State Complexity Approximation

The state complexity of combined operations has been studied in, e. g., [12,4,3]. It has been shown that the state complexity of combined operations is at least as important and practical as the state complexity of individual operations. There is only a limited number of individual operations on regular languages. However, the number of combined operations on regular languages is unlimited and each of them is not simply a mathematical composition of the state complexities of their component individual operations. It appears that the exact state complexity of each combined operation has to be studied specifically.

There are at least the following two problems concerning the state complexities for combined operations. First, the state complexities of many combined operations are extremely difficult to compute. Second, a large proportion of results that have been obtained are pretty complex and impossible to comprehend. For example, the state complexity of the catenation for four regular languages accepted by m, n, p, q states, respectively, is 9(2m -1)2 n+p+q-5 -3(m -1)2 p+q-2 -(2m -1)2 n+q-2 + (m -1)2 q + (2m -1)2 n-2 .

It is clear that close estimations of state complexities are good enough in many automata applications. In [13,3], estimations of state complexity of combined operations have been proposed and studied. In this paper, we go further in the direction of the study in [13,3] and introduce the concept of state complexity approximation. Briefly speaking, an approximation of a state complexity is an estimate of the state complexity with a ratio bound clearly defined. The ratio bound gives a precise measurement on the quality of the estimate.

The idea of state complexity approximation is from the notion of approximation algorithms which was formalized in early 1970’s by David S. Johnson et al. [5,9,10]. Many polynomial-time approximation algorithms have been designed for a quite large number of NP-complete problems, which include the well-known travelling-salesman problem, the set-covering problem, and the subset-sum problem. Obtaining an optimal solution for an NP-complete problem is considered intractable. Near optimal solutions are often good enough in practice. Assuming that the problem is a maximization or a minimization problem, an approximation algorithm is said to have a ratio bound of ρ(n) if for any input of size n, the cost C of the solution produced by the algorithm is within a factor of ρ(n) of the cost C * of an optimal solution [1]:

The concept of state complexity approximation is in many ways similar to that of approximation algorithms. A state complexity approximation is close to the exact state complexity and normally not equal to it. The ratio bound shows the error range of the approximation. In addition to the property of having a small ratio bound in general, we also consider that a state complexity approximation should be in a simple and intuitive form.

In spite of the similarities, there are fundamental differences between a state complexity approximation and an approximation algorithm. The efforts in the area of approximation algorithms are in finding polynomial algorithms for NP-complete problems such that the results of the algorithms approximate the optimal results. In comparison, the efforts in the state complexity approximation are in searching directly for the estimations of state complexities such that they satisfy certain ratio bounds. The aim of designing an approximation algorithm is to transform an intractable problem into one that is easier to compute and the result is acceptable although not optimal. In comparison, a state complexity approximation result may have two different effects: (1) it gives a reasonable estimation of certain state complexity, with some bound, the exact value of which is difficult or impossible to compute; or (2) it gives a simpler and more comprehensible formula that approximates a known state complexity.

In the next section, we give some basic definitions and notation including the formal definition of state complexity approximation. In Section 3, we show the state complexity approximation results on four basic combined operations: the star of union, the star of intersection, the star of catenation, and the star of reversal. In Section 4, we show that state complexity approximation results can be easily obtained for some operations the exact state complexities of which may be very difficult to obtain. In Section 5, we show that certain state complexity can be very complex in formulation. A state complexity approximation is clearly more intuitive and comprehensible than the exact state complexity. In Section 6, we conclude the paper.

A deterministic finite automaton (DFA) is denoted by a 5-tuple A = (Q, Σ, δ, s, F ), where Q is the finite and nonempty set of states, Σ is the finite and nonempty set of input symbols, δ : Q × Σ → Q is the state transition function, s ∈ Q is the initial state, and F ⊆ Q is the set of final states. A DFA

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