Abstract
An operation M + constructing from a given structure M a tree-like structure which domain consists of the sequences of elements of M is considered. A notion of automata running on such tree-like structures is defined. This notion is parametrised by a set of basic formulas. It is shown that if basic formulas satisfy some conditions then the class of languages recognised by automata is closed under disjunction, complementation and projection. For one choice of basic formulas we obtain a characterisation of MSOL over tree-like structures. This characterisation allows us to show that MSOL theory of tree-like structures is effectively reducible to that of the original structures. For a different choice of basic formulas we obtain a characterisation of MSOL on trees of arbitrary degree and the proof that it is equivalent to the first order logic extended with the unary least fixpoint operator.
This work was partially supported by Polish KBN grant No. 2 P301 009 06.
On leave from: Institute of Informatics, Warsaw University, Banacha 2, 02-097 Warsaw, POLAND.
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References
J. Büchi. State strategies for games in Fσδ∩Gσδ. Journal of Symbolic Logic, 48:1171–1198, 1983.
B. Courcelle. The monadic second-order logic on graphs IX: machines and behaviours. Theoretical Computer Science, 149, 1995.
B. Courcelle and I. Walukiewicz. Monadic second-order logic, graphs and unfoldings of transition systems. University of Aarhus BRICS report RS-95-44. Presented at CSL'95.
E. Emerson and C. Jutla. Tree automata, mu-calculus and determinacy. In Proc. FOCS 91, 1991.
Y. Gurevich and L. Harrington. Trees, automata and games. In 14th Symp. on Theory of Computations, ACM, pages 60–65, 1982.
D. Janin and I. Walukiewicz. Automata for the μ-calculus and related results. In MFCS'95, volume 969 of LNCS, pages 552–562, 1995.
A. W. Mostowski. Regular expressions for infinite trees and a standard form of automata. In A. Skowron, editor, Fifth Symposium on Computation Theory, volume 208 of LNCS, pages 157–168, 1984.
A. W. Mostowski. Games with forbidden positions. Technical Report 78, University of Gdansk, 1991.
D. Niwiński. Fixed points vs. infinite generation. In Proc. 3rd. IEEE LICS, pages 402–409, 1988.
M. Rabin. Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc., 141:1–35, 1969.
A. Semenov. Decidability of monadic theories. In MFCS'84, volume 176 of LNCS, pages 162–175. Springer-Verlag, 1984.
S. Shelah. The monadic second order theory of order. Annals of Mathematics, 102:379–419, 1975.
W. Thomas. Automata on infinite objects. In J. Leeuven, editor, Handbook of Theoretical Computer Science Vol. B, pages 995–1072. Elesvier, 1990.
W. Thomas. On logics, tilings, and automata. In ICALP '91, volume 510 of LNCS, pages 441–454, 1991.
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Walukiewicz, I. (1996). Monadic second order logic on tree-like structures. In: Puech, C., Reischuk, R. (eds) STACS 96. STACS 1996. Lecture Notes in Computer Science, vol 1046. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60922-9_33
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DOI: https://doi.org/10.1007/3-540-60922-9_33
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