In this work we use a framework of finite-state automata constructions based on equivalences over words to provide new insights on the relation between well-known methods for computing the minimal deterministic automaton of a language.
Some of these connections have already been studied in order to offer a better understanding of Brzozowski's double-reversal method @cite_10 @cite_3 @cite_8 @cite_9 . In particular, @cite_10 and @cite_3 offer an alternative view of minimization and determinization methods in a uniform way from a category-theoretical perspective. In contrast, our work revisits these well-known minimization techniques relying on simple language-theoretical notions.
@cite_9 @cite_10 @cite_3 @cite_8
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a b s t r a c t We here propose a polynomial-time deterministic finite automaton minimization algorithm directly derived from Brzozowski’s double reversal algorithm. To do so, we take into account the framework by Brzozowski and Tamm, to propose an atomization algorithm that allows us to achieve polynomial time complexity. We show that every regular language defines a unique nondeterministic finite automaton (NFA), which we call "atomaton", whose states are the "atoms" of the language, that is, non-empty intersections of complemented or uncomplemented left quotients of the language. We describe methods of constructing the atomaton, and prove that it is isomorphic to the normal automaton of Sengoku, and to an automaton of Matz and Potthoff. We study "atomic" NFA's in which the right language of every state is a union of atoms. We generalize Brzozowski's double-reversal method for minimizing a deterministic finite automaton (DFA), showing that the result of applying the subset construction to an NFA is a minimal DFA if and only if the reverse of the NFA is atomic. In this paper, we study decomposition methods based on separable approximations for minimizing the augmented Lagrangian. In particular, we study and compare the diagonal quadratic approximation method DQAM of Mulvey and Ruszczynski [A diagonal quadratic approximation method for large scale linear programs, Oper. Res. Lett. 12 1992, pp. 205–215] and the parallel coordinate descent method PCDM of Richtarik and Takac [Parallel coordinate descent methods for big data optimization. Technical report, November 2012. arXiv:1212.0873]. We show that the two methods are equivalent for feasibility problems up to the selection of a step-size parameter. Furthermore, we prove an improved complexity bound for PCDM under strong convexity, and show that this bound is at least 8L′ Lω−12 times better than the best known bound for DQAM, where ω is the degree of partial separability and L’ and L are the maximum and average of the block Lipschitz constants of the gradient of the quadratic penalty appearing in the augmented Lagrangian. We study the number of states of the minimal automaton of the mirror of a rational language recognized by a random deterministic automaton with n states. We prove that, for any d > 0, the probability that this number of states is greater than n d tends to 1 as n tends to infinity. As a consequence, the generic and average complexities of Brzozowski minimization algorithm are super-polynomial for the uniform distribution on deterministic automata.