Adjoint machines, state-behavior machines, and duality

A process X: K→K is output if Dyn(X)→K has a right adjoint; state-behavior if Dyn(X)→X has both left and right adjoints; and adjoint if X has a right adjoint and K has countable coproducts. Output processes provide the proper setting for a general theory of state observability. We give a minimal realization theory using image factorization of a total response map. We give an adjointness theory for state-behavior machines and a duality theory for adjoint machines which clarifies classical linear system duality and yields an improved duality for nondeterministic automata. Adjoint machines (machines with adjoint input processes) provide the first integration of classical sequential machines (the only state-behavior machines in the category, Set, of sets), metric machines, topological machines, linear systems, nondeterministic automata and Boolean machines. There exist state-behavior machines which are not adjoint (but not in Set).