Further improvements of determinization methods for fuzzy finite automata

Abstract

In this paper we provide further improvements of determinization methods for fuzzy finite automata. These methods perform better than all previous determinization methods for fuzzy finite automata, developed by Bělohlávek [3], Li and Pedrycz [21], Ignjatović et al. [12], and Jančić et al. [16], in the sense that they produce smaller automata, while require the same computation time. The only exception is the Brzozowski type determinization algorithm developed recently by Jančić andĆirić [17], which produces a minimal crispdeterministic fuzzy automaton, but the algorithms created here can also be used within the Brzozowski type algorithm and improve its performance.

Determinization of Fuzzy Automata by Means of the Degrees of Language Inclusion

“…Moreover, if ψ wli is the greatest weakly left invariant fuzzy relation and ψ li is the greatest left invariant fuzzy relation on A, then A ψ wli A ψ li A (cf. [12]). Now, define inductively a family {∆ u } u∈X * of fuzzy subsets of A, as follows:…”

“…It is easy to check that Theorems 3.1, 3.2 and 3.3 remain valid when d u is replaced by ∆ u , which can significantly improve our canonization method since the cardinality of the family {ψ w } w∈X * is smaller than or equal to the cardinality of {τ w } w∈X * , and it may be significantly smaller. Furthermore, even in some cases where the family {τ w } w∈X * is infinite, the family {ψ w } w∈X * may be finite (see Example 4.13 [12]).…”

“…As the Nerode automaton A N has at most k n states, the resulting transition tree for A d has at most k n internal vertices, and the total number of vertices does not exceed mk n + 1. In contrast to other determinization algorithms provided in [8], [10], [11], [12], where the most time-demanding part is the check whether the just computed fuzzy set is a copy of some previously computed fuzzy set, here most of the time is spent on computing the fuzzy sets d u , u ∈ X * . Namely, we can write (15) as…”

“…The proposed canonization procedure is based on the degrees of inclusion of the right fuzzy languages associated with states of A into the left derivatives of the fuzzy language recognized by A. The computation time of this procedure is generally better than the computation time of the Brzozowski type determinization, and if the basic operations in the underlying residuated lattice can be performed in constant time, it has the same computational time as all other determinization procedures provided in [8], [11], [12].…”

Application of Fuzzy Automata Decision-Making System in Target Control

“…With the development of information technology, more and more scholars had studied FA, and had achieved fruitful results in theories [9] [10] and applications [11] [12]. Giles et al (1992) [13] used a complete gradient algorithm to derive a Tomita language.…”