A Complex Measure for Linear Grammars

Abstract

Abstract. The signed real measure of regular languages, introduced and validated in recent literature, has been the driving force for quantitative analysis and synthesis of discrete-event supervisory (DES) control systems dealing with finite state automata (equivalently, regular languages). However, this approach relies on memoryless state-based tools for supervisory control synthesis and may become inadequate if the transitions in the plant dynamics cannot be captured by finitely many states. From this perspective, the measure of regular languages needs to be extended to that of non-regular languages, such as Petri nets or other higher level languages in the Chomsky hierarchy. Measures for non-regular languages has not apparently been reported in open literature and is an open area of research. As a step toward achieving this goal, this paper introduces a complex measure of linear context free grammars (LCFG) that belong to the class of non-regular languages. The proposed complex measure becomes equivalent to the signed reed measure, reported in recent literature, if the LCFG is degenerated to a regular grammar. IntroductionFinite state automata (FSA) (equivalently, regular languages) have been widely used to model and synthesize supervisory control laws for discreteevent plants [4] because the task of discrete-event supervisory (DES) control synthesis becomes mathematically tractable and computationally efficient due to simplicity of regular languages. According to the paradigm of DES control, a finite-state automaton (e.g., the discrete-event model of a physical plant) is a language generator whose behavior is constrained by the supervisor to meet a given specification. The (controlled) sublanguage of the plant behavior could be different under different supervisors that satisfy their own respective specifications. Such a partially ordered set of sublanguages re-1991 Mathematics Subject Classification: 93A05.