Abstract
According to Barthelman and Blumensath, the following families of infinite graphs are isomorphic: (1) prefix-recognisable graphs, (2) graph solutions of VR equational systems and (3) MS interpretations of regular trees. In this paper, we consider the extension of prefix-recognisable graphs to prefix-recognisable structures and of graphs solutions of VR equational systems to structures solutions of positive quantifier free definable (PQFD) equational systems. We extend Barthelman and Blumensath’s result to structures parameterised by infinite graphs by proving that the following families of structures are equivalent: (1) prefix-recognisable structures restricted by a language accepted by an infinite deterministic automaton, (2) solutions of infinite PQFD equational systems and (3) MS interpretations of the unfoldings of infinite deterministic graphs. Furthermore, we show that the addition of a fuse operator, that merges several vertices together, to PQFD equational systems does not increase their expressive power.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
K. Barthelmann. On equational simple graphs. Technical Report 9, Universität Mainz, Institut fĂ¼r Informatik, 1997.
A. Blumensath. Automatic structures. Diploma thesis, RWTH-Aachen, 1999.
A. Blumensath. Prefix-recognisable graphs and monadic second-order logic. Technical Report AIB-06-2001, RWTH Aachen, May 2001.
D. Caucal. On the regular structure of prefix rewriting. TCS, 106:61–86, 1992.
D. Caucal. On infinite transition graphs having a decidable monadic theory. In ICALP’96, volume 1099 of LNCS, pages 194–205, 1996.
B. Courcelle and J.A Makowsky. Fusion in relational structures and the verification of msologic. In MSCS, volume 12, pages 203–235, 2002.
T. Colcombet. On families of graphs having a decidable first order theory with reachability. In ICALP’02, 2002.
B. Courcelle. The monadic second-order logic of graphs ii: infinite graphs of bounded tree width. Math. Systems Theory, 21:187–221, 1989.
B. Courcelle. Handbook of Theoretical Computer Science, chapter Graph rewriting: an algebraic and logic approach. Elsevier, 1990.
B. Courcelle. Monadic-second order definable graph transductions: a survey. TCS, vol. 126:pp. 53–75, 1994.
B. Courcelle and I. Walukiewicz. Monadic second-order logic, graph coverings and unfoldings of transition systems. In Annals of Pure and Applied Logic, 1998.
M. Dauchet and S. Tison. The theory of ground rewrite systems is decidable. In Fifth Annual IEEE Symposium on Logic in Computer Science, pages 242–248, 1990.
C. L ding. Ground tree rewriting graphs of bounded tree width. In Stacs O2. LNCS, 2002.
C. Morvan. On rational graphs. In J. Tiuryn, editor, FOSSACS’00, volume 1784 of LNCS, pages 252–266, 2000.
D. Muller and P. Schupp. The theory of ends, pushdown automata, and second-order logic. Theoretical Computer Science, 37:51–75, 1985.
G. S nizergues. Definability in weak monadic second-order logic of some infinite graphs. In Dagstuhl seminar on Automata theory: Infinite computations, Warden, Germany, volume 28, page 16, 1992.
W. Thomas. Languages, automata, and logic. Handbook of Formal Language Theory, 3:389–455, 1997.
T. Urvoy. On abstract families of graphs. In DLT 02, 2002.
I. Walukiewicz. Monadic second order logic on tree-like structures. In STACS’96, volume 1046 of LNCS, pages 401–414, 1996.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Carayol, A., Colcombet, T. (2003). On Equivalent Representations of Infinite Structures. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_48
Download citation
DOI: https://doi.org/10.1007/3-540-45061-0_48
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40493-4
Online ISBN: 978-3-540-45061-0
eBook Packages: Springer Book Archive