First-order and counting theories of ω-automatic structures
Dietrich Kuske and Markus Lohrey
Source: J. Symbolic Logic Volume 73, Issue 1
(2008), 129-150.
Abstract
The logic ℒ(𝒬u) extends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the set C”). This logic is investigated for structures with an injectively ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there are κ many elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of ℒ(𝒬u) that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.
First Page:
Show
Hide
Full-text: Access denied (no subscription
detected)
We're sorry, but we are unable to provide
you with the full text of this article because we are not able to identify
you as a subscriber.
If you have a personal subscription to
this journal, then please login. If you are already logged in, then you
may need to update your profile to register your subscription. Read more about accessing full-text
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.jsl/1208358745
Mathematical Reviews number (MathSciNet): MR2387935
Digital Object Identifier: doi:10.2178/jsl/1208358745
References
V. Bárány, Automatic presentations of infinite structures, Ph.D. thesis, RWTH Aachen, Fakultät für Mathematik, Informatik und Naturwissenschaften, 2007.
M. Benedikt, L. Libkin, T. Schwentick, and L. Segoufin, Definable relations and first-order query languages over strings, Journal of the ACM, vol. 50 (2003), no. 5, pp. 694--751.
Mathematical Reviews (MathSciNet): MR2146995
Digital Object Identifier: doi:10.1145/876638.876642
A. Bès, Undecidable extensions of Büchi arithmetic and Cobham-Semenov theorem, Journal of Symbolic Logic, vol. 62 (1997), no. 4, pp. 1280--1296.
Mathematical Reviews (MathSciNet): MR1617949
Digital Object Identifier: doi:10.2307/2275643
Project Euclid: euclid.jsl/1183745382
A. Blumensath, Automatic structures, Technical report, RWTH Aachen, 1999.
A. Blumensath and E. Grädel, Automatic structures, Proceedings of the IEEE Symposium on Logic in Computer Science (LICS), EEE Computer Society, 2000, pp. 51--62.
Mathematical Reviews (MathSciNet): MR1946085
--------, Finite presentations of infinite structures: Automata and interpretations, Theory of Computing Systems, vol. 37 (2004), no. 6, pp. 641--674.
Mathematical Reviews (MathSciNet): MR2093606
Digital Object Identifier: doi:10.1007/s00224-004-1133-y
Zentralblatt MATH: 1061.03038
J. R. Büchi, Weak second-order arithmetic and finite automata, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 6 (1960), pp. 66--92.
Mathematical Reviews (MathSciNet): MR125010
Y. Choueka, Theories of automata on $\omega$-tapes: A simplified approach, Journal of Computer and System Sciences, vol. 8 (1974), pp. 117--141.
Mathematical Reviews (MathSciNet): MR342378
Digital Object Identifier: doi:10.1016/S0022-0000(74)80051-6
Zentralblatt MATH: 0292.02033
K. J. Compton and C. W. Henson, A uniform method for proving lower bounds on the computational complexity of logical theories, Annals of Pure and Applied Logic, vol. 48 (1990), pp. 1--79.
Mathematical Reviews (MathSciNet): MR1061862
Digital Object Identifier: doi:10.1016/0168-0072(90)90080-L
Zentralblatt MATH: 0705.03017
D. B. A. Epstein, J. W. Cannon, D. F. Holt, S. V. F. Levy, M. S. Paterson, and W. P. Thurston, Word Processing in Groups, Jones and Bartlett, Boston, 1992.
Mathematical Reviews (MathSciNet): MR1161694
Zentralblatt MATH: 0764.20017
H. Gaifman, On local and nonlocal properties, Logic Colloquium '81 (J. Stern, editor), North-Holland, 1982, pp. 105--135.
Mathematical Reviews (MathSciNet): MR757024
K. Härtig, Über einen Quantifikatur mit zwei Wirkungsbereichen, Colloquium on the Foundations of Mathematics, Mathematical Machines and their Applications (Tihany, 1962), Akademiai Kiadó, Budapest, 1965, pp. 31--36.
Mathematical Reviews (MathSciNet): MR189980
W. Hodges, Model Theory, Cambridge University Press, 1993.
Mathematical Reviews (MathSciNet): MR1221741
Zentralblatt MATH: 0789.03031
B. R. Hodgson, On direct products of automaton decidable theories, Theoretical Computer Science, vol. 19 (1982), pp. 331--335.
