Automatic Sequences and Zip-Specifications

Abstract

We consider infinite sequences of symbols, also known as streams, and the decidability question for equality of streams defined in a restricted format. This restricted format consists of prefixing a symbol at the head of a stream, of the stream function `zip', and recursion variables. Here `zip' interleaves the elements of two streams in alternating order, starting with the first stream; e.g., for the streams defined by {zeros = 0 : zeros, ones = 1 : ones, alt = 0 : 1 : alt} we have zip(zeros, ones) = alt. The celebrated Thue-Morse sequence is obtained by the succinct `zip-specification' {M = 0:X, X = 1:zip(X,Y), Y = 0:zip(Y,X)}.

Our analysis of such systems employs both term rewriting and coalgebraic techniques. We establish decidability for these zip-specifications, employing bisimilarity of observation graphs based on a suitably chosen cobasis. The importance of zip-specifications resides in their intimate connection with automatic sequences. The analysis leading to the decidability proof of the `infinite word problem' for zip-specifications, yields a new and simple characterization of automatic sequences. Thus we obtain for the binary zip that a stream is 2-automatic iff its observation graph using the cobasis (hd,even,odd) is finite. Here odd and even have the usual recursive definition: even(a : s) = a : odd(s), and odd(a : s) = even(s). The generalization to zip-k specifications and their relation to k-automaticity is straightforward. In fact, zip-specifications can be perceived as a term rewriting syntax for automatic sequences. Our study of zip-specifications is placed in an even wider perspective by employing the observation graphs in a dynamic logic setting, leading to an alternative characterization of automatic sequences.

We further obtain a natural extension of the class of automatic sequences, obtained by `zip-mix' specifications that use zips of different arities in one specification (recursion system). The corresponding notion of automaton employs a state-dependent input-alphabet, with a number representation (n)_A = d_m ... d_0 where the base of digit d_i is determined by the automaton A on input d_(i−1)...d_0.

We also show that equivalence is undecidable for a simple extension of the zip-mix format with projections like even and odd. However, it remains open whether zip-mix specifications have a decidable equivalence problem.