Algebraic representations for finite-state machines. II. Module formulation Original Research Article

We show that finite-state machines can be represented as unique elements of special modules of functions. We obtain a module representation for the machine with the least number of states over a class of equivalent machines. We present a unique factorization of this representation. We construct an array which characterizes all state transitions and is identical for all machines in the equivalence class. Further, we show that the module representation for any finite-state machine is contained in a free submodule, and can be written as a linear combination of elements of submodules obtained from equivalent machine states. Module representations and associated arrays are given for two examples.