Hopcroft's automaton minimization algorithm and Sturmian words

International audience This paper is concerned with the analysis of the worst case behavior of Hopcroft's algorithm for minimizing deterministic finite state automata. We extend a result of Castiglione, Restivo and Sciortino. They show that Hopcroft's algorithm has a worst case behavior for the automata recognizing Fibonacci words. We prove that the same holds for all standard Sturmian words having an ultimately periodic directive sequence (the directive sequence for Fibonacci words is $(1,1,\ldots)$).

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Continuant polynomials and worst-case behavior of Hopcroft’s minimization algorithm

Theoretical Computer Science ◽

10.1016/j.tcs.2009.01.039 ◽

2009 ◽

Vol 410 (30-32) ◽

pp. 2811-2822 ◽

Cited By ~ 11

Author(s):

Jean Berstel ◽

Luc Boasson ◽

Olivier Carton

Keyword(s):

Minimization Algorithm ◽

Worst Case ◽

Worst Case Behavior

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BERGMAN under MS-DOS and Anick's resolution

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.237 ◽

1997 ◽

Vol Vol. 1 ◽

Author(s):

S. Cojocaru ◽

V. Ufnarovski

Keyword(s):

Hilbert Series ◽

The Other ◽

Finite State Automata ◽

Universal Enveloping Algebras ◽

Generators And Relations ◽

Enveloping Algebras ◽

Finite State ◽

Algorithmic Problems ◽

Noncommutative Algebras ◽

International Audience

International audience Noncommutative algebras, defined by the generators and relations, are considered. The definition and main results connected with the Gröbner basis, Hilbert series and Anick's resolution are formulated. Most attention is paid to universal enveloping algebras. Four main examples illustrate the main concepts and ideas. Algorithmic problems arising in the calculation of the Hilbert series are investigated. The existence of finite state automata, defining thebehaviour of the Hilbert series, is discussed. The extensions of the BERGMAN package for IBM PC compatible computers are described. A table is provided permitting a comparison of the effectiveness of the calculations in BERGMAN with the other systems.

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Stable Encoding of Large Finite-State Automata in Recurrent Neural Networks with Sigmoid Discriminants

Neural Computation ◽

10.1162/neco.1996.8.4.675 ◽

1996 ◽

Vol 8 (4) ◽

pp. 675-696 ◽

Cited By ~ 32

Author(s):

Christian W. Omlin ◽

C. Lee Giles

Keyword(s):

Neural Networks ◽

Recurrent Neural Networks ◽

Case Analysis ◽

Network Size ◽

Worst Case Analysis ◽

Small Subset ◽

Finite State Automata ◽

State Behavior ◽

Worst Case ◽

Finite State

We propose an algorithm for encoding deterministic finite-state automata (DFAs) in second-order recurrent neural networks with sigmoidal discriminant function and we prove that the languages accepted by the constructed network and the DFA are identical. The desired finite-state network dynamics is achieved by programming a small subset of all weights. A worst case analysis reveals a relationship between the weight strength and the maximum allowed network size, which guarantees finite-state behavior of the constructed network. We illustrate the method by encoding random DFAs with 10, 100, and 1000 states. While the theory predicts that the weight strength scales with the DFA size, we find empirically the weight strength to be almost constant for all the random DFAs. These results can be explained by noting that the generated DFAs represent average cases. We empirically demonstrate the existence of extreme DFAs for which the weight strength scales with DFA size.

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