Hostname: page-component-76fb5796d-qxdb6 Total loading time: 0 Render date: 2024-04-29T17:43:28.012Z Has data issue: false hasContentIssue false
  • English
  • Français

Unary automatic graphs: an algorithmic perspective

Published online by Cambridge University Press:  01 February 2009

BAKHADYR KHOUSSAINOV ,
JIAMOU LIU  and
MIA MINNES
BAKHADYR KHOUSSAINOV
Affiliation:
Department of Computer Science, University of Auckland, Auckland, New Zealand Emails: bmk@cs.auckland.ac.nz; jliu036@aucklanduni.ac.nz
JIAMOU LIU
Affiliation:
Department of Computer Science, University of Auckland, Auckland, New Zealand Emails: bmk@cs.auckland.ac.nz; jliu036@aucklanduni.ac.nz
MIA MINNES
Affiliation:
Department of Mathematics, Cornell University, Ithaca, NY, USA Email: minnes@math.mit.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper studies infinite graphs produced from a natural unfolding operation applied to finite graphs. Graphs produced using such operations are of finite degree and automatic over the unary alphabet (that is, they can be described by finite automata over the unary alphabet). We investigate algorithmic properties of such unfolded graphs given their finite presentations. In particular, we ask whether a given node belongs to an infinite component, whether two given nodes in the graph are reachable from one another and whether the graph is connected. We give polynomial-time algorithms for each of these questions. For a fixed input graph, the algorithm for the first question is in constant time and the second question is decided using an automaton that recognises the reachability relation in a uniform way. Hence, we improve on previous work, in which non-elementary or non-uniform algorithms were found.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 19 , Special Issue 1: Theory and Applications of Models of Computation (TAMC) , February 2009 , pp. 133 - 152
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Blumensath, A. (1999) Automatic Structures, Diploma Thesis, RWTH Aachen.Google Scholar
Blumensath, A. and Grädel, E. (2004) Finite presentations of infinite structures: Automata and interpretations. Theory of Computing Systems 37 642–674.CrossRefGoogle Scholar
Bouajjani, A., Esparza, J. and Maler, O. (1997) Reachability analysis of pushdown automata: Application to model-checking. In: Proceedings of CONCUR'97. Springer-Verlag Lecture Notes in Computer Science 1243 135–150.Google Scholar
Büchi, J. R. (1960) On a decision method in restricted second-order arithmetic. In: Nagel, E., Suppes, P. and Tarski, A. (eds.) Proc. International Congress on Logic, Methodology and Philosophy of Science, Stanford University Press 1–11.Google Scholar
Caucal, D. (2002) On infinite graphs having a decidable monadic theory. In: Diks, K. and Rytter, W. (eds.) Proc. 27th MFCS. Springer-Verlag Lecture Notes in Computer Science 2420 165–176.Google Scholar
Esparza, J., Hansel, D., Rossmanith, P. and Schwoon, S. (2000) Efficient algorithms for model checking pushdown systems. In: Proc. CAV 2000. Springer-Verlag Lecture Notes in Computer Science 1855 232–247.Google Scholar
Hell, P. and Nešetřil, J. (2004) Graphs and Homomorphisms, Oxford University Press.Google Scholar
Hodgson, B. R. (1976) Théories décidables par automate fini., Ph.D. thesis, University of Montréal.Google Scholar
Khoussainov, B. and Minnes, M. (2008) Automatic structures and their complexity (extended abstract). In: Proc. TAMC'08. Springer-Verlag Lecture Notes in Computer Science (to appear).Google Scholar
Khoussainov, B. and Nerode, A. (1995) Automatic presentation of structures. Springer-Verlag Lecture Notes in Computer Science 960 367–392.CrossRefGoogle Scholar
Khoussainov, B., Nies, A., Rubin, S. and Stephan, F. (2004) Automatic structures: richness and limitations. In: Proc. 19th LICS 44–53.CrossRefGoogle Scholar
Khoussainov, B. and Rubin, S. (2001) Graphs with automatic presentations over a unary alphabet. Journal of Automata, Languages and Combinatorics 6 (4)467–480.Google Scholar
Khoussainov, B., Rubin, S. and Stephan, F. (2005) Automatic linear orders and trees. ACM Trans. Comput. Log. 6 (4)675–700.CrossRefGoogle Scholar
Libkin, L. (2004) Elements of finite model theory, Springer-Verlag.Google Scholar
Lohrey, M. (2003) Automatic structures of bounded degree. In: Proc. 10th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning (LPAR). Springer-Verlag Lecture Notes in Artificial Intelligence 2850 344–358.Google Scholar
Oliver, G. P. and Thomas, R. M. (2005) Automatic presentations for finitely generated groups. In: Diekert, V. and Durand, B. (eds.) Proc. 22nd STACS. Springer-Verlag Lecture Notes in Computer Science 3404 693–704.CrossRefGoogle Scholar
Rabin, M. O. (1969) Decidability of second-order theories and automata on infinite trees. Trans. Amer. Math. Soc. 141 1–35.Google Scholar
Rubin, S. (2004) Automatic Structures, Ph.D. Thesis, University of Auckland.Google Scholar
Thomas, W. (2002) A short introduction to infinite automata. In: Proceedings of the 5th International Conference Development in Language Theory. Springer-Verlag Lecture Notes in Computer Science 2295 130–144.CrossRefGoogle Scholar
Wöhrle, S. and Thomas, W. (2004) Model Checking Synchronized Products of Infinite Transition Systems. In Proc. 19th Annual IEEE Symposium on Logic in Computer Science (LICS'04) 2–11.Google Scholar