Abstract
We propose a new, logical, approach to the decidability prob- lem for the Straubing and Brzozowski hierarchies based on the preser- vation theorems from model theory, on a theorem of Higman, and on the Rabin tree theorem. In this way, we get purely logical, short proofs for some known facts on decidability, which might be of methodological interest.
Our approach is also applicable to some other similar situations, say to “words” over dense orderings which is relevant to the continuous time and hybrid systems.
Partly supported by the Alexander von Humboldt Foundation, by a grant of the Russian Ministry of Education and by RFBR Grant 00-01-00810.
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.
Similar content being viewed by others
References
J. Addison. The method of alternating chains. In: The theory of models, North Holland, Amsterdam, 1965, 1–16.
C. Choffrut and J. Karumäki. Combinatirics of words. In: Handbook of Formal Languages (G. Rozenberg and A. Salomaa, ed.), v. 1 Springer, 1996, 329–438.
C.C. Chang, H.J. Keisler. Model theory, North Holland, Amsterdam, 1973.
C. Glasser, H. Schmitz. The Boolean structure of dot-depth one. J. Automata, Languages and Combinatorics, to appear.
A.I. Malcev. Algebraic Systems, Springer, Berlin, 1971.
R. McNaughton and S. Papert. Counter free automata. MIT Press, Cambridge, Massachusets, 1971.
D. Perrin and J.E. Pin. First order logic and star-free sets. J. Comp. and Syst. Sci., 32 (1986), 393–406.
J.E. Pin. Varieties of Formal Languages. Plenum, London, 1986.
J.E. Pin. Logic on words. Bulletin of the EATCS, 54 (1994), 145–165.
M.O. Rabin. Decidability of second order theories and automata on infinite trees. Trans. Amer. Math. Soc., 141 (1969), 1–35.
A. Robinson. Introduction to Model Theory and to the Metamathematics of Algebra, North Holland, Amsterdam, 1963.
H. Schmitz. Restricted temporal logic and deterministic languages. J. Automata, Languages and Combinatorics, 5 (2000), 325–342.
V.L. Selivanov. Fine hierarchy of formulas. Algebra i Logika, 30 (1991), 568–583 (Russian, English translation: Algebra and Logic, 30 (1991), 368-379).
V.L. Selivanov. Fine hierarchies and definable index sets. Algebra i logika, 30, No6 (1991), 705–725 (Russian, English translation: Algebra and logic, 30 (1991), 463-475).
V.L. Selivanov. Computing degrees of definable classes of sentences. Contemporary Math., 131, part 3 (1992), 657–666.
V.L. Selivanov. Fine hierarchies and Boolean terms. J. Symbolic Logic, 60 (1995), 289–317.
J.R. Shoenfield. Mathematical Lodic, Addison-Wesley, 1967.
J. Stern. Characterizations of some classes of regular events. Theor. Comp. Science, 35 (1985), 17–42.
H. Schmitz and K. Wagner. The Boolean hierarchy over level 1/2 of the Sraubing-Therien hierarchy, to appear (currently available at http://www.informatik.uni-wuerzburg.de).
W. Thomas. Classifying regular events in symbolic logic. J. Comp. and Syst. Sci.,25 (1982), 360–376.
A.N. Trahtman. Computing the order of local testability. In: Proc. 1st Int. Workshop on Descriptional Complexity of Automata, Grammars and Related Structures, Magdeburg, 1999, 197–206.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Selivanov, V.L. (2001). A Logical Approach to Decidability of Hierarchies of Regular Star—Free Languages. In: Ferreira, A., Reichel, H. (eds) STACS 2001. STACS 2001. Lecture Notes in Computer Science, vol 2010. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44693-1_47
Download citation
DOI: https://doi.org/10.1007/3-540-44693-1_47
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41695-1
Online ISBN: 978-3-540-44693-4
eBook Packages: Springer Book Archive