Trees, grids, and MSO decidability: From graphs to matroids

Theoretical Computer Science

Volumn 351, Issue 3, 2006, Pages 372-393

  Hliněný, Petr ^a,b   Seese, Detlef ^c  

^a VSB TECHNICAL UNIVERSITY OF OSTRAVA   (Czech Republic)

^b MASARYK UNIVERSITY   (Czech Republic)

^c UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Branch width; Clique width; Decidability; Graph; Grids; Interpretability; Matroid; MSO logic; Spikes; Tree width

Indexed keywords

COMPUTABILITY AND DECIDABILITY; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL METHODS; FORMAL LOGIC; MATRIX ALGEBRA; PARALLEL PROCESSING SYSTEMS; PROBLEM SOLVING;

BRANCH-WIDTH; CLIQUE-WIDTH; GRAPHS; GRIDS; INTERPRETABILITY; MSO LOGIC; SPIKES; TREE-WIDTH;

TREES (MATHEMATICS);

EID: 32144462149     PISSN: 03043975     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.tcs.2005.10.006     Document Type: Conference Paper

References (88)

1. S. Arnborg, D.G. Corneil, and A. Proskurowski Complexity of finding embeddings in a k-tree SIAM J. Algebraic Discrete Methods 8 1987 277 284
2. S. Arnborg, J. Lagergren, D. Seese, Problems easy for tree-decomposable graphs (extended abstract), Proc. 15th ICALP, Lecture Notes in Computer Science, Vol. 317, Springer, Berlin, 1988, pp. 38-51.
3. S. Arnborg, J. Lagergren, and D. Seese Easy problems for tree-decomposable graphs J. Algorithms 12 1991 308 340
4. K. Barthelmann, On equational graphs, Technical Report 9, Universität Mainz, Institut für Informatik, 1997.
5. A. Baudisch, Zur Methatheorie abelscher Gruppen, Dissertation, Humboldt-Universitaet zu Berlin, 1974 (translation: To the meta-theory of abelian groups).
6. A. Baudisch, D.G. Seese, P. Tuschik, M. Weese, Decidability and Generalized Quantifiers, Akademie-Verlag, Berlin, 1980, xii+235pp.
7. A. Baudisch, D.G. Seese, P. Tuschik, and M. Weese Decidability and quantifier-elimination J. Barwise S. Feferman Model-Theoretic Logics 1985 Springer New York 235 268 (Chapter VII)
8. A. Blumensath, Axiomatising tree-interpretable structures, Technical Report AIB-2001-10, RWTH Aachen, 2001, pp. 1-23, .
9. A. Blumensath, A model theoretic characterization of clique width, RWTH Aachen, 2002, preprint, .
10. A. Blumensath, Axiomatising tree-interpretable structures, in: Proc. 19th Internat. Symp. Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Vol. 2285, Springer, Berlin, 2002, pp. 596-607.
11. H.L. Bodlaender A linear time algorithm for finding tree-decompositions of small treewidth SIAM J. Comput. 25 1996 1305 1317
12. E. Börger, E. Grädel, and Y. Gurevich The Classical Decision Problem 1997 Springer Berlin
13. J.R. Büchi Weak second-order arithmetic and finite automata Z. Math. Logik Grundlag. Math. 6 1960 66 92
14. J.R. Büchi, The monadic second order theory of ω 1, in: G.H. Müller, D. Siefkes (Eds.), Decidable Theories, II: Monadic Second Order Theory of All Countable Ordinals, Springer Lecture Notes in Mathematics, Vol. 328, Springer, Berlin, 1973, pp. 1-127.
15. J.R. Büchi, D. Siefkes, Axiomatization of the monadic second order theory of ω 1, in: G.H. Müller, D. Siefkes (Eds.), Decidable Theories, II: Monadic Second Order Theory of All Countable Ordinals, Springer, Lecture Notes in Mathematics, Vol. 328, Springer, Berlin, 1973, pp. 129-217.
16. B. Courcelle, The decidability of the monadic second order theory of certain sets of finite and infinite graphs, LICS'88, Logic in Computer Science, Edinburgh, 1988.
17. B. Courcelle, The monadic second order theory of graphs III: Tree decompositions, minors, and complexity issues, Inform. and Comput. 85 (1990) 12-75
