Regular expressions are a well-known tool to specify the behaviour of
finite sequential automata. Kleene's classical theorem states that the
semantics of every finite automaton can be expressed by a regular
expression. Furthermore, Kleene algebras provide a sound and complete
calculus for the behavioural equivalence of finite automata.

Recently, Bonsangue, Rutten and Silva provided an analog of these
classical results pertaining to coalgebras for endofunctors of the
category of sets. From every Kripke polynomial functor one can derive
a calculus of expressions that is sound and complete with respect to
the behavioral equivalence of finite coalgebras.

Based on these ideas I will present a sound and complete expression
calculus for finite dimensional linear systems. These systems are
coalgebras for the functor $H = R \times -$ on the category of real
vector spaces, where $R$ are the reals. Finite dimensional linear
systems are equivalently presented by finite stream circuits. So the
calculus allows one to reason about the equivalence of finite closed
stream circuits.

The main technical result of my talk is that expressions modulo the
laws of the calculus form the final locally finite dimensional
coalgebra for $H$. This gives a new syntactic characterization of the
coalgebra of rational streams known from Rutten's stream calculus.

Time permitting, I will show that, more generally the final locally
finite (dimensional) coalgebra is, equivalently, the initial iterative
algebra for a set (vector space) endofunctor.