Strong Stability Problems for Probabilistic Sequential Machines

The strong stability problem for a machine A and some behavioral equivalence ≡ is to characterize the input symbol transition matrices of all machines A′ formed from the input symbol transition matrices of A such that from any initial distribution A ≡ A′ . The strong stability problem is studied herein for the behavioral equivalences ≡ I (indistinguishability) and ≡ N ( N -moment equivalence). The concepts of strong stability transformation and error matrix are introduced to formalize the perturbations which are allowed in the input symbol transition matrices. Necessary conditions for the existence of any strongly stable transformations are studied using the concept of invariant error matrix. A strong stability transformation T is called “eigenstate behavior preserving” for an equivalence ≡ if for any distribution v over the states of A and any input symbol x (non-null) such that v·A ( x ) = v , we have the trajectory of distributions { v·T[ A ( x r )]}, r = 1, 2, ⋯ in A′ equivalent by ≡ to v in A′ . It is shown that strong stability transformations for ≡ I and ≡ N are eigenstate behavior preserving.