Abstract
The problem considered in this paper is whether an inequality of ω-terms is valid in a given level of a concatenation hierarchy of star-free languages. The main result shows that this problem is decidable for all (integer and half) levels of the Straubing–Thérien hierarchy.
Funding source: Centro de Matemática Universidade do Porto
Award Identifier / Grant number: UID/MAT/00144/2013
Funding source: Fundação para a Ciência e a Tecnologia
Award Identifier / Grant number: UID/MAT/00144/2013
Funding source: Ministério da Ciência, Tecnologia e Ensino Superior
Award Identifier / Grant number: UID/MAT/00144/2013
Funding source: European Regional Development Fund
Award Identifier / Grant number: UID/MAT/00144/2013
Funding statement: The first author was partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MCTES) and European structural funds (FEDER), under the partnership agreement PT2020. The second and third authors were partially supported by the grant 15-02862S of the Grant Agency of the Czech Republic.
Acknowledgements
The authors wish to thank the anonymous referee for comments that helped improving the presentation of the paper.
References
[1]
J. Almeida,
Implicit operations on finite
[2] J. Almeida, Finite Semigroups and Universal Algebra, Ser. Algebra 3, World Scientific, River Edge, 1994. 10.1142/2481Search in Google Scholar
[3] J. Almeida, Hyperdecidable pseudovarieties and the calculation of semidirect products, Internat. J. Algebra Comput. 9 (1999), no. 3–4, 241–261, 10.1142/S0218196799000163Search in Google Scholar
[4] J. Almeida, Profinite semigroups and applications, Structural Theory of Automata, Semigroups and Universal Algebra (Montreal 2003), Kluwer Academic Publishers, Dordrecht (2005), 1–45. 10.1007/1-4020-3817-8_1Search in Google Scholar
[5] J. Almeida, J. Bartoňová, O. Klíma and M. Kunc, On decidability of intermediate levels of concatenation hierarchies, Developments in Language Theory, Lecture Notes in Comput. Sci. 9168, Springer, Cham (2015) 58–70. 10.1007/978-3-319-21500-6_4Search in Google Scholar
[6] J. Almeida, A. Cano, O. Klíma and J.-E. Pin, On fixed points of the lower set operator, Internat. J. Algebra Comput. 25 (2015), no. 1–2, 259–292. 10.1142/S021819671540010XSearch in Google Scholar
[7] J. Almeida and A. Costa, Infinite-vertex free profinite semigroupoids and symbolic dynamics, J. Pure Appl. Algebra 213 (2009), no. 5, 605–631. 10.1016/j.jpaa.2008.08.009Search in Google Scholar
[8] J. Almeida, J. C. Costa and M. Zeitoun, Iterated periodicity over finite aperiodic semigroups, European J. Combin. 37 (2014), 115–149. 10.1016/j.ejc.2013.07.011Search in Google Scholar
[9] J. Almeida, J. C. Costa and M. Zeitoun, McCammond’s normal forms for free aperiodic semigroups revisited, LMS J. Comput. Math. 18 (2015), no. 1, 130–147. 10.1112/S1461157014000448Search in Google Scholar
[10] J. Almeida, J. C. Costa and M. Zeitoun, Factoriality and the Pin–Reutenauer procedure, Discrete Math. Theor. Comput. Sci. 18 (2016), no. 3, Paper No. 1. 10.46298/dmtcs.650Search in Google Scholar
[11] J. Almeida and B. Steinberg, On the decidability of iterated semidirect products with applications to complexity, Proc. Lond. Math. Soc. (3) 80 (2000), no. 1, 50–74. 10.1112/S0024611500012144Search in Google Scholar
[12] S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Grad. Texts in Math. 78, Springer, New York, 1981. 10.1007/978-1-4613-8130-3Search in Google Scholar
[13] R. S. Cohen and J. A. Brzozowski, Dot-depth of star-free events, J. Comput. System Sci. 5 (1971), 1–16. 10.1016/S0022-0000(71)80003-XSearch in Google Scholar
[14] S. Eilenberg, Automata, Languages, and Machines. Vol. B, Academic Press, New York, 1976. Search in Google Scholar
[15] H. J. Keisler, Fundamentals of model theory, Handbook of Mathematical Logic, Stud. Logic Found. Math. 90, North Holland, Amsterdam (1977), 47–104. 10.1016/S0049-237X(08)71098-XSearch in Google Scholar
[16] J. P. McCammond, Normal forms for free aperiodic semigroups, Internat. J. Algebra Comput. 11 (2001), no. 5, 581–625. 10.1142/S0218196701000693Search in Google Scholar
[17] J. D. McKnight, Jr. and A. J. Storey, Equidivisible semigroups, J. Algebra 12 (1969), 24–48. 10.1016/0021-8693(69)90015-5Search in Google Scholar
[18] V. Molchanov, Nonstandard characterization of pseudovarieties, Algebra Universalis 33 (1995), no. 4, 533–547. 10.1007/BF01225473Search in Google Scholar
[19] Z.-E. Pèn, Eilenberg’s theorem for positive varieties of languages, Izv. Vyssh. Uchebn. Zaved. Mat. (1995), no. 1, 80–90. Search in Google Scholar
