Comparing the succinctness of monadic query languages over finite trees

Grohe, Martin; Schweikardt, Nicole

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 38 (2004) no. 4, pp. 343-373.

We study the succinctness of monadic second-order logic and a variety of monadic fixed point logics on trees. All these languages are known to have the same expressive power on trees, but some can express the same queries much more succinctly than others. For example, we show that, under some complexity theoretic assumption, monadic second-order logic is non-elementarily more succinct than monadic least fixed point logic, which in turn is non-elementarily more succinct than monadic datalog. Succinctness of the languages is closely related to the combined and parameterised complexity of query evaluation for these languages.

[1] S. Abiteboul, P. Buneman and D. Suciu, Data on the Web: From Relations to Semistructured Data and XML. Morgan Kaufmann (1999).

[2] S. Abiteboul, R. Hull and V. Vianu, Foundations of databases. Addison-Wesley (1995). | Zbl

[3] M. Adler and N. Immerman, An $n!$ lower bound on formula size. ACM Trans. Comput. Logic 4 (2003) 296-314.

[4] N. Alechina and N. Immerman, Reachability logic: An efficient fragment of transitive closure logic. Logic Journal of the IGPL 8 (2000) 325-338. | Zbl

[5] A. Chandra and D. Harel, Structure and complexity of relational queries. J. Comput. Syst. Sci. 25 (1982) 99-128. | Zbl

[6] E. Dantsin, A. Goerdt, E.A. Hirsch, R. Kannan, J. Kleinberg, C. Papadimitriou, P. Raghavan and U. Schöning, A deterministic ${(2-2/(k+1))}^n$ algorithm for $k$-SAT based on local search. Theor. Comput. Sci. 289 (2002) 69-83. Revised version of: Deterministic algorithms for $k$-SAT based on covering codes and local search, ICALP'00. Lect. Notes Comput. Sci. 1853. | Zbl

[7] H.-D. Ebbinghaus and J. Flum, Finite Model Theory. Springer-Verlag, 2nd edition (1999). | MR | Zbl

[8] K. Etessami, M.Y. Vardi and T. Wilke, First-order logic with two variables and unary temporal logic. Inform. Comput. 179 (2002) 279-295. | Zbl

[9] R. Fagin, Monadic generalized spectra. Zeitschrift für mathematische Logik und Grundlagen der Mathematik 21 (1975) 89-96. | Zbl

[10] M.F. Fernandez, J. Siméon and P. Wadler, An algebra for XML query, in Proc. of the 20th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS'00), edited by S. Kapoor and S. Prasad, Springer-Verlag. Lect. Notes Comput. Sci. 1974 (2000) 11-45. | Zbl

[11] M. Frick and M. Grohe, The complexity of first-order and monadic second-order logic revisited. Ann. Pure Appl. Logic, accepted (2004). | MR | Zbl

[12] M. Frick, M. Grohe and C. Koch, Query evaluation on compressed trees, in Proc. of the 18th IEEE Symposium on Logic in Computer Science (LICS'03) (2003) 188-197.

[13] G. Gottlob and C. Koch, Monadic datalog and the expressive power of web information extraction languages. J. ACM 51 (2004) 74-113.

[14] E. Grädel and M. Otto, On Logics with Two Variables. Theor. Comput. Sci. 224 (1999) 73-113. | Zbl

[15] M. Grohe and N. Schweikardt, Comparing the succinctness of monadic query languages over finite trees, in Proc. of the 17th International Workshop on Computer Science Logic (CSL'03), Springer-Verlag. Lect. Notes Comput. Sci. 2803 (2003) 226-240. | Zbl

[16] M. Grohe and N. Schweikardt, Comparing the succinctness of monadic query languages over finite trees. Technical Report EDI-INF-RR-0168, School of Informatics, University of Edinburgh, Scotland, UK (2003). | MR | Zbl

[17] M. Grohe and N. Schweikardt, The succinctness of first-order logic on linear orders, in Proc. of the 19th IEEE Symposium on Logic in Computer Science (LICS'04) (2004) 438-447. | Zbl

[18] H. Hosoya and B.C. Pierce, XDuce: A typed XML processing language (preliminary report), in International Workshop on the Web and Databases, edited by D. Suciu and G. Vossen (2000). Reprinted in The Web and Databases, Selected Papers, Springer. Lect. Notes Comput. Sci. 1997 (2001).

[19] N. Immerman, Descriptive Complexity. Springer-Verlag (1999). | MR | Zbl

[20] H. Kamp, Tense Logic and the theory of linear order. Ph.D. Thesis, University of California, Los Angeles (1968).

[21] C. Koch, Efficient processing of expressive node-selecting queries on XML data in secondary storage: A tree-automata based approach, in VLDB'03: 29th Conference on Very Large Databases, Berlin, September (2003) 249-260.

[22] F. Neven, Design and Analysis of Query Languages for Structured Documents - A Formal and Logical Approach. Ph.D. Thesis, Limburgs Universitair Centrum (1999).

[23] F. Neven and T. Schwentick, Query automata over finite trees. Theor. Comput. Sci. 275 (2002) 633-674. | Zbl

[24] W. Thomas, Languages, automata, and logic, in Handbook of formal languages 3 (1996), edited by G. Rozenberg and A. Salomaa, Springer, New York. | MR

[25] M.Y. Vardi, Reasoning about the past with two-way automata, in 25th International Colloquium on Automata, Languages and Programming (ICALP'98), edited by K.G. Larsen, S. Skyum and G. Winskel, Springer-Verlag. Lect. Notes Comput. Sci. 1443 (1998) 628-641. | Zbl

[26] T. Wilke, CTL${}^{+}$ is exponentially more succinct than CTL, in Proc. of the 19th Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS'99), Springer-Verlag. Lect. Notes Comput. Sci. 1738 (1999) 110-121. | Zbl