The power of synchronizing operations on strings

Abstract

Operations on strings and languages, such as shuffle, iterated shuffle, inverse shuffle and cancellation, have been used to describe sequentialized execution histories of concurrent processes. The power of these operations and their relation to the usual AFL-operations is studied and it is shown that flow expressions [11, 12], event expressions [8–10] and even very restricted variants of them define all the recursively enumerable sets. The family of recursively enumerable languages is equal to the least full trio which is in addition closed under iterated shuffle, and it is also equals the the smallest family of languages containing the finite sets and closed under (a) shuffle, iterated shuffle, and inverse shuffle; (b) shuffle, iterated shuffle, and cancellation; (c) product, iterated shuffle, and cancellation with finite sets; (d) product, iterated shuffle, and inverse shuffle with regular sets; (e) product, iterated shuffle, homomorphisms, and inverse homomorphisms. The family of languages definable by shuffle expressions [6, 12] is incomparable with the family of computation sequence sets [2–5].