Derivation of the Cascade Parallel Composition of Timed Finite State Machines Using BALM-II

Abstract

The paper considers the problem of deriving a cascade parallel composition of timed finite state machines (TFSMs). It can be reduced to a step-by-step derivation of a binary parallel composition. It is known that if each component of the binary parallel composition is a TFSM with constant output delays, then the composition can result in a TFSM with an infinite set of output delays given by a finite set of linear functions. Therefore, the problem of deriving a cascade composition of TFSMs with constant output delays is reduced to deriving a series of binary parallel compositions of TFSMs with output delays given either as constants or as a set of linear functions. In this paper, we refine the definition of the TFSM with particular attention to the description of the output delay. BALM-II is used as an instrument for deriving the composition. Therefore, we consider the transition from a TFSM with output delays given as a set of linear functions to the corresponding automaton. We propose a new procedure for deriving an automaton. Unlike the known procedure, it does not require subsequent determinization of the derived automaton. In addition, we provide a step-by-step description of how to derive the composition of the corresponding automata using BALM-II, discuss the procedure of reverse transformation from the automaton of the composition to the TFSM, and note some aspects associated with the composition of TFSMs with output delays as a set of linear functions. An example that illustrates the derivation of the cascade parallel composition of TFSMs is given.