Symbolic Dynamics to Coding , Automata and System Theory

Algorithms for sliding block codes | an application of symbolic dynamics to information theory, IEEE Trans. then (w; x)j (?1;0) (w; x)j 0;1) 2 B s. The external behavior of a state realization is the system (Z; W; B) where B = fw : (w; x) 2 B s for some xg. The state realization may be regarded as a presentation of its external behavior, with the elements of X as states. A state realization of a system is closely related to a presentation of a sooc shift; in fact, any labeled graph may be regarded as a state realization. A realization of an external behavior is minimal if (roughly speaking) no other realization of the same external behavior can be obtained by \merging" states 47]. Willems characterized the time-invariant systems with unique minimal realization, using the notion of futures F(xj (?1;0)) that we described in Section 4. We state this in Theorem 8.1 below. The past induced canonical realization of a time-invariant system (Z; W; B) is the state realization p = Likewise there is the notion of future induced canonical realization. Theorem 8.1 (Willems 47], and references therein). A time-invariant system has a unique minimal realization if and only if its future induced and past induced canonical realizations are equivalent in the sense that there is a bijection from states of one realization to states of the other that transforms one realization to the other. The unique minimal realization is the past induced canonical realization. Note how similar this result is to the characterization of irreducible sooc shifts with unique I-minimal presentation (Theorem 4.2 (1 , 2)). The Dictionary gives many concepts in system theory which are closely related to concepts in symbolic dynamics. For instance, an input-state-output system is deened to be a state realization (Z; U Y; X; B) subject to certain natural restrictions (see Deenition VI-1 in the Dictionary). Here, U; Y; and X play the role of input, output and state. Such systems are closely related to nite equivalences in symbolic dynamics and transducers in automata theory (as discussed in Section 7). Also, the notions of observability (Deenitions II-3 and II-4) are very similar to the idea of shift of nite type, and the notions of controllability (Deenitions II-7 and II-8) are very similar to the idea of irreducibility for sooc shifts; see the remarks in the Dictionary. We have given an introduction to …