18. (Theoret. Inform. Appl. 26(2) (1992) 257-286).
19. B. Courcelle, The monadic second order logic of graphs VI: On several representations of graphs by relational structures, Discrete Appl. Math. 54 (1994) 117-149, Erratum 63 (1995) 199-200.
20. B. Courcelle, Monadic second order definable graph transductions: a survey, Theoret. Comput. Sci. 126(1) (1994) 53-75, Seventh Colloq. Trees in Algebra and Programming (CAAP'92) and European Symp. Programming (ESOP), Rrennes, 1992.
21. B. Courcelle Structural properties of context-free sets of graphs generated by vertex replacement Inform. and Comput. 116 1995 275 293
22. B. Courcelle The monadic second order theory of graphs X: Linear orderings Theoret. Comput. Sci. 160 1996 87 144
23. B. Courcelle The expression of graph properties and graph transformations in monadic second-order logic Handbook of Graph Grammars and Computing by Graph Transformation Vol. 1 1997 World Scientific River Edge, NJ 313 400
24. B. Courcelle The monadic second-order logic of graphs XII: Planar graphs and planar maps Theoret. Comput. Sci. 237 2000 1 32
25. B. Courcelle, Clique-width of countable graphs: a compactness property, Internat. Conf. Graph Theory, Marseille, 2000 (available from ).
26. B. Courcelle The monadic second-order logic of graphs XIV: Uniformly sparse graphs and edge set quantification Theoret. Comput. Sci. 299 2003 1 36
27. B. Courcelle, The monadic second-order logic of graphs XV: On a conjecture by D. Seese, manuscript, LaBri, Bordeaux 1 University, March 8, 2004, pp. 1-40.
28. B. Courcelle, and J. Engelfriet A logical characterization of the sets of hypergraphs defined by hyperedge replacement grammars Math. Systems Theory 28 6 1995 515 552
29. B. Courcelle, J. Engelfriet, and G. Rozenberg Handle-rewriting hypergraph grammars J. Comput. System Sci. 46 1993 218 270
30. B. Courcelle, J.A. Makowsky, and U. Rotics Linear time solvable optimization problems on graphs of bounded clique-width Theory Comput. Systems 33 2000 125 150
31. B. Courcelle, and M. Mosbah Monadic second-order evaluations on tree-decomposable graphs Theoret. Comput. Sci. 109 1993 49 82
32. B. Courcelle, and M. Olariu Upper bounds to the clique-width of graphs Discrete Appl. Math. 101 2000 77 114
33. B. Courcelle, S.-I. Oum, Vertex-minors, monadic second-order logic, and a conjecture by Seese, July 23, 2004, pp. 37, preprint.
34. F. Drake Set Theory, an Introduction to Large Cardinals 1974 North-Holland Amsterdam
35. H.-D. Ebbinghaus, and J. Flum Finite Model Theory 1995 Springer Berlin
36. H.-D. Ebbinghaus, J. Flum, and W. Thomas Mathematical Logic second ed. 1994 Springer Berlin
37. S. Feferman, and R.L. Vaught The first-order properties of algebraic systems Fund. Math. 47 1959 57 103
38. M.R. Garey, and D.S. Johnson Computers and Intractability 1979 Bell Telephone Laboratories
39. S. Garfunkel, and J.H. Schmerl The undecidability of theories of groupoids with an extra predicate Proc. Amer. Math. Soc. 42 1974 286 289
40. J.F. Geelen, A.H.M. Gerards, N. Robertson, and G.P. Whittle On the excluded minors for the matroids of branch-width k J. Combin. Theory Ser. B 88 2003 261 265
41. J.F. Geelen, A.H.M. Gerards, and G.P. Whittle Branch-width and well-quasi-ordering in matroids and graphs J. Combin. Theory Ser. B 84 2002 270 290
42. J.F. Geelen, A.H.M. Gerards, G.P. Whittle, Excluding a planar graph from a GF(q)-representable matroid, manuscript, 2003.
