Abstract
Causality is one of the most fundamental properties of dynamical systems. Clarification of its nature is an important subject in the systems theory.
In our previous paper we characterized the causality for a rather restricted class of time systems.
In this paper we present a new condition for causality; that is,
(∀x, x')(∀t)(xt=x't→S(x)|Tt=S(x')|Tt).
We show that this condition is a necessary and sufficient condition for the time systems with output completeness to be causal. The output completeness is introduced here as one of the basic properties of time systems. It is shown that the class of time systems with output completeness is sufficiently large, including finite automata, the systems described by linear ordinary differential equations and also the systems treated in our previous paper.
In this sense this paper extends our previous results and establishes a practically complete characterization of causality for general time systems.