[20] J.-E. Pin, Syntactic semigroups, Handbook of Formal Languages, Vol. 1, Springer, Berlin (1997), 679–746. 10.1007/978-3-642-59136-5_10Search in Google Scholar
[21] J.-E. Pin and P. Weil, A Reiterman theorem for pseudovarieties of finite first-order structures, Algebra Universalis 35 (1996), no. 4, 577–595. 10.1007/BF01243597Search in Google Scholar
[22] J.-E. Pin and P. Weil, Profinite semigroups, Mal’cev products, and identities, J. Algebra 182 (1996), no. 3, 604–626. 10.1006/jabr.1996.0192Search in Google Scholar
[23] J.-E. Pin and P. Weil, Polynomial closure and unambiguous product, Theory Comput. Syst. 30 (1997), no. 4, 383–422. 10.1007/BF02679467Search in Google Scholar
[24] T. Place, and M. Zeitoun, Going higher in the first-order quantifier alternation hierarchy on words, Automata, Languages, and Programming. Part II (Copenhagen 2014), Lecture Notes in Comput. Sci. 8573, Springer, Heidelberg (2014), 342–353. 10.1007/978-3-662-43951-7_29Search in Google Scholar
[25] T. Place and M. Zeitoun, Separating regular languages with first-order logic, Log. Methods Comput. Sci. 12 (2016), no. 1, Paper No. 5. 10.1145/2603088.2603098Search in Google Scholar
[26] J. Reiterman, The Birkhoff theorem for finite algebras, Algebra Universalis 14 (1982), no. 1, 1–10. 10.1007/BF02483902Search in Google Scholar
[27] J. Rhodes and B. Steinberg, The q-Theory of Finite Semigroups, Springer Monogr. Math., Springer, New York, 2009. 10.1007/b104443Search in Google Scholar
[28] H. Straubing, A generalization of the Schützenberger product of finite monoids, Theoret. Comput. Sci. 13 (1981), no. 2, 137–150. 10.1016/0304-3975(81)90036-0Search in Google Scholar
[29]
H. Straubing,
Finite semigroup varieties of the form
[30] D. Thérien, Classification of finite monoids: The language approach, Theoret. Comput. Sci. 14 (1981), no. 2, 195–208. 10.1016/0304-3975(81)90057-8Search in Google Scholar
[31] W. Thomas, Classifying regular events in symbolic logic, J. Comput. System Sci. 25 (1982), no. 3, 360–376. 10.1016/0022-0000(82)90016-2Search in Google Scholar
[32] S. J. van Gool and B. Steinberg, Pro-aperiodic monoids via saturated models, 34th Symposium on Theoretical Aspects of Computer Science (STACS 2017), LIPIcs. Leibniz Int. Proc. Inform. 66, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern (2017), 39:1–39:14. 10.1007/s11856-019-1947-6Search in Google Scholar
[33] S. Willard, General Topology, Addison-Wesley, Reading, 1970. Search in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
- Positive solutions for nonlinear nonhomogeneous parametric Robin problems
- Period relations for cusp forms of GSp4
- A generalization of a graph theory Mertens’ theorem: Galois covering case
- Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
- Vojta’s conjecture on rational surfaces and the abc conjecture
- The least unramified prime which does not split completely
- The ω-inequality problem for concatenation hierarchies of star-free languages
- Higher weight on GL(3). I: The Eisenstein series
- Fourier transforms of powers of well-behaved 2D real analytic functions
- Parabolic conformally symplectic structures I; definition and distinguished connections
- The quasi-arithmetic means and Cartan barycenters of compactly supported measures
- Estimates of lattice points in the discriminant aspect over abelian extension fields
- On Hecke eigenvalues of Siegel modular forms in the Maass space
- Harmonicity and minimality of complex and quaternionic radial foliations
Articles in the same Issue
- Frontmatter
- A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
- Positive solutions for nonlinear nonhomogeneous parametric Robin problems
- Period relations for cusp forms of GSp4
- A generalization of a graph theory Mertens’ theorem: Galois covering case
- Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
- Vojta’s conjecture on rational surfaces and the abc conjecture
- The least unramified prime which does not split completely
- The ω-inequality problem for concatenation hierarchies of star-free languages
- Higher weight on GL(3). I: The Eisenstein series
- Fourier transforms of powers of well-behaved 2D real analytic functions
- Parabolic conformally symplectic structures I; definition and distinguished connections
- The quasi-arithmetic means and Cartan barycenters of compactly supported measures
- Estimates of lattice points in the discriminant aspect over abelian extension fields
- On Hecke eigenvalues of Siegel modular forms in the Maass space
- Harmonicity and minimality of complex and quaternionic radial foliations