Mathematical Reviews (MathSciNet): MR671875
Digital Object Identifier: doi:10.1016/0304-3975(82)90042-1
Zentralblatt MATH: 0493.03002
H. Ishihara, B. Khoussainov, and S. Rubin, Some results on automatic structures, Proceedings of the IEEE Symposium on Logic in Computer Science (LICS), EEE Computer Society, 2002, pp. 235--244.
H. J. Keisler and W. B. Lotfallah, A local normal form theorem for infinitary logic with unary quantifiers, Mathematical Logic Quarterly, vol. 51 (2005), no. 2, pp. 137--144.
Mathematical Reviews (MathSciNet): MR2121206
Digital Object Identifier: doi:10.1002/malq.200410013
Zentralblatt MATH: 1060.03060
B. Khoussainov, G. Hjorth, A. Montalban, and A. Nies, From automatic structures to Borel structures, 2007, manuscript.
B. Khoussainov and A. Nerode, Automatic presentations of structures, LCC: International Workshop on Logic and Computational Complexity (Daniel Leivant, editor), Lecture Notes in Computer Science, vol. 960, Springer, 1994, pp. 367--392.
Mathematical Reviews (MathSciNet): MR1449670
B. Khoussainov and S. Rubin, Graphs with automatic presentations over a unary alphabet, Journal of Automata, Languages and Combinatorics, vol. 6 (2001), no. 4, pp. 467--480.
Mathematical Reviews (MathSciNet): MR1897055
B. Khoussainov, S. Rubin, and F. Stephan, Automatic partial orders, Proceedings of the IEEE Symposium on Logic in Computer Science (LICS), EEE Computer Society, 2003, pp. 168--177.
--------, Definability and regularity in automatic structures, Proceedings of the Symposium on Theoretical Aspects of Computer Science (STACS) (Volker Diekert and Michel Habib, editors), Lecture Notes in Computer Science, vol. 2996, Springer, 2004, pp. 440--451.
Mathematical Reviews (MathSciNet): MR2093979
D. Kuske, Is Cantor's theorem automatic, Proceedings of the International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR) (Moshe Y. Vardi and Andrei Voronkov, editors), Lecture Notes in Artificial Intelligence, vol. 2850, Springer, 2003, pp. 332--345.
D. Kuske and M. Lohrey, First-order and counting theories of $\omega$-automatic structures, Proceedings of the 9th International Conference on Foundations of Software Science and Computation Structures (FOSSACS) (Luca Aceto and Anna Ingólfsdóttir, editors), Lecture Notes in Computer Science, vol. 3921, Springer, 2006, pp. 322--336.
Mathematical Reviews (MathSciNet): MR2246804
Digital Object Identifier: doi:10.1007/11690634_22
Zentralblatt MATH: 1138.03032
R. Lindner and L. Staiger, Algebraische Codierungstheorie, Akademie-Verlag Berlin, 1977.
Mathematical Reviews (MathSciNet): MR469495
Zentralblatt MATH: 0363.94016
M. Lohrey, Automatic structures of bounded degree, Proceedings of the 10th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR) (Moshe Y. Vardi and Andrei Voronkov, editors), Lecture Notes in Artificial Intelligence, vol. 2850, Springer, 2003, pp. 344--358.
C. H. Papadimitriou, Computational Complexity, Addison Wesley, 1994.
Mathematical Reviews (MathSciNet): MR1251285
Zentralblatt MATH: 0833.68049
D. Perrin and J.-E. Pin, Infinite Words, Pure and Applied Mathematics, vol. 141, Elsevier, 2004.
L.J. Stockmeyer and A. R. Meyer, Word problems requiring exponential time (preliminary report), Proceedings of the ACM Symposium on Theory of Computing (STOC), 1973, pp. 1--9.
Mathematical Reviews (MathSciNet): MR418518
Zentralblatt MATH: 0359.68050
Digital Object Identifier: doi:10.1145/800125.804029
Y. Tao, Infinity problems and countability problems for $\omega$-automata, Information Processing Letters, vol. 100 (2006), no. 4, pp. 151--153.
Mathematical Reviews (MathSciNet): MR2256770
Digital Object Identifier: doi:10.1016/j.ipl.2006.06.011
Zentralblatt MATH: 05664708
W. Thomas, Automata on infinite objects, Handbook of Theoretical Computer Science (J. van Leeuwen, editor), Elsevier Science Publishers B. V., 1990, pp. 133--191.
Mathematical Reviews (MathSciNet): MR1127189
Zentralblatt MATH: 0900.68316
