Closure Properties of Synchronized Relations

Authors María Emilia Descotte, Diego Figueira, Santiago Figueira

Thumbnail PDF


  • Filesize: 0.54 MB
  • 17 pages

Document Identifiers

Author Details

María Emilia Descotte
  • LaBRI, Université de Bordeaux, France
Diego Figueira
  • CNRS & LaBRI, Université de Bordeaux, France
Santiago Figueira
  • CONICET & Universidad de Buenos Aires, Argentina


Work supported by ANR project DELTA, grant ANR-16-CE40-0007, grant PICT-2016-0215, and LIA INFINIS.

Cite AsGet BibTex

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)


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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • synchronized word relations
  • rational
  • closure
  • characterization
  • intersection
  • complement
  • Kleene star
  • concatenation


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Parosh Aziz Abdulla, Bengt Jonnson, Marcus Nilsson, and Mayank Saksena. A survey of regular model checking. In International Conference on Concurrency Theory (CONCUR), pages 35-48, 2003. Google Scholar
  2. Renzo Angles and Claudio Gutiérrez. Survey of graph database models. ACM Comput. Surv., 40(1), 2008. URL:
  3. Kemafor Anyanwu and Amit Sheth. ρ-Queries: enabling querying for semantic associations on the semantic web. In 12th International World Wide Web Conference (WWW'03), pages 690-699, 2003. Google Scholar
  4. Pablo Barceló, Diego Figueira, and Leonid Libkin. Graph Logics with Rational Relations. Logical Methods in Computer Science (LMCS), 9(3:01), 2013. URL:
  5. Pablo Barceló, Leonid Libkin, Anthony Widjaja Lin, and Peter T. Wood. Expressive Languages for Path Queries over Graph-Structured Data. ACM Trans. Database Syst., 37(4):31, 2012. URL:
  6. Pablo Barceló and Pablo Muñoz. Graph Logics with Rational Relations: The Role of Word Combinatorics. ACM Trans. Comput. Log., 18(2):10:1-10:41, 2017. URL:
  7. Jean Berstel. Transductions and Context-Free Languages. B. G. Teubner, 1979. Google Scholar
  8. Achim Blumensath and Erich Grädel. Automatic Structures. In Annual IEEE Symposium on Logic in Computer Science (LICS), pages 51-62. IEEE Computer Society Press, 2000. URL:
  9. Mikołaj Bojańczyk. Transducers with origin information. In International Colloquium on Automata, Languages and Programming (ICALP), volume 8573 of Lecture Notes in Computer Science, pages 26-37. Springer, 2014. URL:
  10. Ahmed Bouajjani, Bengt Jonsson, Marcus Nilsson, and Tayssir Touili. Regular Model Checking. In International Conference on Computer Aided Verification (CAV), pages 403-418, London, UK, 2000. Springer-Verlag. Google Scholar
  11. Julius Richard Büchi. Weak Second-order Arithmetic and Finite Automata. Mathematical Logic Quarterly, 6(1-6):66-92, 1960. Google Scholar
  12. Olivier Carton. The growth ratio of synchronous rational relations is unique. Theor. Comput. Sci., 376(1-2):52-59, 2007. URL:
  13. Christian Choffrut. Relations over Words and Logic: A Chronology. Bulletin of the EATCS, 89:159-163, 2006. Google Scholar
  14. María Emilia Descotte, Diego Figueira, and Gabriele Puppis. Resynchronizing Classes of Word Relations. In International Colloquium on Automata, Languages and Programming (ICALP), Leibniz International Proceedings in Informatics (LIPIcs), pages 381:1-381:13. Leibniz-Zentrum für Informatik, 2018. URL:
  15. Calvin C. Elgot and Jorge E. Mezei. On relations defined by generalized finite automata. IBM Journal of Research and Development, 9(1):47-68, 1965. URL:
  16. Ronald Fagin, Benny Kimelfeld, Frederick Reiss, and Stijn Vansummeren. Document Spanners: A Formal Approach to Information Extraction. Journal of the ACM, 62(2):12:1-12:51, 2015. URL:
  17. Diego Figueira and Leonid Libkin. Path Logics for Querying Graphs: Combining Expressiveness and Efficiency. In Annual IEEE Symposium on Logic in Computer Science (LICS), pages 329-340. IEEE Computer Society Press, 2015. URL:
  18. Diego Figueira and Leonid Libkin. Synchronizing Relations on Words. Theory of Computing Systems, 57(2):287-318, 2015. URL:
  19. Emmanuel Filiot, Ismaël Jecker, Christof Löding, and Sarah Winter. On Equivalence and Uniformisation Problems for Finite Transducers. In International Colloquium on Automata, Languages and Programming (ICALP), volume 55 of Leibniz International Proceedings in Informatics (LIPIcs), pages 125:1-125:14. Leibniz-Zentrum für Informatik, 2016. URL:
  20. Christiane Frougny and Jacques Sakarovitch. Synchronized Rational Relations of Finite and Infinite Words. Theoretical Computer Science, 108(1):45-82, 1993. URL:
  21. Seymour Ginsburg and Edwin Spanier. Semigroups, Presburger formulas, and languages. Pacific Journal of Mathematics, 16(2):285-296, 1966. Google Scholar
  22. Bengt Jonsson and Marcus Nilsson. Transitive Closures of Regular Relations for Verifying Infinite-State Systems. In International Conference on Tools and Algorithms for the Construction and Analysis of Systems (TACAS), pages 220-234. Springer, 2000. Google Scholar
  23. R. Milo, S. Shen-Orr, S. Itzkovitz, N. Kashtan, D. Chklovskii, and U. Alon. Network motifs: simple building blocks of complex networks. Science, 298(5594):824-827, 2002. Google Scholar
  24. Maurice Nivat. Transduction des langages de Chomsky. Annales de l'Institut Fourier, 18:339-455, 1968. Google Scholar
  25. Rohit Parikh. On Context-Free Languages. Journal of the ACM, 13(4):570-581, 1966. URL:
  26. Anthony Widjaja To and Leonid Libkin. Algorithmic Metatheorems for Decidable LTL Model Checking over Infinite Systems. In International Conference on Foundations of Software Science and Computational Structures (FOSSACS), pages 221-236, 2010. URL:
  27. Sarah Winter. Uniformization Problems for Synchronizations of Automatic Relations on Words. In International Colloquium on Automata, Languages and Programming (ICALP), Leibniz International Proceedings in Informatics (LIPIcs). Leibniz-Zentrum für Informatik, 2018. URL: