Abstract
A formula from monadic second-order (MSO) logic can be used to specify a binary relation on the set of nodes of a tree. It is proved that, equivalently, such a relation can be computed by a finite-state tree-walking automaton, provided the automaton can test MSO properties of the nodes of the tree. For graphs, if a binary relation on the nodes of a graph can be computed by a finite-state graph-walking automaton, then it can be specified by an MSO formula, but, in general, not vice versa.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
S. Arnborg, J. Lagergren, D. Seese; Easy problems for tree-decomposable graphs, J. of Algorithms 12 (1991), 308–340
A. V. Aho, J. D. Ullman; Translations on a context-free grammar, Inf. and Control 19 (1971), 439–475
R. Bloem; Attribute Grammars and Monadic Second Order Logic, Master's Thesis, Leiden University, June 1996
J. Büchi; Weak second-order arithmetic and finite automata, Z. Math. Logik Grundlag. Math. 6 (1960), 66–92
B. Courcelle, J. Engelfriet; A logical characterization of the sets of hyper-graphs defined by hyperedge replacement grammars, Math. Systems Theory 28 (1995), 515–552
B. Courcelle; Graph rewriting: an algebraic and logic approach, in Handbook of Theoretical Computer Science, Vol.B (J. van Leeuwen, ed.), Elsevier, 1990, 193–242
B. Courcelle; The monadic second-order logic of graphs V: On closing the gap between definability and recognizability, Theor. Comput. Sci. 80 (1991), 153–202
B. Courcelle; Monadic second-order definable graph transductions: a survey, Theor. Comput. Sci. 126 (1994), 53–75
B. Courcelle; The expression of graph properties and graph transformations in monadic second-order logic, Chapter 5 of the Handbook of Graph Grammars and Computing by Graph Transformation, Volume 1: Foundations (G. Rozenberg, ed.), World Scientific, 1997
J. Doner; Tree acceptors and some of their applications, J. of Comp. Syst. Sci. 4 (1970), 406–451
C. C. Elgot; Decision problems of finite automata and related arithmetics, Trans. Amer. Math. Soc. 98 (1961), 21–51
J. Engelfriet; Simple Program Schemes and Formal Languages, Lecture Notes in Computer Science 20, Springer-Verlag, Berlin, 1974
J. Engelfriet; A characterization of context-free NCE graph languages by monadic second-order logic on trees, in Graph Grammars and their Application to Computer Science (H. Ehrig, H.-J. Kreowski, G. Rozenberg, eds.), Lecture Notes in Computer Science 532, Springer-Verlag, Berlin, 1991, 311–327
J. Engelfriet; A regular characterization of graph languages definable in monadic second-order logic, Theor. Comput. Sci. 88 (1991), 139–150.
J. Engelfriet; Context-free graph grammars, Chapter 3 of the Handbook of Formal Languages, Volume 3: Beyond Words (G. Rozenberg, A. Salomaa, eds.), Springer-Verlag, 1997
J. Engelfriet, V. van Oostrom; Regular description of context-free graph languages, J. of Comp. Syst. Sci. 53 (1996), 556–574
J. Engelfriet, V. van Oostrom; Logical description of context-free graph languages, Tech. Report 96-22, Leiden University, August 1996
J. Engelfriet, G. Rozenberg, G. Slutzki; Tree transducers, L systems, and two-way machines, J. of Comp. Syst. Sci. 20 (1980), 150–202
F. Gécseg, M. Steinby; Tree automata, Akadémiai Kiadó, Budapest, 1984
J. E. Hopcroft, J. D. Ullman; Introduction to Automata Theory, Languages, and Computation, Addison-Wesley, Reading, Mass., 1979
N. Klarlund, M. L. Schwartzbach; Graph Types, Proc. of the 20th Conference on Principles of Programming Languages, 1993, 196–205
W. Thomas; Automata on infinite objects, in Handbook of Theoretical Computer Science, Vol.B (J. van Leeuwen, ed.), Elsevier, 1990, 133–192
J. W. Thatcher, J. B. Wright; Generalized finite automata theory with an application to a decision problem of second-order logic, Math. Systems Theory 2 (1968), 57–81
Author information
Authors and Affiliations
Corresponding author
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Bloem, R., Engelfriet, J. (1997). Monadic second order logic and node relations on graphs and trees. In: Mycielski, J., Rozenberg, G., Salomaa, A. (eds) Structures in Logic and Computer Science. Lecture Notes in Computer Science, vol 1261. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63246-8_9
Download citation
DOI: https://doi.org/10.1007/3-540-63246-8_9
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63246-7
Online ISBN: 978-3-540-69242-3
eBook Packages: Springer Book Archive