Abstract

Motivated by enumeration problems, we define linear orders ${\le }_Z$ on Cartesian products $Z:=X_1\times X_2\times \cdots \times X_n$ and on subsets of $X_1\times X_2$ where each component set $X_i$ is $\left[0,p\right]$ or $\mathbb{N}$, ordered in the natural way. We require that $\left(Z,{\le }_Z\right)$ be isomorphic to $\left(\mathbb{N},\le \right)$ 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.

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Mathematical Subject Classification 2010

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