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Applications of Transducers in Independent Languages, Word Distances, Codes

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Descriptional Complexity of Formal Systems (DCFS 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10316))

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Abstract

A (nondeterministic) transducer t is an operator mapping an input word to a set of possible output words. A few types of transducers are important in this work: input-altering, input-preserving, and input-decreasing. Two words are t-dependent, if one is the output of t when the other one is used as input. A t-independent language is one containing no two t-dependent words. Examples of independent languages are found in noiseless coding theory, noisy coding theory and DNA computing. We discuss how the above transducer types can provide elegant solutions to some cases of the following broad problems: (i) computing two minimum distance witness words of a given regular language; (ii) computing witness words for the non-satisfaction, or non-maximality, of a given regular language with respect to the independence specified by a given transducer t; (iii) computing, for any given t and language L, a maximal t-independent language containing L; (iv) computing, for any given positive integer n and transducer t, a t-independent language of n words. The descriptional complexity cost of converting between transducer types is discussed, when this conversion is possible. We also explore methods of defining more independences in a way that some of the above problems can still be computed.

Research supported by NSERC.

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Notes

  1. 1.

    UD means Uniquely Decodable/Decipherable.

  2. 2.

    In this order, the alphabet \(\varSigma \) is totally ordered. Then, \(u\prec v\) if and only if, either \(|u|<|v|\), or \(|u|=|v|\) and \(u=x\sigma _1y_1,v=x\sigma _2y_2\) such that \(\sigma _1,\sigma _2\in \varSigma \) and \(\sigma _1\prec \sigma _2\).

  3. 3.

    This is when \(\theta (w)\notin \mathbf {t} (w)\), for all nonempty words w.

  4. 4.

    The edit distance \(\delta _\mathrm {ed} (x,y)\) is the minimum number of single-symbol substitutions/insertions/deletions in x that turn x to y. The Levenshtein distance \(\delta _\mathrm {L} (x,y)\) is similar by considering only insertions/deletions.

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  1. Department of Mathematics and Computing Science, Saint Mary’s University, 923 Robie Street, Halifax, NS, B3H 3C3, Canada

    Stavros Konstantinidis

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  1. Stavros Konstantinidis
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  1. Università degli Studi di Milano, Milan, Italy

    Giovanni Pighizzini

  2. University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada

    Cezar Câmpeanu

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Konstantinidis, S. (2017). Applications of Transducers in Independent Languages, Word Distances, Codes. In: Pighizzini, G., Câmpeanu, C. (eds) Descriptional Complexity of Formal Systems. DCFS 2017. Lecture Notes in Computer Science(), vol 10316. Springer, Cham. https://doi.org/10.1007/978-3-319-60252-3_4

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  • DOI: https://doi.org/10.1007/978-3-319-60252-3_4

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