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DOI: 10.4230/LIPIcs.STACS.2019.22

Closure Properties of Synchronized Relations

Authors: María Emilia Descotte, Diego Figueira, and Santiago Figueira

Published in: LIPIcs, Volume 126, 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)

Abstract

A standard approach to define k-ary word relations over a finite alphabet A is through k-tape finite state automata that recognize regular languages L over {1, ..., k} x A, where (i,a) is interpreted as reading letter a from tape i. Accordingly, a word w in L denotes the tuple (u_1, ..., u_k) in (A^*)^k in which u_i is the projection of w onto i-labelled letters. While this formalism defines the well-studied class of rational relations, enforcing restrictions on the reading regime from the tapes, which we call synchronization, yields various sub-classes of relations. Such synchronization restrictions are imposed through regular properties on the projection of the language L onto {1, ..., k}. In this way, for each regular language C subseteq {1, ..., k}^*, one obtains a class Rel({C}) of relations. Synchronous, Recognizable, and Length-preserving rational relations are all examples of classes that can be defined in this way. We study basic properties of these classes of relations, in terms of closure under intersection, complement, concatenation, Kleene star and projection. We characterize the classes with each closure property. For the binary case (k=2) this yields effective procedures.

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María Emilia Descotte, Diego Figueira, and Santiago Figueira. Closure Properties of Synchronized Relations. In 36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 126, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{descotte_et_al:LIPIcs.STACS.2019.22,
  author =	{Descotte, Mar{\'\i}a Emilia and Figueira, Diego and Figueira, Santiago},
  title =	{{Closure Properties of Synchronized Relations}},
  booktitle =	{36th International Symposium on Theoretical Aspects of Computer Science (STACS 2019)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-100-9},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{126},
  editor =	{Niedermeier, Rolf and Paul, Christophe},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2019.22},
  URN =		{urn:nbn:de:0030-drops-102614},
  doi =		{10.4230/LIPIcs.STACS.2019.22},
  annote =	{Keywords: synchronized word relations, rational, closure, characterization, intersection, complement, Kleene star, concatenation}
}

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Closure Properties of Synchronized Relations

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