## Journal of Symbolic Logic

### On spectra of sentences of monadic second order logic with counting

#### Abstract

We show that the spectrum of a sentence φ in Counting Monadic Second Order Logic (CMSOL) using one binary relation symbol and finitely many unary relation symbols, is ultimately periodic, provided all the models of φ are of clique width at most k, for some fixed k. We prove a similar statement for arbitrary finite relational vocabularies τ and a variant of clique width for τ-structures. This includes the cases where the models of φ are of tree width at most k. For the case of bounded tree-width, the ultimate periodicity is even proved for Guarded Second Order Logic GSOL. We also generalize this result to many-sorted spectra, which can be viewed as an analogue of Parikh’s Theorem on context-free languages, and its analogues for context-free graph grammars due to Habel and Courcelle. Our work was inspired by Gurevich and Shelah (2003), who showed ultimate periodicity of the spectrum for sentences of Monadic Second Order Logic where only finitely many unary predicates and one unary function are allowed. This restriction implies that the models are all of tree width at most 2, and hence it follows from our result.

#### Article information

Source
J. Symbolic Logic, Volume 69, Issue 3 (2004), 617-640.

Dates
First available in Project Euclid: 4 October 2004

https://projecteuclid.org/euclid.jsl/1096901758

Digital Object Identifier
doi:10.2178/jsl/1096901758

Mathematical Reviews number (MathSciNet)
MR2078913

Zentralblatt MATH identifier
1070.03018

#### Citation

Fischer, E.; Makowsky, J. A. On spectra of sentences of monadic second order logic with counting. J. Symbolic Logic 69 (2004), no. 3, 617--640. doi:10.2178/jsl/1096901758. https://projecteuclid.org/euclid.jsl/1096901758

