Linearization of Automatic Arrays and Weave Specifications

Grabmayer, Endrullis, Hendriks, Klop, and Moss [C. Grabmayer, J. Endrullis, D. Hendriks, J.W. Klop, and L.S. Moss. Automatic sequences and zipspecifications. In Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science, pages 335–344. IEEE Computer Society, 2012] developed a method for defining automatic sequences in terms of ʼzip specificationsʼ and proved that a sequence is automatic [J.-P. Allouche and J. Shallit. Automatic Sequences. Cambridge University Press, 2003] iff it has a zip specification where all zip terms have the same arity. This paper begins by investigating a similar definitional scheme for the higher-dimensional counterpart of automatic sequences, automatic arrays. In the course of establishing the results required for this machinery, we find an isomorphism–closely related to the z-order curve [G.M. Morton. A computer oriented geodetic data base and a new technique in file sequencing. International Business Machines Company, 1966]–between a final coalgebra for arrays and the standard final coalgebra for sequences. This isomorphism preserves automaticity properties: an array is k,l-automatic iff its corresponding sequence is kl-automatic. The former notion of automaticity (k,l-automatic, note the comma) is defined for arrays as in [J.-P. Allouche and J. Shallit. Automatic Sequences. Cambridge University Press, 2003], and the latter notion is the standard notion of automaticity for sequences. It also provides a convenient way to translate between stream zip specifications and array zip specifications.