Interleaving of path sets

TL;DR

This paper explores the properties of path sets, focusing on decimation and interleaving operations, providing algorithms for their presentation, and analyzing their closure properties and factorizations.

Contribution

It introduces algorithms for computing presentations of decimated and interleaved path sets, and characterizes path sets with infinite interleaving factorizations.

Findings

Path sets are closed under all interleaving operations.

Decimation operations preserve the minimal right-resolving presentation with at most one additional vertex.

Path sets have finitely many distinct decimations.

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. Path sets are a generalization of one-sided sofic shifts. This paper studies decimation operations $\psi_{j, n}(\cdot)$ 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, one for each arity n, which act on an ordered set $(X_0, X_1, ..., X_{n-1})$ of one-sided symbol sequences on a finite alphabet A, to produce a set $X$ of all output sequences obtained by interleaving the symbols of words $x_i$ in each $X_i$ in arithmetic progressions (mod n). It studies a set of closure operations relating interleaving and decimation. It reviews 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 $\psi_{j,n}(X)$ 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 all interleaving operations, and gives algorithms for computing presentations of n-fold interleavings of given sets $X_i$. It studies interleaving factorizations and classifies path sets that have infinite interleaving factorizations, and gives an algorithm to recognize them. It shows a finiteness of a process of iterated interleaving factorizations, which "freezes" factors that have infinite interleavings.