Modeling of symbolic systems: Part I - Vector space representation of probabilistic finite state automata

This paper, which is the first of two parts, brings in the notions of vector addition and the associated scalar multiplication operations on probabilistic finite state automata (PFSA). A class of PFSA is shown to constitute a vector space over the real field R, where the zero element is semantically equivalent to a subclass of PFSA, referred to as symbolic white noise. A norm is introduced on the vector space of PFSA and it quantifies the non-probabilistic behavior of a PFSA. The second part constructs a family of inner products on this vector space and presents numerical examples and applications.