43. Y. Gurevich Monadic theory of order and topology, I Israel J. Math. 27 1977 299 319
44. Y. Gurevich Monadic theory of order and topology, II Israel J. Math. 34 1979 45 71
45. Y. Gurevich Monadic second-order theories J. Barwise S. Feferman Model-Theoretic Logics 1985 Springer New York 479 506 (Chapter XIII)
46. Y. Gurevich, M. Magidor, and S. Shelah The monadic theory of ω 2 J. Symbolic Logic 48 1983 387 398
47. Y. Gurevich, and S. Shelah Monadic theory of order and topology in ZFC Ann. Math. Logic 23 1982 179 182
48. Y. Gurevich, and S. Shelah The monadic theory of the "next world" Israel J. Math. 49 1984 55 68
49. Y. Gurevich, and S. Shelah On the strength of the interpretation method J. Symbolic Logic 54 1989 305 323
50. H. Hauschild, H. Herre, W. Rautenberg, eit der monadischen Theorie 2. Stufe der n -separierten Graphen (translation: Decidability of the monadic second order theory of n -separated graphs), Wiss. Z. Humboldt-Univ. Math.-Natur. Reihe. XXI(5) (1972) 507-511.
51. H. Hauschild, and W. Rautenberg Interpretierbarkeit in der Gruppentheorie (translation: Interpretability in the theory of groups) Algebra Universalis 1 2 1971 136 151
52. H. Herre, Unentscheidbarkeit in der Graphentheorie (translated: Undecidability in graph theory), in: Bulletin de L'Academie Polonaise des Sciences, Serie des Sciences Mathematiques, Astronomiques et Physiques, Vol. XXI(3), 1973, pp. 201-208.
53. P. Hliněný, Branch-Width, Parse Trees, and Monadic Second-Order Logic for Matroids, J. Comb. Theory B, 2005, to appear.
54. P. Hliněný, On some hard problems on matroid spikes, Theor. Comput. Syst., 2005, accepted.
55. P. Hliněný, A parametrized algorithm for matroid branch-width, SIAM J. Comput. 35 (2005) 259-277 (electronic).
56. P. Hliněný, On matroid properties definable in the MSO logic, in: Math Foundations of Computer Science MFCS 2003, Lecture Notes in Computer Science, Vol. 2747, Springer, Berlin, 2003, pp. 470-479.
57. P. Hliněný, On hardness of the matroid minor problem, 2005, submitted.
58. P. Hliněný, D. Seese, Decidability of MSO theories of representable matroids, in: R. Downey, M. Fellows, F. Dehne (Eds.), Parameterized and Exact Computation, Proc. First Internat. Workshop on Parameterized and Exact Computation, IWPEC 2004, Bergen, Norway, September 14-17, 2004, Lecture Notes in Computer Science, Vol. 3162, Springer, Berlin, 2004, pp. 96-107.
59. P. Hliněný, D. Seese, Interpretability and decidability for monadic theories of graphs and matroids, 2005, in preparation.
60. P. Hliněný, G.P. Whittle, Matroid Tree-Width, 2003, submitted for publication (extended abstract in: Eurocomb'03, ITI Series, Vol. 2003-145, Charles University, Prague, Czech Republic, pp. 202-205).
61. K. Kunen, Combinatorics, in: J. Barwise (Ed.), Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977, pp. 371-401 (Chapter B.3).
62. P. Lapoire, Recognizability equals monadic second-order definability for sets of graphs of bounded tree-width, in: M. Morvan, Ch. Meinel, Da. Krob (Eds.), Proc. 15th Annual Symp. Theoretical Aspects of Computer Science Paris, France, February 1998, STACS, 1998, Springer Lecture Notes in Computer Science, Springer, Berlin, Vol. 1373, pp. 618-628.
63. J.A. Makowsky Algorithmic uses of the Feferman-Vaught theorem Ann. Pure Appl. Logic 126 2004 159 213
64. S.-I. Oum, Rank-width and well-quasi-ordering, manuscript, 2004.
65. S.-I. Oum, Approximating rank-width and clique-width quickly, manuscript, 2005.
66. S.-I. Oum, P. Seymour, Approximation algorithm to the clique-width of a graph, manuscript, 2003.
