## Subset construction complexity for homogeneous automata, position automata and ZPC-structures (2001)

Venue: | Theoretical Computer Science |

Citations: | 7 - 2 self |

### BibTeX

@ARTICLE{Champarnaud01subsetconstruction,

author = {J. -m. Champarnaud},

title = {Subset construction complexity for homogeneous automata, position automata and ZPC-structures},

journal = {Theoretical Computer Science},

year = {2001},

volume = {267},

pages = {2001}

}

### OpenURL

### Abstract

The aim of this paper is to investigate how subset construction performs on specific families of automata. A new upper bound on the number of states of the subset-automaton is established in the case of homogeneous automata. The complexity of the two basic steps of subset construction, i.e. the computation of deterministic transitions and the set equality tests, is examined depending on whether the nondeterministic automaton is an unrestricted one, an homogeneous one, a position one or a ZPC-structure, which is an implicit construction for a position automaton.

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Citation Context ...der of the definitions and notations which are useful for the description of our results will be found in this section. For further details about regular languages and finite automata, the references =-=[5, 18, 12, 19]-=- should be consulted. A finite automaton is a 5-tuple M = (Q, Σ, δ, I, F) where Q is a (finite) set of states, Σ is a (finite) alphabet, I ⊆ Q is the set of initial states, F ⊆ Q is the set of final s... |

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Citation Context ...der of the definitions and notations which are useful for the description of our results will be found in this section. For further details about regular languages and finite automata, the references =-=[5, 18, 12, 19]-=- should be consulted. A finite automaton is a 5-tuple M = (Q, Σ, δ, I, F) where Q is a (finite) set of states, Σ is a (finite) alphabet, I ⊆ Q is the set of initial states, F ⊆ Q is the set of final s... |

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Citation Context ...utomaton; ZPC-structure. 1 Introduction Automata determinization may be exponential, whereas most of automata operations are polynomial. It is not possible to avoid this behaviour in the general case =-=[16]-=-. It is nevertheless important to carefully handle the implementation of the determinization algorithm when designing automata software tools, in order to preserve as well as possible the performances... |

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Citation Context ...vestigate how subset construction performs on specific automata, in particular on position automata. The position automaton of a regular expression is defined by the algorithm described in [4] and in =-=[11]-=-. The author and his co-researchers [20, 13] have designed a linear space and time representation of the position automaton of a regular expression, which is based on two state forests connected by a ... |

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Citation Context ...der of the definitions and notations which are useful for the description of our results will be found in this section. For further details about regular languages and finite automata, the references =-=[5, 18, 12, 19]-=- should be consulted. A finite automaton is a 5-tuple M = (Q, Σ, δ, I, F) where Q is a (finite) set of states, Σ is a (finite) alphabet, I ⊆ Q is the set of initial states, F ⊆ Q is the set of final s... |

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Citation Context ...er is to investigate how subset construction performs on specific automata, in particular on position automata. The position automaton of a regular expression is defined by the algorithm described in =-=[4]-=- and in [11]. The author and his co-researchers [20, 13] have designed a linear space and time representation of the position automaton of a regular expression, which is based on two state forests con... |

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Citation Context ...(a) (b) Figure 2: a. The nondeterministic automaton PE. b. The deterministic automaton DE which is the subset automaton of PE. Example 3 E = (a + b) ∗ (babab(a + b) ∗ bab + bba(a + b) ∗ bab)(a + b) ∗ =-=[2]-=-: the standard bound is 8.3 × 10 6 and the new bound is 8.7 × 10 3 . 6 ZPC-structures An implicit construction of the position automaton, the so-called ZPC-structure [20, 21, 13, 14], has been develop... |

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Citation Context ...eterministic transitions, the choice of the data structure implementing the set of nondeterministic transitions has much influence on the complexity. Softwares such as AUTOMATE [3], INR [6] and GRAIL =-=[17]-=- make use of a representation in which both transitions with ∗ This work is a contribution to the Automate software development project carried on by A.I.A. (Algorithmics and Implementation of Automat... |

