Synchronization and simplification

TL;DR: The main result of the paper states that a recognizable subset X of a free semigroup A + is strictly locally testable if and only if all the idempotents of the syntactic semigroup S ( X ) of X are constants for X′ = XΦ , where Φ : A + → S (X ) is the syntact morphism.

TL;DR: The authors show that a necessary and sufficient condition for the existence of a statistically self-synchronizing decoder and, therefore, for a code to be statistically synchronizable, is that the code has a synchronizing sequence.

TL;DR: It is shown that three- element codes have some special properties which do not hold even for four-element codes, and each three-element code has a bounded delay in at least one direction.

TL;DR: In this paper, the syntactic algebra of a code is defined as the algebra of the sub-monoid of the code generated by the code's generator, which is a special class of languages: free submonoids of the free monoid.

TL;DR: A particular class of finite-state automata, christened by the authors "counter-free," is shown here to behave like a good actor: it can drape itself so thoroughly in the notational guise and embed itself so deeply in the conceptual character of several quite different approaches to automata theory that on the surface it is hard to believe that all these roles are being assumed by the same class.

TL;DR: This class is shown to attain the upper bound on codebook size previously encountered (but not always attained) for the familiar subclass of “comma-free codes”, which has the property that every message can be uniquely decoded by examining a segment of bounded length, from any starting point.

TL;DR: The solution relates the concept of local testability with that of synchronization in a code and the algebraic notion of conjugacy in a monoid to show that under what conditions a free submonoid of a free monoid is locally testable.