Abstract
We search for the largest syntactic semigroup of a star-free language having n left quotients; equivalently, we look for the largest transition semigroup of an aperiodic finite automaton with n states.
We first introduce unitary semigroups generated by transformations that change only one state. In particular, we study complete unitary semigroups which have a special structure, and we show that each maximal unitary semigroup is complete. For \(n \geqslant 4\) there exists a complete unitary semigroup that is larger than any aperiodic semigroup known to date.
We then present even larger aperiodic semigroups, generated by transformations that map a non-empty subset of states to a single state; we call such transformations and semigroups semiconstant. In particular, we examine semiconstant tree semigroups which have a structure based on full binary trees. The semiconstant tree semigroups are at present the best candidates for largest aperiodic semigroups.
This work was supported by the Natural Sciences and Engineering Research Council of Canada grant No. OGP000087 and by Polish NCN grant DEC-2013/09/N/ST6/01194.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Brzozowski, J.: Quotient complexity of regular languages. J. Autom. Lang. Comb. 15(1/2), 71–89 (2010)
Brzozowski, J.: In search of the most complex regular languages. Internat. J. Found. Comput. Sci. 24(6), 691–708 (2013)
Brzozowski, J., Li, B.: Syntactic complexity of \(\mathcal R\)- and \(\mathcal J\)-trivial languages. In: Jurgensen, H., Reis, R. (eds.) DCFS 2013. LNCS, vol. 8031, pp. 160–171. Springer, Heidelberg (2013)
Brzozowski, J., Li, B., Liu, D.: Syntactic complexities of six classes of star-free languages. J. Autom. Lang. Comb. 17(2-4), 83–105 (2012)
Brzozowski, J., Liu, B.: Quotient complexity of star-free languages. Internat. J. Found. Comput. Sci. 23(6), 1261–1276 (2012)
Brzozowski, J., Szykuła, M.: Large aperiodic semigroups (2014), http://arxiv.org/abs/1401.0157
Diestel, R.: Graph Theory, Graduate Texts in Mathematics, 4th edn., vol. 173. Springer, Heidelberg (2010), http://diestel-graph-theory.com
Gomes, G., Howie, J.: On the ranks of certain semigroups of order-preserving transformations. Semigroup Forum 45, 272–282 (1992)
Howie, J.M.: Products of idempotents in certain semigroups of transformations. Proc. Edinburgh Math. Soc. 17(2), 223–236 (1971)
Kisielewicz, A., Szykuła, M.: Generating small automata and the Černý conjecture. In: Konstantinidis, S. (ed.) CIAA 2013. LNCS, vol. 7982, pp. 340–348. Springer, Heidelberg (2013)
McNaughton, R., Papert, S.A.: Counter-Free Automata, MIT Research Monographs, vol. 65. MIT Press (1971)
Pin, J.E.: Syntactic semigroups. In: Handbook of Formal Languages. Word, Language, Grammar, vol. 1, pp. 679–746. Springer, New York (1997)
Schützenberger, M.: On finite monoids having only trivial subgroups. Inform. and Control 8, 190–194 (1965)
Yu, S.: State complexity of regular languages. J. Autom. Lang. Comb. 6, 221–234 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Brzozowski, J., Szykuła, M. (2014). Large Aperiodic Semigroups. In: Holzer, M., Kutrib, M. (eds) Implementation and Application of Automata. CIAA 2014. Lecture Notes in Computer Science, vol 8587. Springer, Cham. https://doi.org/10.1007/978-3-319-08846-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-08846-4_9
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08845-7
Online ISBN: 978-3-319-08846-4
eBook Packages: Computer ScienceComputer Science (R0)