Automatic Graphs and Graph D0L-Systems

The concept of end is a classical mean of understanding the behavior of a graph at infinity. In this respect, we show that the problem of deciding whether an infinite automatic graph has more than one end is recursively undecidable. The proof is based on the analysis of some global topological properties of the configuration graph of a self-stabilizing Turing machine. Next, this result is applied to show the undecidability of connectivity of all the finite graphs produced by iterating a graph DOL-system. We also prove that the graph DOL-systems with which we deal can emulate hyperedge replacement systems for which the above connectivity problem is decidable.

[1]  J. Peyriére.  FREQUENCY OF PATTERNS IN CERTAIN GRAPHS AND IN PENROSE TILINGS , 1986 .

[2]  Aristid Lindenmayer,et al.  An introduction to parallel map generating systems , 1986, Graph-Grammars and Their Application to Computer Science.

[3]  Grzegorz Rozenberg,et al.  Handbook of Graph Grammars and Computing by Graph Transformations, Volume 1: Foundations , 1997 .

[4]  Grzegorz Rozenberg,et al.  The mathematical theory of L systems , 1980 .

[5]  Grzegorz Rozenberg,et al.  Parallel Generation of Maps: Developmental Systems for Cell Layers , 1978, Graph-Grammars and Their Application to Computer Science and Biology.

[6]  David B. A. Epstein,et al.  Word processing in groups , 1992 .

[7]  Bruno Courcelle,et al.  The Expression of Graph Properties and Graph Transformations in Monadic Second-Order Logic , 1997, Handbook of Graph Grammars.

[8]  Anil Nerode,et al.  Automatic Presentations of Structures , 1994, LCC.

[9]  J. Peyrière,et al.  Processus de naissance avec interaction des voisins, évolution de graphes , 1981 .

[10]  Bruno Courcelle,et al.  Handle-Rewriting Hypergraph Grammars , 1993, J. Comput. Syst. Sci..

[11]  Joost Engelfriet,et al.  Node Replacement Graph Grammars , 1997, Handbook of Graph Grammars.

[12]  Aristid Lindenmayer,et al.  Algorithms for the generation and drawing of maps representing cell clones , 1982, Graph-Grammars and Their Application to Computer Science.

[13]  S. Semmes,et al.  A Graphic Apology for Symmetry and Implicitness , 2000 .

[14]  Bruno Courcelle,et al.  Graph Rewriting: An Algebraic and Logic Approach , 1991, Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics.

[15]  John Stallings,et al.  On Torsion-Free Groups with Infinitely Many Ends , 1968 .

[16]  Jeffrey D. Ullman,et al.  Introduction to Automata Theory, Languages and Computation , 1979 .

[17]  R. Halin,et al.  Über unendliche Wege in Graphen , 1964 .

[18]  David E. Muller,et al.  The Theory of Ends, Pushdown Automata, and Second-Order Logic , 1985, Theor. Comput. Sci..

[19]  B. Courcelle.  Fundamental properties of infinite trees , 1983 .

[20]  Géraud Sénizergues.  An Effective Version of Stallings' Theorem in the Case of Context-Free Groups , 1993, ICALP.

[21]  William S. Massey,et al.  Algebraic Topology: An Introduction , 1977 .

[22]  Bruno Courcelle,et al.  A Logical Characterization of the Sets of Hypergraphs Defined by Hyperedge Replacement Grammars , 1995, Math. Syst. Theory.

[23]  Achim Blumensath,et al.  Automatic structures , 2000, Proceedings Fifteenth Annual IEEE Symposium on Logic in Computer Science (Cat. No.99CB36332).

[24]  P. Seymour,et al.  Some New Results on the Well-Quasi-Ordering of Graphs , 1984 .

[25]  Warren Dicks,et al.  Groups Acting on Graphs , 1989 .

[26]  É. Ghys,et al.  Sur Les Groupes Hyperboliques D'Apres Mikhael Gromov , 1990 .

[27]  Bruno Courcelle,et al.  The Monadic Second-Order Logic of Graphs IV: Definability Properties of Equational Graphs , 1990, Ann. Pure Appl. Log..