#### References

• G. Asser Das Repräsentationsproblem im Prädikatenkalkül der ersten Stufe mit Identität, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 252--263.
• E. W. Beth Observations métamathématiques sur les structures simplement ordonneés, Applications scientifiques de la logique mathématique, Collection de Logique Mathématique, Serie A, vol. 5, Paris and Louvain,1954, pp. 29--35.
• C. Blatter and E. Specker First order spectra with one variable, Computation theory and logic, Lecture Notes in Computer Science, vol. 270, Springer-Verlag,1987, pp. 166--180.
• H. Bodlaender A tourist guide through tree width, Acta Cybernetica, vol. 11 (1993), pp. 1--23.
• A. Brandstädt, V. B. Le, and J. Spinrad Graph classes: A survey, SIAM Monographs on Discrete Mathematics and Applications, SIAM,1999.
• C. A. Christen Spektralproblem und Komplexitätstherie, Komplexität von Entscheidungsproblemen (E. Specker and V. Strassen, editors), Lecture Notes in Computer Science, vol. 43, Springer-Verlag,1976, pp. 102--126.
• D. G. Corneil, M. Habib, J.-M. Lanlignel, B. Reed, and U. Rotics Polynomial time recognition of clique-width $\leq 3$ graphs, Proceedings of LATIN 2000, Lecture Notes in Computer Science, vol. 1776, Springer-Verlag,2000, pp. 126--134.
• B. Courcelle The monadic second--order logic of graphs I: Recognizable sets of finite graphs, Information and Computation, vol. 85 (1990), pp. 12--75.
• B. Courcelle and J. Engelfriet A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars, Mathematical Systems Theory, vol. 28 (1995), pp. 515--552.
• B. Courcelle, J. Engelfriet, and G. Rozenberg Handle-rewriting hypergraph grammars, Journal of Computer and System Sciences, vol. 46 (1993), pp. 218--270.
• B. Courcelle and J. A. Makowsky Fusion on relational structures and the verification of monadic second order properties, Mathematical Structures in Computer Science, vol. 12 (2002), no. 2, pp. 203--235.
• B. Courcelle, J. A. Makowsky, and U. Rotics On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic, Discrete Applied Mathematics, vol. 108 (2001), no. 1--2, pp. 23--52.
• B. Courcelle and S. Olariu Upper bounds to the clique-width of graphs, Discrete Applied Mathematics, vol. 101 (2000), pp. 77--114.
• R. Diestel Graph theory, Graduate Texts in Mathematics, Springer-Verlag,1996.
• R. G. Downey and M. F. Fellows Parametrized complexity, Springer-Verlag,1999.
• A. Durand, R. Fagin, and B. Loescher Spectra with only unary function symbols, CSL '97 (M. Nielsen and W. Thomas, editors), Lecture Notes in Computer Science, vol. 1414, Springer-Verlag,1997, pp. 189--202.
• A. Durand and S. Ranaivoson First order spectra with one binary predicate, Theoretical Computer Science, vol. 160 (1996), pp. 305--320.
• H. D. Ebbinghaus and J. Flum Finite model theory, Perspectives in Mathematical Logic, Springer-Verlag,1995.
• H. D. Ebbinghaus, J. Flum, and W. Thomas Mathematical logic, Undergraduate Texts in Mathematics, Springer-Verlag,1980.
• J. Engelfriet and V. van Oostrom Logical description of context-free graph-languages, Journal of Computer and System Sciences, vol. 55 (1997), pp. 489--503.
• R. Fagin Generalized first-order spectra and polynomial time recognizable sets, Complexity of computation (R. Karp, editor), American Mathematical Society Proceedings, vol. 7, Society for Industrial and Applied Mathematics,1974, pp. 27--41.
• S. Feferman and R. Vaught The first order properties of algebraic systems, Fundamenta Mathematicae, vol. 47 (1959), pp. 57--103.
• E. Fischer The Specker-Blatter theorem does not hold for quaternary relations, Journal of Combinatorial Theory, Series A, vol. 103 (2003), pp. 121--136.
• E. Fischer and J. A. Makowsky The Specker-Blatter theorem revisited, Cocoon '03, Lecture Notes in Computer Science, vol. 2697, Springer-Verlag,2003, pp. 90--101.
• F. Gécseg and M. Steinby Tree languages, Handbook of formal languages (G. Rozenberg and A. Salomaa, editors), vol. 3: Beyond words, Springer-Verlag, Berlin,1997, pp. 1--68.
• S. Ginsburg and E. H. Spanier Bounded Algol-like languages, Transactions of the American Mathematical Society, vol. 113 (1966), pp. 333--368.
• A. Glikson and J. A. Makowsky NCE graph grammars and clique-width, Graph-theoretic concepts in computer science (H. Bodlaender, editor), Lecture Notes in Computer Science, vol. 2880, Springer-Verlag,2003, pp. 237--248.
• M. C. Golumbic and U. Rotics On the clique-width of some perfect graph classes, International Journal of Foundations of Computer Science, vol. 11 (2000), pp. 423--443.
• T. Grädel, C. Hirsch, and M. Otto Back and forth between guarded and modal logics, LiCS 2000, IEEE,2000, pp. 217--228.
• É. Grandjean First order spectra with one variable, Journal of Computer and System Sciences, vol. 40 (1990), no. 2, pp. 136--153.
• Y. Gurevich Modest theory of short chains, I, Journal of Symbolic Logic, vol. 44 (1979), pp. 481--490.
• Y. Gurevich and S. Shelah Spectra of monadic second-order formulas with one unary function, LiCS '03, IEEE,2003, pp. 291--300.
• Annegret Habel Hyperedge replacement: Grammars and languages, Lecture Notes in Computer Science, vol. 643, Springer-Verlag, Berlin,1992.
• N. Immerman Descriptive complexity, Graduate Texts in Computer Science, Springer-Verlag,1999.
• T. Johnson, N. Robertson, P. Seymour, and R. Thomas Directed tree-width, Journal of Combinatorial Theory, Serie B, vol. 81 (2001), no. 1, pp. 138--154.
• N. G. Jones and A. L. Selman Turing machines and spectra of first order formulas, Journal of Symbolic Logic, vol. 39 (1972), pp. 139--150.
• C. Kim A hierarchy of eNCE families of graph languages, Theoretical Computer Science, vol. 186 (1997), pp. 157--169.
• L. Lovász and P. Gácz Some remarks on generalized spectra, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), pp. 547--554.
• J. F. Lynch Complexity classes and theories of finite models, Mathematical Systems Theory, vol. 15 (1982), pp. 127--144.
• J. A. Makowsky Logical methods in graph algorithms, lecture notes of a course given at ESSLLI '99 in Utrecht, August, 1999.
• J. A. Makowsky and J. P. Mariño Tree-width and the monadic quantifier hierarchy, Theoretical Computer Science, vol. 303 (2003), pp. 157--170.
• J. A. Makowsky and Y. Pnueli Arity vs. alternation in second order logic, Annals of Pure and Applied Logic, vol. 78 (1996), no. 2, pp. 189--202.
• J. A. Makowsky and U. Rotics On the cliquewidth of graphs with few $P_4$'s, International Journal on Foundations of Computer Science, vol. 10 (1999), pp. 329--348.
• S. K. Mo Solutions to the Scholz problem, Chinese Annals of Mathematics, Series A, vol. 12 (1991), pp. 89--97.
• M. More Investigations of binary spectra by explicit polynomial transformations of graphs, Theoretical Computer Science, vol. 124 (1994), no. 2, pp. 221--272.
• R. Parikh On context-free languages, Journal of the Association for Computing Machinery, vol. 13 (1966), pp. 570--581.
• N. Robertson and P. Seymour Graph minors. IV. Tree-width and well-quasi-ordering, Journal of Combinatorial Theory, Serie B, vol. 48 (1990), pp. 227--254.
• U. Rotics Efficient algorithms for generally intractable graph problems restricted to specific classes of graphs, Ph.,D. thesis, Technion-Israel Institute of Technology,1998.
• H. Scholz Problem #1: Ein ungelöstes Problem in der symbolischen Logik, Journal of Symbolic Logic, vol. 17 (1952), p. 160.
• S. Shelah The monadic theory of order, Annals of Mathematics, vol. 102 (1975), pp. 379--419.
• E. Specker Application of logic and combinatorics to enumeration problems, Trends in theoretical computer science (E. Börger, editor), Computer Science Press,1988, pp. 141--169, reprinted in Ernst Specker, series Selecta, Birkhäuser, 1990, pp. 324--350.
• H. Straubing Finite automata, formal logic, and circuit complexity, Progress in Theoretical Computer Science, Birkhäuser,1994.