An inner product space on irreducible and synchronizable probabilistic finite state automata

Probabilistic finite state automata (PFSA) have found their applications in diverse systems. This paper presents the construction of an inner-product space structure on a class of PFSA over the real field via an algebraic approach. The vector space is constructed in a stationary setting, which eliminates the need for an initial state in the specification of PFSA. This algebraic model formulation avoids any reference to the related notion of probability measures induced by a PFSA. A formal language-theoretic and symbolic modeling approach is adopted. Specifically, semantic models are constructed in the symbolic domain in an algebraic setting. Applicability of the theoretical formulation has been demonstrated on experimental data for robot motion recognition in a laboratory environment.