Vol. 9, No. 3, 2020

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On defining linear orders by automata

Bruno Courcelle, Irène Durand and Michael Raskin

Vol. 9 (2020), No. 3, 253–291
Abstract

Motivated by enumeration problems, we define linear orders ${\le }_{Z}$ on Cartesian products $Z:={X}_{1}×{X}_{2}×\cdots ×{X}_{n}$ and on subsets of ${X}_{1}×{X}_{2}$ where each component set ${X}_{i}$ is $\left[0,p\right]$ or $ℕ$, ordered in the natural way. We require that $\left(Z,{\le }_{Z}\right)$ be isomorphic to $\left(ℕ,\le \right)$ if it is infinite. We want linear orderings of $Z$ such that, in two consecutive tuples $z$ and ${z}^{\prime }$, at most two components differ, and they differ by at most 1.

We are interested in algorithms that determine the next tuple in $Z$ by using local information, where “local” is meant with respect to certain graphs associated with $Z$. We want these algorithms to work as well for finite and infinite components ${X}_{i}$. We will formalise them by deterministic graph-walking automata and compare their enumeration powers according to the finiteness of their sets of states and the kinds of moves they can perform.

Keywords
enumeration algorithm, diagonal enumeration, graph-walking automaton, linear order
Mathematical Subject Classification 2010
Primary: 06A05, 05C38, 68R10, 68P10