67. J.G. Oxley Matroid Theory 1992 Oxford University Press Oxford
68. M.O. Rabin A simple method for undecidability proofs and some applications Bar-Hillel Logic, Methodology and Philosophy of Sciences, Vol. 1 1964 North-Holland Amsterdam 58 68
69. M.O. Rabin Decidability of second order theories and automata on infinite trees Trans. Amer. Math. Soc. 141 1969 1 35
70. M.O. Rabin Decidable theories J. Barwise Handbook of Mathematical Logic 1977 North-Holland Amsterdam 595 629 (Chapter C.3)
71. W. Rautenberg, Lectures on decidability theory, Logical Semester Warsaw 1973, Lecture I-VII, Banach Center Warsaw, Warsaw, 1973, p. 67.
72. N. Robertson, P.D. Seymour, Graph minors - a survey, Surveys in Combinatorics, Cambridge University Press, Cambridge, 1985, pp. 153-171.
73. N. Robertson, and P.D. Seymour Graph minors X. Obstructions to tree-decomposition J. Combin. Theory Ser. B 52 1991 153 190
74. N. Robertson, P.D. Seymour, Graph minors XX. Wagner's conjecture, J. Comb. Theory B 92(2) (2004) 325-357.
75. D. Seese, Entscheidbarkeitsfragen der Theorien 2. Stufe einiger "netzartiger" Graphen (translation: Questions of decidability of second order theories of some "netlike" graphs), Diploma Thesis, Humboldt-University, 1972, p. 45.
76. D. Seese, Entscheidbarkeits- und Definierbarkeitsfragen der Theorie "netzartiger" Graphen - I (translation: Questions of decidability and definability of the theory of "netlike" graphs - I), Wiss. Z. Humboldt-Univ. Math.-Natur. Reihe XXI (1972) 513-517.
77. D. Seese, Zur Entscheidbarkeit der monadischen Theorie 2. Stufe baumartiger Graphen (translation: To the decidability of the monadic second order theory of graphs with a similarity to trees), Wiss. Z. Humboldt-Univ. Berlin Math.-Natur. Reihe. XXIV (6) (1975) 768-772.
78. D. Seese, Ein Unentscheidbarkeitskriterium (translation: A criterion of undecidability), Wiss. Z. Humboldt-Univ. Berlin Math.-Natur. Reihe XXIV (6) (1975) 772-780.
79. D. Seese, Entscheidbarkeits- und Interpretierbarkeitsfragen monadischer Theorien zweiter Stufe gewisser Klassen von Graphen (translation to English: Questions of decidability and interpretability of monadic second order theories of certain classes of graphs), Dissertation, Humboldt-Universität zu Berlin, 1976, pp. iv+172.
80. D. Seese, Decidability of ω -trees with bounded sets - a survey, in: M. Karpinski (Ed.), Fundamentals of Computation Theory, Lecture Notes in Computer Science, Vol. 56, Springer, Berlin, 1977, pp. 511-515.
81. D. Seese, Decidability of ω -trees with bounded sets, Akademie der Wissenschaften der DDR, Zentralinstitut für Mathematik und Mechanik, Report R-01/78, 1978, pp. 1-52.
82. D. Seese, Ordered tree representations of infinite graphs, Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut fuer Mathematik, P-MATH-12/88, Berlin, 1988, pp. 1-50, preprint.
83. D. Seese The structure of the models of decidable monadic theories of graphs Ann. Pure Appl. Logic 53 1991 169 195
84. D. Seese Interpretability and tree automata: a simple way to solve algorithmic problems on graphs closely related to trees M. Nivat A. Podelski Tree Automata and Languages 1992 Elsevier Amsterdam 83 114
85. P.D. Seymour Recognizing graphic matroids Combinatorica 1 1981 75 78
86. S. Shelah The monadic theory of order Ann. of Math. 102 1975 379 419
87. S. Shelah Notes on monadic logic, part B: Complexity of linear orders Israel J. Math. 69 1990 99 116
88. H. Wang, Proving theorems by pattern recognition II, Bell System Technical J. 40 (1961) 1-41.