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A complex measure for linear grammars

Chattopadhyay I., Ray A.

Demonstratio Mathematica

2005

Vol. 38, nr 3

761--775

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 real measure, reported in recent literature, if the LCFG is degenerated to a regular grammar.

Optimal supervisory control of regular languages

Ray A., Fu J., Lagoa C.

Demonstratio Mathematica

2004

Vol. 37, nr 4

991--1013

This paper presents an algorithm for optimal control of regular languages, realized as deterministic finite state automata (DFSA), with (possible) penalty on event disabling. A signed real measure quantifies the behavior of controlled sublanguages based on a state transition cost matrix and a characteristic vector as reported in an earlier publication. The performance index for the proposed optimal policy is obtained by combining the measure of the supervised plant language with the cost of disabled controllable event(s). Synthesis of this optimal control policy requires at most n iterations, where n is the number of states of the DFSA model generated from the unsupervised regular language. The computational complexity of the optimal control synthesis is polynomial in n. The control algorithms are illustrated with an application example of a twin-engine surveillance aircraft.

Measure of regular languages

Surana A., Ray A.

Demonstratio Mathematica

2004

Vol. 37, nr 2

485--503

This paper reviews and extends the recent work on signed real measure of regular languages within a unified framework. The language measure provides total ordering of partially ordered sets of sublanguages of a regular language to allow quantitative evaluation of the controlled behavior of deterministic finite state automata under different supervisors. The paper presents a procedure by which performance of different supervisors can be evaluated based on a common quantitative tool. Two algorithms are provided for computation of the language measure and their equivalence is established along with a physical interpretation from the probabilistic perspective.