automata
- Berstel, Perrin, et al.
(Show Context)
Citation Context ...the pairs ht 1 ; t 2 i such that L(t 1 ) = L(t 2 ). Recall that Reg[A] is an idempotent semiring, i.e. satisfies the SR-axioms (1--9). There are further axioms concerning Kleene star (see e.g. [8] or =-=[16]-=-). Thus, we have the following chain of quotients of T Reg related by surjective homomorphisms: T Reg ! T Reg = E(ACI) ! T Reg = E(ACIZ) ! T Reg = E(SR) ! T Reg = E(Reg) (10) where the last quotient i... |

39 |
On equations for regular languages, finite automata, and sequential networks
- Brzozowski, Leiss
- 1980
(Show Context)
Citation Context ...ed by a finite system of state equations of the form �� := o(��) + x 1 \Delta �� 1 + : : : + x k \Delta �� k (18) for each state �� 2 M where x i 2 A and �� i 2 �� (��;=-= x i ), i = 1 : : : k (see e.g. [6]). Here -=-o(��) issif �� 2 F , or ; otherwise -- in the latter case it is omitted from the sum. The components x i \Delta ; (if any) can also be omitted from the righthand sides of (18), so that the res... |

34 | From regular expressions to DFAs using compressed NFAs
- Chang, Paige
- 1992
(Show Context)
Citation Context ...r, in many cases our NFA have actually fewer states than this upper bound. Moreover, there are examples when our NFA turn out to be smaller than those produced by a tricky Chang and Paige's algorithm =-=[7]-=- (which involves several non-trivial optimizations of the representation of NFA). The second procedure in this section provides several improvements to Brzozowski's algorithm [5] due to the use of par... |

19 | P.D.: Rewriting extended regular expressions
- Antimirov, Mosses
- 1995
(Show Context)
Citation Context ... further development of this procedure is provided by Mizoguchi et al [15]. Yet another procedure for proving equivalence of extended regular expressions is suggested by the present author and Mosses =-=[2, 3]-=- and is also based on some constructions closely related to derivatives. In the present paper we come up with a new notion of a partial derivative which is, in a sense, a non-deterministic generalizat... |

18 |
Theory of Automata, Pergamon
- Salomaa
- 1969
(Show Context)
Citation Context ...et A = fx 1 ; : : : ; xn g can be represented in the following "linear" form: r = o(r) + x 1 \Delta r 1 + \Delta \Delta \Delta + xn \Delta r n (11) where all the r i are some regular express=-=ions (see [5, 18, 8]-=-). In particular, one can take each r i to be the derivative x i \Gamma1 r. We are going to generalize this linear factorization in several ways to make it non-deterministic in a sense; this will lead... |

15 | A taxonomy of finite automata construction algorithms
- Watson
- 1993
(Show Context)
Citation Context ...ll-known algorithms performing this task [14, 19, 4]. Nevertheless, new algorithms, aimed at reducing sizes of resulting automata, improving their performance, etc., keep appearing (see e.g. a survey =-=[20]-=-). Using partial derivatives, we get yet another new algorithms. 4.1 From regular expressions to small NFA. In this subsection we describe a new algorithm turning a regular term t into an NFA having n... |

9 |
A procedure for checking equality of regular expressions
- Ginzburg
- 1967
(Show Context)
Citation Context ...lean networks. Berry and Sethi [4] give a solid theoretical background for McNaughton and Yamada's algorithm [14] through the notion of continuation which is a particular kind of derivative. Ginzburg =-=[9]-=- uses derivatives to develop a procedure for proving equivalence of regular expressions; further development of this procedure is provided by Mizoguchi et al [15]. Yet another procedure for proving eq... |

3 |
Differentation of K-rational expressions
- Krob
- 1992
(Show Context)
Citation Context ...ful and productive tool. Conway [8] uses derivatives to present various computational procedures in the algebra of regular expressions and to investigate some logical properties of this algebra. Krob =-=[12]-=- extends this differential calculus to a more general algebra of K-rational expressions. Brzozowski and Leiss [6] employ the idea of derivatives to ascertain relations between regular expressions, fin... |

2 |
Rewriting extended regular expressions (short version
- Antimirov, Mosses
- 1994
(Show Context)
Citation Context ... further development of this procedure is provided by Mizoguchi et al [15]. Yet another procedure for proving equivalence of extended regular expressions is suggested by the present author and Mosses =-=[2, 3]-=- and is also based on some constructions closely related to derivatives. In the present paper we come up with a new notion of a partial derivative which is, in a sense, a non-deterministic generalizat... |

1 |
On Ground-Reducibility Problem for Word Rewriting Systems with Variables
- Kucherov, Rusinowitch
- 1994
(Show Context)
Citation Context ...is due to Gregory Kucherov: he proposed us to construct an automaton for the following regular expression (which is an example of those appearing in the study of word-rewriting systems with variables =-=[13]-=-): t = (a + b) (babab(a + b) bab + bba(a + b) bab)(a + b) Our algorithm turns this expression into the following NFA with 11 states: State equations Partial derivatives S 1 := a \Delta S 1 + b \Delta ... |

1 |
A symbolic calculus of regular expressions
- Mizoguchi, Ohtsuka, et al.
- 1987
(Show Context)
Citation Context ...articular kind of derivative. Ginzburg [9] uses derivatives to develop a procedure for proving equivalence of regular expressions; further development of this procedure is provided by Mizoguchi et al =-=[15]-=-. Yet another procedure for proving equivalence of extended regular expressions is suggested by the present author and Mosses [2, 3] and is also based on some constructions closely related to derivati... |