A language measure for discrete-event automata

This paper presents the concept and formulation of a signed real measure of regular languages for analysis of discrete-event supervisory control systems. The measure is constructed based upon the principles of language theory and real analysis for quantitative evaluation and comparison of the controlled behavior for discrete-event automata. The marked (i.e., accepted) states of finite-state automata are classified in different categories such that the event strings leading to good and bad marked states have positive and negative measures, respectively. In this setting, a controlled language attempts to disable as many bad strings as possible and as few good strings as possible. Different supervisors may achieve this goal in different ways and generate a partially ordered set of controlled languages. The language measure creates a total ordering on the performance of the controlled languages, which provides a precise quantitative comparison of the controlled plant behavior under different supervisors. The total variation of this language measure induces a norm on the vector space of sublanguages of the given regular language over the Galois field GF(2).