## Minimizing local automata

Citations: | 6 - 2 self |

### BibTeX

@MISC{Béal_minimizinglocal,

author = {Marie-pierre Béal},

title = {Minimizing local automata},

year = {}

}

### OpenURL

### Abstract

Abstract — We design an algorithm that minimizes irreducible deterministic local automata by a sequence of state mergings. Two states can be merged if they have exactly the same outputs. The running time of the algorithm is O(min(m(n −r +1), m log n)), where m is the number of edges, n the number of states of the automaton, and r the number of states of the minimized automaton. In particular, the algorithm is linear when the automaton is already minimal and contrary to Hopcroft’s minimisation algorithm that has a O(kn log n) running time in this case, where k is the size of the alphabet, and that applies only to complete automata. (Note that kn ≥ m.) While Hopcroft’s algorithm relies on a “negative strategy”, starting from a partition with a single class of all states, and partitioning classes when it is discovered that two states cannot belong to the same class, our algorithm relies on a “positive strategy”, starting from the trivial partition for which each class is a singleton. Two classes are then merged when their leaders have the same outputs. The algorithm applies to irreducible deterministic local automata, where all states are considered both initial and final. These automata, also called covers, recognize symbolic dynamical shifts of finite type. They serve to present a large class of constrained channels, the class of finite memory systems, used for channel coding purposes. The algorithm also applies to irreducible deterministic automata that are left-closing and have a synchronizing word. These automata present shifts that are called almost of finite type. Almost-of-finite-type shifts make a meaningful class of shifts, intermediate between finite type shifts and sofic shifts.

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(Show Context)
Citation Context ...(a) have to be merged. Classes of the partition of Im(a) are implemented as trees where the leader of the class is at the root of the tree. A sequence of m Union-Find operations with path compression =-=[7]-=- is performed in time O(m min{α(m), log n}), where α is the inverse of the Ackermann function. Let us denote by p (resp. q) the leader of the class of x (resp. the class of y). Assume that the number ... |

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Citation Context ...g radix sort for sorting the list whose elements belong to the integer interval [1,2,...,m]. The implementation of lists arc and succ(x) for all nodes x is done with a sparse list implementation (see =-=[2]-=- Exercise 2.12 p. 71 and [13] Exercise 1.14 “Implantation de fonctions partielles” Chapter 1). These lists are indexed by (a,p) where ap is the label of some arc of the trie. As a consequence, it is p... |

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Citation Context ...erministic automaton. It is known that, when A is irreducible, the shift presented by A has a minimal deterministic irreducible automaton called the right Fischer cover of the shift (see for instance =-=[12]-=-, [11], [4]). It is unique up to a renumbering of its states. It can be obtained from A by computing the Nerode partition of the states: two states p, q belong to the same class of this partition if a... |

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Citation Context ...ion. The algorithm consists first in building a digital tree, called the signature trie, which stores the lexicographically-sorted outputs of states. This technique, called multiset discrimination in =-=[15]-=-, [6] is used to avoid hashing and to produce deterministic algorithms. In the second step the tree is updated after each state merging. Our algorithm applies to the class of almost-of-finite-type aut... |

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Citation Context ...stic automaton. It is known that, when A is irreducible, the shift presented by A has a minimal deterministic irreducible automaton called the right Fischer cover of the shift (see for instance [12], =-=[11]-=-, [4]). It is unique up to a renumbering of its states. It can be obtained from A by computing the Nerode partition of the states: two states p, q belong to the same class of this partition if and onl... |

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Citation Context ...The trie can be constructed after a lexicographical sort of the signatures. This technique is analogue to the multiset discrimination described in [15], [6], which avoids using hashing. It is used in =-=[17]-=- to minimize acyclic automata. The structure of a node x of the trie is the following. It contains the (non-sorted) list succ(x) of the arcs leaving x. The list is indexed by all (a, p) for which ap i... |

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Citation Context ...shifts. I. INTRODUCTION Local automata, also called definite automata or definite covers [26], present a large class of systems: the class of finite memory systems used in coding constrained channels =-=[1]-=-. These channels have a canonical minimal deterministic presentation, called the Fischer cover, which can be computed from an n-state local deterministic presentation using Hopcroft’s minimisation alg... |

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Citation Context ...he algorithm consists first in building a digital tree, called the signature trie, which stores the lexicographically-sorted outputs of states. This technique, called multiset discrimination in [15], =-=[6]-=- is used to avoid hashing and to produce deterministic algorithms. In the second step the tree is updated after each state merging. Our algorithm applies to the class of almost-of-finite-type automata... |

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Citation Context ...ible automata that are also left-closing (or co-deterministic with a finite delay), and synchronizing. The automata present channels called almost of finite type (AFT). They were introduced by Marcus =-=[13]-=- for coding purposes. Indeed, the theory of modulation codes provides many natural AFT examples. The AFT sofic shifts is a meaningful class, intermediate between shifts of finite type and sofic shifts... |

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Citation Context ...utomaton. It is known that, when A is irreducible, the shift presented by A has a minimal deterministic irreducible automaton called the right Fischer cover of the shift (see for instance [12], [11], =-=[4]-=-). It is unique up to a renumbering of its states. It can be obtained from A by computing the Nerode partition of the states: two states p, q belong to the same class of this partition if and only if ... |

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Citation Context ...ost-of-finite-type shifts make a meaningful class of shifts, intermediate between finite type shifts and sofic shifts. I. INTRODUCTION Local automata, also called definite automata or definite covers =-=[16]-=-, present a large class of systems: the class of finite memory systems used in coding for constrained channels [1]. These channels have a canonical minimal deterministic presentation, called the Fisch... |

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Citation Context ...t whose elements belong to the integer interval [1, 2, . . .,n]. The implementation of lists arc and succ(x) for all nodes x is done with a sparse list implementation (see [2] Exercise 2.12 p. 71 and =-=[8]-=- Exercise 1.14 “Implantation de fonctions partielles” Chapter 1). These lists are indexed by (a, p) where ap is the label of some arc of the trie. As a consequence, it is possible to find, add, or rem... |

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Citation Context ...the sense of symbolic dynamics. It is possible to encode an unconstrained source into an AFT constrained channel having a capacity not less that the entropy of the source with a sliding block decoder =-=[10]-=-, while it is not possible to build such a code for a general finite-state constrained channel in the case of equality of the capacity and entropy. From the algorithmic point of view, our method is a ... |

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Citation Context ...ollowing lemma from [3, p.41] provides a efficient characterization of a left-closing automata. Efficient characterizations of AFT shifts where obtained by Boyle, Kitchens, and Marcus [5] and by Nasu =-=[14]-=-. The lemma uses the notion of product of two automata defined as follows. The product of the automaton A by itself is the automaton A 2 = A × A = (Q × Q, F) where ((p, q), a, (r, s)) ∈ F if and only ... |

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5 |
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(Show Context)
Citation Context ...utomaton. It is known that, when A is irreducible, the shift presented by A has a minimal deterministic irreducible automaton called the right Fischer cover of the shift (see for instance [22], [20], =-=[5]-=-). It is unique up to a renumbering of its states. It can be obtained from A by computing the Nerode partition of the states: two states p,q belong to the same class of this partition if and only if t... |

3 |
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Citation Context ... AFT sofic shifts is a meaningful class, intermediate between shifts of finite type and sofic shifts from the point of view of symbolic dynamics. Indeed, it was shown by Boyle, Kitchens and Marcus in =-=[5]-=- that the class of AFT shifts is the unique class of shifts having a minimal cover in the sense of symbolic dynamics. It is possible to encode an unconstrained source into an AFT constrained channel h... |