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By Vandenbussche D., Nemhauser G. L.

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L. Pitt. Inductive Inference, DFAs, and Computational Complexity. In Proceedings of AII-89 Workshop on Analogical and Inductive Inference; Lecture Notes in Artificial Intelligence 397, pages 18-44, Heidelberg, October 1989. Springer-Verlag. Valiant. A theory of the learnable. Commun. ACM, 27(11):1134-1142, November 1984. Smoothing Probabilistic Automata: An Error-Correcting Approach Pierre Dupont1 and Juan-Carlos Amengual2 1 EURISE, Universit´e Jean Monnet 23, rue P. es Abstract. In this paper we address the issue of smoothing the probability distribution defined by a probabilistic automaton.

5th ACM workshop on Computation Learning Theorie, 1992, pp. 45 – 52. Lang (K. ), Pearlmutter (B. ) et Price (R. ). – Results of the abbadingo one DFA learning competition and a new evidence-driven state merging algorithm. Lecture Notes in Computer Science, vol. 1433, 1998, pp. 1–12. ). – Inferring regular languages in polynomial update time. Pattern Recognition and Image Analysis, 1992, pp. 49 – 61. Oliveira (A. ) et Silva (J. P. ). – Efficient search techniques for the inference of minimum size finite automata.

1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18: Incompatible States(A = (Σ, Γ, Q, Q0 , δ, ρ)): /* Search of the set of incompatible states of A */ /* and ambiguity detection of A */ E ∼ ← ∅ /* set of incompatible states */ for all {qi , qj } ∈ Q × Q, ρ(qi ) ∼ρ(qj ) do if {qi , qj } ∈ E∼ then Set Incompatible And Propagate(qi , qj ) return E ∼ Set Incompatible And Propagate(q1 ,q2 ): /* ambiguity detection */ if (q1 = q2 ) ∨ (q1 ∈ Q0 ∧ q2 ∈ Q0 ) then throw exception(“ambiguous C-NFA”) /* Incompability memorization */ E ∼ ← E ∼ ∪ {q1 , q2 } /* Propagation */ for all a ∈ Σ, {p1 , p2 } ∈ δ −1 (q1 , a) × δ −1 (q2 , a) do if {p1 , p2 } ∈ E∼ then Set Incompatible And Propagate(p1 , p2 ) In the worst case, the complexity of algorithm 2 is O(|Σ|n4 ) : O(n2 ) calls of the function Set Incompatible And Propagate, whose body needs O(|Σ|n2 ) steps.

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