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

Motivated by enumeration problems, we define linear orders Z on Cartesian products Z := X1×X2××Xn and on subsets of X1 × X2 where each component set Xi is [0,p] or , ordered in the natural way. We require that (Z,Z) be isomorphic to (,) if it is infinite. We want linear orderings of Z such that, in two consecutive tuples z and z , 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 Xi. 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.

enumeration algorithm, diagonal enumeration, graph-walking automaton, linear order
Mathematical Subject Classification 2010
Primary: 06A05, 05C38, 68R10, 68P10
Received: 27 November 2019
Revised: 2 April 2020
Accepted: 17 April 2020
Published: 15 October 2020
Bruno Courcelle
University of Bordeaux and CNRS
Irène Durand
University of Bordeaux and CNRS
Michael Raskin
Technische Universität München
Garching bei München