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The last ten years saw the emergence of some results about recognizable subsets of a free monoid with multiplicities in the Min-Plus semiring. An interesting aspect of this theoretical body is that its discovery was motivated throughout by applications such as the finite power property, Eggan's classical star height problem and the measure of the nondeterministic complexity of finite automata. We review here these results, their applications and point out some open problems.

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Let 2 be a finite alphabet, Z* the free monoid generated by ~ and Ixl the length of x E Z*. For any integer k >_ O, fk(x) (tk(X)) is x if Ixl < k+ 1, and it is the prefix (suffix) of x of length k, otherwise. Also let mk+l(X) = {olx = uuw and Ivl = k+l}. For x,y ~ Z* define x ~ k+ lY ifffk(x) = fk(Y), tk(X) = tkO:) and ink+ 1 (x) = mk+ 1 fY)" The relation ~k+l is a congruence of finite index over ~*. An event E _C ~* is (k+l)-testable iff it is a union of congruence classes of ~k+l" E is locally testable (LT) if it is k+l-testable for some k. (This definition differs from that of [6] but is equivalent.) We show that the family of LT events is a proper sub-family of star-free events of dot-depth 1. LT events and k-testable events are characterized in terms of (a) restricted star-free expressions based on finite and cofinite events; (b) finite automata accepting these events; (c) semigroups; and (d) structural decomposition of such automata. Algorithms are given for deciding whether a regular event is (a) LT and (b) k+l-testable. Generalized definite events are also characterized.