## State Complexity of Regular Languages (2000)

Venue: | Journal of Automata, Languages and Combinatorics |

Citations: | 35 - 6 self |

### BibTeX

@ARTICLE{Yu00statecomplexity,

author = {Sheng Yu},

title = {State Complexity of Regular Languages},

journal = {Journal of Automata, Languages and Combinatorics},

year = {2000},

volume = {6},

pages = {221--234}

}

### OpenURL

### Abstract

State complexity is a descriptive complexity measure for regular languages. We investigate the problems related to the state complexity of regular languages and their operations. In particular, we compare the state complexity results on regular languages with those on finite languages.

### Citations

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(Show Context)
Citation Context ...ll z 2 \Sigma : Clearly, jL is an equivalence relation, which partitions \Sigma into equivalence classes. The number of equivalence classes of jL is called the index of jL . The Myhill-Nerode Theorem =-=[9]-=- states that L is regular if and only if jL has a finite index and the minimal number of states of a complete DFA that accepts L is equal to the index of jL . For a rather complete background knowledg... |

2563 |
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Citation Context ...tages over other representations of regular languages such as nondeterministic finite automata (NFA) and regular expressions: (1) Checking the equivalence of two DFA can be done in almost linear time =-=[1]-=-, while the same problem for NFA, respectively for regular expressions, is PSPACE complete. (2) For each regular language, the minimal DFA that accepts the language is unique up to an isomorphism. Thi... |

2288 |
The Unified Modeling Language User Guide
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Citation Context ...struct in many recently-created programming languages such as Perl and Python, and the adoption of the statecharts as part of the object-oriented modeling and design methodologies such as OMT and UML =-=[19, 2]-=-. In recent years, quite a few software systems for manipulating formal language objects, with the emphasis on regular-language objects, have been developed. Examples include AMoRE, Automate, FIRE Eng... |

1824 |
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Citation Context ...struct in many recently-created programming languages such as Perl and Python, and the adoption of the statecharts as part of the object-oriented modeling and design methodologies such as OMT and UML =-=[19, 2]-=-. In recent years, quite a few software systems for manipulating formal language objects, with the emphasis on regular-language objects, have been developed. Examples include AMoRE, Automate, FIRE Eng... |

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Citation Context ...ny extension 2 of state complexity in this article. Examining the state complexity results on the basic operations, e.g., catenation, union, intersection, and complementation, on regular languages in =-=[27]-=-, one would notice that all the worst cases are given by using infinite languages only. This observation raises the question: Are finite languages significantly different from (infinite) regular langu... |

54 |
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Citation Context ...ages. Finite languages are exactly the languages accepted by acyclic finite automata. It has been shown that there is a linear (time) algorithm for the minimization of an acyclic DFA by Revuz in 1992 =-=[15]-=-. However, for the minimization of a general DFA, the best known algorithm has a time complexity O(n log n) by Hopcropt in 1971 [8]. In this article, we compare the state complexity results for finite... |

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Citation Context ... #Q 1 = m, #Q 2 = n, and #F 1 = t, where t ? 0 is a constant. Then there exists a DFA A = (Q; \Sigma; ffi; s; F ) of O(mn t\Gamma1 + n t ) states such that L(A) = L(A 1 )L(A 2 ). It has been shown in =-=[3]-=- that the bound given in Theorem 3 can be reached in the case j\Sigmaj = 2. About the state complexity of the reversal of an m-state DFA language, one may easily have a misconception. Many thought, wi... |

28 |
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Citation Context ... State complexity is a complexity measure only for regular languages. However, it can be extended to cover other families of languages as well. For example, the automaticity studied by Shallit et al. =-=[24]-=- can be considered as an extension of the state complexity. We will not consider any extension 2 of state complexity in this article. Examining the state complexity results on the basic operations, e.... |

20 |
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Citation Context ...s a finite index and the minimal number of states of a complete DFA that accepts L is equal to the index of jL . For a rather complete background knowledge in automata theory, the reader may refer to =-=[9, 22]-=-. The following lemmas will be used in the subsequent sections. They can be proved rather easily. Thus, we omit the proofs to concentrate on our main results. Lemma 1 Let R ` \Sigma be a regular langu... |

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19 |
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Citation Context ...investigation of the state complexity of regular languages and their operations was already going on in the sixties and seventies. Examples of early studies concentrated on this topic can be found in =-=[17, 20, 21]-=-. However, many problems remain. We are now back to those basic problems with much renewed motivation and interest. The DFA model has at least the following advantages over other representations of re... |

16 | Minimal coverautomata for finite languages - Câmpeanu, Santean, et al. |

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9 |
personal communication
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Citation Context ... in this paper. It is more interesting to show that the state complexities of those two operations are indeed of the order of mn but not lower. The following examples were originally given by Shallit =-=[23]-=-. Automatonbased examples are given in [4], which give better lower bounds than the examples below. We choose to present the following examples due to their clarity and intuitiveness. For the intersec... |

5 |
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Citation Context ...r languages, finite languages, deterministic finite automata, state complexity. 1 Introduction Regular languages and their implementations have been attracting more and more attention in recent years =-=[18, 25]-=- due to the increased applications of regular languages and finite automata in software engineering, programming languages, and other practical areas of computer science. Evidences of the increased ap... |

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3 |
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(Show Context)
Citation Context ...investigation of the state complexity of regular languages and their operations was already going on in the sixties and seventies. Examples of early studies concentrated on this topic can be found in =-=[17, 20, 21]-=-. However, many problems remain. We are now back to those basic problems with much renewed motivation and interest. The DFA model has at least the following advantages over other representations of re... |

3 |
Chapter 2: Regular Languages", in Handbook of Formal Languages
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Citation Context ...trast, for infinite regular languages, there are examples in which A 1 has only one final state but any DFA accepting the catenation of the two languages needs at least (2m \Gamma 1)2 n\Gamma1 states =-=[27, 26]-=-. This is one of a few cases in which the state complexities for finite and infinite regular languages, respectively, are in different orders. We now give the proof for the finite language case. For t... |

2 |
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(Show Context)
Citation Context ...investigation of the state complexity of regular languages and their operations was already going on in the sixties and seventies. Examples of early studies concentrated on this topic can be found in =-=[17, 20, 21]-=-. However, many problems remain. We are now back to those basic problems with much renewed motivation and interest. The DFA model has at least the following advantages over other representations of re... |

1 |
Finite languages and cover automata", in preparation
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(Show Context)
Citation Context ...languages accepted by an m-state and an n-state DFA, respectively. However, this is a very rough upper bound. Much tighter upper bounds for the union and intersection of finite languages are given in =-=[4]-=-, which unfortunately are in a very complicated and highly incomprehensible form. Thus, we will not quote them in this paper. It is more interesting to show that the state complexities of those two op... |

1 |
An n log n algorithm for minimizing the states in a finite automaton", The Theory
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(Show Context)
Citation Context ...) algorithm for the minimization of an acyclic DFA by Revuz in 1992 [15]. However, for the minimization of a general DFA, the best known algorithm has a time complexity O(n log n) by Hopcropt in 1971 =-=[8]-=-. In this article, we compare the state complexity results for finite and infinite regular languages. We first consider the relatively simple cases, i.e., the operations on languages over a one-letter... |

1 | Some applications of a technique of Sakoda and Sipser - Ravikumar - 1990 |