Interleaving of path sets

Abstract

Path sets are spaces of one-sided infinite symbol sequences corresponding to the one-sided infinite walks beginning at a fixed initial vertex in a directed labeled graph G. Path sets are a generalization of one-sided sofic shifts. This paper studies decimation operations ψj,n(•) which extract symbol sequences in infinite arithmetic progressions (mod n) starting with the symbol at position j. It also studies a family of n-ary interleaving operations ⊛n, one for each n ≥ 1, which act on an ordered set (X0, X1, ..., Xn−1) of one-sided symbol sequences X0⊛X1⊛ • • • ⊛Xn−1 on an alphabet A, by interleaving the symbols of each Xi in arithmetic progressions (mod n), It studies a set of closure operations relating interleaving and decimation. This paper gives basic algorithmic results on presentations of path sets and existence of a minimal right-resolving presentation. It gives an algorithm for computing presentations of decimations of path sets from presentations of path sets, showing the minimal right-resolving presentation of ψj,n(X) has at most one more vertex than a minimal right-resolving presentation of X. It shows that a path set has only finitely many distinct decimations. It shows the class of path sets on a fixed alphabet is closed under interleavings and gives an algorithm to compute presentations of interleavings of path sets. It studies interleaving factorizations and classifies path sets that have infinite interleaving factorizations and gives an algorithm to recognize them. It shows the finiteness of a process of iterated interleaving factorizations, which "freezes" factors that have infinite interleavings.