Finite Step Conditions

Abstract

Arbib and Zeiger [9] in giving a unified (but “pre-categorical”) view of linear systems and sequential machines, observed that the subspaces

$${\text{S}}_{\text{1}} \subset {\text{S}}_{\text{2}} \subset {\text{S}}_{\text{3}} \subset ...$$

with S_j being the set of states reachable in at most k steps from the initial state had the property that if ever S_k equalled S_k+1, then all the S_j’s were equal for j ≥ k. They observed that whenever S_j was not equal to S_j+1, then S_j+1 exceeded S_j by at least 1 in cardinality for sequential machines, and by at least 1 in dimensionality for linear systems. They could then exhibit the commonality of the results that for a sequential machine of n states the set of reachable states was S_n-1, while for a linear system of dimension n the space of reachable states was S_n. We now provide a general categorical setting for these results, and the dual observability results. In this section, we provide a theory applicable, in particular, to adjoint processes (which include our classic examples of sequential machines and linear systems); while Section 5 gives a more general theory applicable to tree automata, which are not even state-behavior.