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Citation Context ...utation of the set of deterministic transitions, the choice of the data structure implementing the set of nondeterministic transitions has much influence on the complexity. Softwares such as AUTOMATE =-=[3]-=-, INR [6] and GRAIL [17] make use of a representation in which both transitions with ∗ This work is a contribution to the Automate software development project carried on by A.I.A. (Algorithmics and I... |

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Citation Context ...2 2n ) when using lists and O(n 2 log(n)2 n ) when using balanced search trees [15]. There exist investigations that are motivated by the will to improve such complexities, for instance Leslie et al. =-=[9, 10]-=-. The aim of this paper is to investigate how subset construction performs on specific automata, in particular on position automata. The position automaton of a regular expression is defined by the al... |

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Citation Context ...ms on specific automata, in particular on position automata. The position automaton of a regular expression is defined by the algorithm described in [4] and in [11]. The author and his co-researchers =-=[20, 13]-=- have designed a linear space and time representation of the position automaton of a regular expression, which is based on two state forests connected by a set of links. The so-called ZPCstructure lea... |

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Citation Context ...ion of Automata) Working Group, L.I.F.A.R. Contact: {Champarnaud, Ziadi}@dir.univ-rouen.fr. 1sthe same origin and transitions with the same origin and the same symbol are contiguous. Johnson and Wood =-=[7]-=- have studied the efficiency of various sorting procedures applied to subsets computed from such data structures. On the other hand, the choice of the data structure encoding subsets has a significant... |

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Citation Context ...ms on specific automata, in particular on position automata. The position automaton of a regular expression is defined by the algorithm described in [4] and in [11]. The author and his co-researchers =-=[20, 13]-=- have designed a linear space and time representation of the position automaton of a regular expression, which is based on two state forests connected by a set of links. The so-called ZPCstructure lea... |

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Citation Context ...f the set of deterministic transitions, the choice of the data structure implementing the set of nondeterministic transitions has much influence on the complexity. Softwares such as AUTOMATE [3], INR =-=[6]-=- and GRAIL [17] make use of a representation in which both transitions with ∗ This work is a contribution to the Automate software development project carried on by A.I.A. (Algorithmics and Implementa... |

2 |
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Citation Context ...b + bba(a + b) ∗ bab)(a + b) ∗ [2]: the standard bound is 8.3 × 10 6 and the new bound is 8.7 × 10 3 . 6 ZPC-structures An implicit construction of the position automaton, the so-called ZPC-structure =-=[20, 21, 13, 14]-=-, has been developed in 1995. It can be obtained in O(|E|) time, it requires O(|E|) space, and it can be converted into a position automaton in O(|E| 2 ) time. The main feature of a ZPC-structure is t... |

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Citation Context ...2 2n ) when using lists and O(n 2 log(n)2 n ) when using balanced search trees [15]. There exist investigations that are motivated by the will to improve such complexities, for instance Leslie et al. =-=[9, 10]-=-. The aim of this paper is to investigate how subset construction performs on specific automata, in particular on position automata. The position automaton of a regular expression is defined by the al... |

1 |
Algorithmique et implémentation des automates. Thèse, Université de
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Citation Context ...g depends on the data structure encoding subsets. The results reported in Theorem 1 concern lists and balanced search trees, improved by a hashing w.r.t. subset size. A complete proof can be found in =-=[14]-=-. 4 Homogeneous automata This section aims at studying the impact of the homogeneity property on subset construction. We first establish a new upper bound on the number of states of the subset-automat... |

1 |
An efficient null-free procedure for deciding regular language membership. Theoret
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Citation Context ...omplexity of set equality testing. The number of integer comparisons involved by the overall set equality tests is O( √ n2 2n ) when using lists and O(n 2 log(n)2 n ) when using balanced search trees =-=[15]-=-. There exist investigations that are motivated by the will to improve such complexities, for instance Leslie et al. [9, 10]. The aim of this paper is to investigate how subset construction performs o... |