Automata-based representations for infinite graphs

Torre, Salvatore La; Napoli, Margherita

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 35 (2001) no. 4, pp. 311-330.

New compact representations of infinite graphs are investigated. Finite automata are used to represent labelled hyper-graphs which can be also multi-graphs. Our approach consists of a general framework where vertices are represented by a regular prefix-free language and edges are represented by a regular language and a function over tuples. We consider three different functions over tuples: given a tuple the first function returns its first difference, the second one returns its suffix and the last one returns its infixes. The first-difference function is substantially a direct generalization to infinite multi-hyper-graphs of the representation introduced by Ehrenfeucht et al. for finite graphs. This representation, though very interesting for finite graphs, turns out to be quite unsatisfactory for infinite graphs. The other two functions we consider while preserving some interesting features of their representation also achieves a high expressive power. As a matter of fact, our formalism either with the suffix or infix function results to be more powerful than the equational graphs introduced by Courcelle and the simple graphs defined by Caucal. The monadic second order theories of these two classes of graphs are undecidable, but still many interesting graph properties are decidable. The use of a regular prefix-free language to represent the vertices allows (fixed the language of the edges) to express a graph by a labelled tree, moreover, the use of finite automata to represent the edges allows the verification of graph properties.

[1] S. Arnborg, J. Lagergren and D. Seese, Easy problems for tree-decomposable graphs. J. Algorithms 12 (1991) 308-340. | MR | Zbl

[2] K. Barthelmann, When Can an Equational Simple Graph Be Generated by Hyperedge Replacement?, in Proc. of the 23th International Symposium on Mathematical Foundations of Computer Science (MFCS'98), edited by L. Brim, J. Gruska and J. Zlatuska. Brno, Czech Republic, August 24-28, 1998, Lecture Notes in Comput. Sci. 1450 (1998) 543-552. | Zbl

[3] M. Bauderon and B. Courcelle, Graph Expressions and Graph Rewritings. Math. Systems Theory 20 (1987) 83-127. | MR | Zbl

[4] H. L. Bodlaender and R. H. Möhring, The pathwidth and treewidth of cographs. SIAM J. Discrete Math. 6 (1993) 181-188. | MR | Zbl

[5] D. Caucal, On infinite transition graphs having a decidable monadic theory, in Proc. of 23rd International Colloquium on Automata, Languages and Programming (ICALP'96), edited by F.M. auf der Heide and B. Monien. Paderborn, Germany, July 8-12, 1996, Lecture Notes in Comput. Sci. 1099 (1996) 194-205. | MR | Zbl

[6] D.G. Corneil, H. Lerchs and L. Stuart Burlingham, Complement reducible graphs. Discrete Appl. Math. 3 (1981) 163-174. | MR | Zbl

[7] B. Courcelle, The monadic second-order logic of graphs. II. Infinite graphs of bounded width. Mathematical Systems Theory 21 (1989) 187-221. | MR | Zbl

[8] B. Courcelle, The monadic second-order logic of graphs. III. Tree-width, forbidden minors and complexity issues. RAIRO: Theoret. Informatics Appl. 26 (1992) 257-286. | EuDML | Numdam | MR | Zbl

[9] A. Ehrenfeucht, J. Engelfriet and G. Rozenberg, Finite Languages for the Representation of Finite Graphs. J. Comput. System Sci. 52 (1996) 170-184. | MR | Zbl

[10] J. Engelfriet, T. Harju, A. Proskurowski and G. Rozenberg, Characterization and Complexity of Uniformly Non-primitive Labeled 2-Structures. Theoretical Comput. Sci. 154 (1996) 247-282. | MR | Zbl

[11] P.C. Fisher, Multi-Tape and Infinite-State Automata - A Survey. Comm. ACM 8 (1965) 799-805. | Zbl

[12] J. Hopcroft and J. Ullman, Introduction to Automata Theory, Formal Languages and Computation. Addison-Wesley Publishing Company, Addison-Wesley Series in Comput. Sci.(1979) | MR | Zbl

[13] S. La Torre and M. Napoli, Representing Hyper-Graphs by Regular Languages. in Proc. of the 23th International Symposium on Mathematical Foundations of Computer Science (MFCS'98), edited by L. Brim, J. Gruska and J. Zlatuska, Brno, Czech Republic, August 24-28, 1998, Lecture Notes in Comput. Sci. 1450 (1998) 571-579. | Zbl

[14] C. Morvan, On Rational Graphs, in Proc. of the 3rd International Conference on Foundations of Software Science and Computation Structures (FOSSACS 2000). Berlin, Germany, March 25 - April 2, 2000, Lecture Notes in Comput. Sci. 1784 (2000) 252-266. | MR | Zbl

[15] N. Robertson and P. Seymour, Graph Minors. II Algorithmic aspects of tree-width. J. Algorithms 7 (1986) 309-322. | MR | Zbl

[16] J. Valdez, R. E. Tarjan and E. Lawler, The recognition of series parallel digraphs. SIAM J. Comput. 11 (1982) 298-313. | MR | Zbl