On genetic algorithms minimizing a class of FSA with fuzzy automata

Finite state automata are crucial for numerous practical algorithms of computer science. We show how to use genetic algorithms and fuzzy automata to simplify a class of FSA defined by labeled graphs and considered in the literature. Finite state automata, FSA, are well-known tools used in coding theory, text processing, image analysis and compression, speech recognition, and bioinformatics. Labeled directed graphs have been applied in [2], [4] and [5] to define and investigate a class of FSA. Throughout the word graph means a finite directed graph without multiple edges but possibly with loops, and D = (V,E) is a graph. Graphs have been used by several authors to define automata and investigate properties of languages accepted by them, see [2]. A language over an alphabet X is a subset of the free monoid X∗ generated by X. For standard concepts of automata and languages theory we refer to [2] and [8]. Let X be an alphabet, f : X → V a mapping, and let T be a subset of V ∪ {0, 1}. The FSA Atm`(D) = Atm`(D,T ) = Atm`(D,T, f) of the graph D is the finite state acceptor with (LA1) the set of states V ∪ {0, 1}; (LA2) the initial state 1; (LA3) the set of terminal states T ; (LA4) the next-state function given, for a state u and a letter x ∈ X, by defining u · x to be equal to f(x), if (f(x), u) ∈ E or u = 1, and be 0 otherwise. Supported by Discovery grant DP0449469 from the Australian Research Council. For motivation and relations to previous results the reader is referred to [2]. A language over X is a set of words that can be formed by the letters of X, i.e., a subset of the free monoid X∗ generated by X. The language recognized or accepted by Atm`(D,T ) is the set {u ∈ X∗ | 1 · u ∈ T}. Combinatorial minimization algorithms for FSA of this type are computationally expensive. Efficient optimization methods, like those based on genetic algorithms, are not directly applicable either, since randomized steps of these algorithms can dramatically change the crisp language recognized by the FSA. In order to apply genetic algorithms for minimizing the FSA, we offer the following general scheme, where it is suggested to replace the original crisp FST with its fuzzy analogue, as summarized in Figure 1. For preliminaries on fuzzy systems theory and genetic algorithms we refer to [1], [6], and [7]. Algorithm 1 Enables the application of genetic algorithms to the minimization of FSA Atm`(D,T, f) with fuzzy automata. . Step 1. Replace Atm`(D,T, f) with equivalent fAtm`(D,T, f). Step 2. Encode each fuzzy set of fAtm`(D,T, f) a minimal set of parameters. Step 3. Define crossover operation on blocks of the encoding. Step 4. Use a genetic algorithm to minimize fAtm`(D,T, f). Step 5. Defuzzify the minimal fuzzy automaton. Figure 1: The general scheme of applying genetic algorithms via fuzzy automata. Fuzzy automata have been investigated by many authors (see [6] for references). This concept is related to the notion of a fuzzy language, that has been introduced explicitly by Zadeh [10]. Next, we follow [3] and [6] and define the concept of fuzzy automaton that can be used to approximate the functions of crisp automata above. The required standard concepts of fuzzy theory, like fuzzy congruences, etc., are explained in [6]. A fuzzy automaton is a system A = (S,Λ, p, F,G), where S is a finite set of states; Λ is a finite set of inputs; p ⊂∼ S is a fuzzy set called a fuzzy initial state; F : S×Λ×S → [0, 1] is a fuzzy transition function, i.e., F (λ) ⊂∼ S×S is a fuzzy transition matrix; and G ⊂∼ S is a fuzzy set called a fuzzy final state. For s, s′ ∈ S and λ ∈ Λ, recall that F (s, λ, s′) denotes the grade of transition of a fuzzy automaton from state s to state s′ when the input is λ ∈ Λ. If we denote the free monoid generated by Λ by Λ∗, then F can be extended to a fuzzy transition function F ∗ : S × Λ∗ × S → [0, 1], where λ = λ1λ2 . . . λn ∈ Λ∗. Then the following diagram commutes