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Transducer degrees: atoms, infima and suprema

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Abstract

Although finite state transducers are very natural and simple devices, surprisingly little is known about the transducibility relation they induce on streams (infinite words). We collect some intriguing problems that have been unsolved since several years. The transducibility relation arising from finite state transduction induces a partial order of stream degrees, which we call Transducer degrees, analogous to the well-known Turing degrees or degrees of unsolvability. We show that there are pairs of degrees without supremum and without infimum. The former result is somewhat surprising since every finite set of degrees has a supremum if we strengthen the machine model to Turing machines, but also if we weaken it to Mealy machines.

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Notes

  1. 1.

    Thereby FSTs generalise the class of Mealy machines. The latter are restricted to output precisely one letter in each step. The transducer shown in Fig. 1 is not a Mealy machine, and there exists no Mealy machine implementing this transformation.

  2. 2.

    If such a transducer exists, then it must be an erasing transducer. That is, at least one of the output words along the edges must be empty. A non-erasing transducer cannot do the transformation since it preserves \(\alpha \)-substitutivity [37].

  3. 3.

    The papers [3, 4] consider regularity preserving operations on regular languages. There spiralling functions are employed in the context of regularity preserving filter operations. See also [10] for functions that preserve but do not reflect regularity.

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Acknowledgements

We thank the editors for inviting us to contribute to this special issue, and the referees for their scrutiny of our paper and many useful suggestions. We thank Hans Zantema for fruitful discussions about the topic.

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Correspondence to Rena Bakhshi.

Additional information

Dedicated to Rob van Glabbeek in celebration of his 60th anniversary.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The first and third author have fond memories to their stay in Australia at NICTA, and the contacts there with Rob in 2013. The second author has equally fond memories to the years around 1985–1990 at CWI Amsterdam, when Rob was completing his groundbreaking Ph.D. thesis about comparative concurrency semantics.

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Endrullis, J., Klop, J.W. & Bakhshi, R. Transducer degrees: atoms, infima and suprema. Acta Informatica (2019). https://doi.org/10.1007/s00236-019-00353-7

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