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Minimizing Deterministic Lattice Automata
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BibTeX
@MISC{Halamish_minimizingdeterministic,
author = {Shulamit Halamish and Orna Kupferman},
title = {Minimizing Deterministic Lattice Automata},
year = {}
}
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Abstract
Abstract. Traditional automata accept or reject their input, and are therefore Boolean. In contrast, weighted automata map each word to a value from a semiring over a large domain. The special case of lattice automata, in which the semiring is a finite lattice, has interesting theoretical properties as well as applications in formal methods. A minimal deterministic automaton captures the combinatoric nature and complexity of a formal language. Deterministic automata are used in run-time monitoring, pattern recognition, and modeling systems. Thus, the minimization problem for deterministic automata is of great interest, both theoretically and in practice. For traditional automata on finite words, a minimization algorithm, based on the Myhill-Nerode right congruence on the set of words, generates in polynomial time a canonical minimal deterministic automaton. A polynomial algorithm is known also for weighted automata over the tropical semiring. For general deterministic weighted automata, the problem of minimization is open. In this paper we study minimization of lattice automata. We show that it is impossible to define a right congruence in the context of lattices, and that no canonical minimal automaton exists. Consequently, the minimization problem is much more complicated, and we prove that it is NP-complete. As good news, we show that while right congruence fails already for finite lattices that are fully ordered, for this setting we are able to combine a finite number of right congruence and generate a minimal deterministic automaton in polynomial time. 1
Keyphrases
deterministic lattice automaton polynomial time deterministic automaton right congruence minimization problem lattice automaton minimal deterministic automaton finite lattice right congruence fails formal method interesting theoretical property myhill-nerode right congruence special case finite number good news canonical minimal automaton exists traditional automaton accept finite word tropical semiring polynomial algorithm formal language weighted automaton run-time monitoring great interest combinatoric nature large domain canonical minimal deterministic automaton automaton map minimization algorithm traditional automaton pattern recognition
