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Automatic Sequences and Zip-Specifications

By C.A. Grabmayer, J. Endrullis, D. Hendriks, J.W. Klop and L.S. Moss


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)}.\ud \ud 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.\ud \ud 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.\ud \ud 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

Topics: Wijsbegeerte
Year: 2012
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