# Decidable Theories of Cayley-Graphs

## Abstract

We prove that a connected graph of bounded degree with only finitely many orbits has a decidable MSO-theory if and only if it is context-free. This implies that a group is context-free if and only if its Cayley-graph has a decidable MSO-theory. On the other hand, the first-order theory of the Cayley-graph of a group is decidable if and only if the group has a decidable word problem. For Cayley-graphs of monoids we prove the following closure properties. The class of monoids whose Cayley-graphs have decidable MSO-theories is closed under free products. The class of monoids whose Cayley-graphs have decidable firstorder theories is closed under general graph products. For the latter result on first-order theories we introduce a new unfolding construction, the factorized unfolding, that generalizes the tree-like structures considered by Walukiewicz. We show and use that it preserves the decidability of the first-order theory. p]Most of the proofs are omitted in this paper, they can be found in the full version [17].

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### References

- 1.A. V. Anisimov. Group languages.
*Kibernetika*, 4:18–24, 1971. In Russian; English translation in:*Cybernetics 4*, 594–601, 1973.MathSciNetGoogle Scholar - 2.A. Blumensath. Prefix-recognizable graphs and monadic second-order logic. Technical Report 2001-06, RWTH Aachen, Department of Computer Science, 2001.Google Scholar
- 3.H. L. Bodlaender. A note on domino treewidth.
*Discrete Mathematics & Theoretical Computer Science*, 3(4):141–150, 1999.MATHMathSciNetGoogle Scholar - 4.C. M. Campbell, E. F. Robertson, N. Ruŝkuc, and R. M. Thomas. Automatic semigroups.
*Theoretical Computer Science*, 250(1–2):365–391, 2001.MATHCrossRefMathSciNetGoogle Scholar - 5.B. Courcelle. The monadic second-order logic of graphs, II: Infinite graphs of bounded width.
*Mathematical Systems Theory*, 21:187–221, 1989.MATHCrossRefMathSciNetGoogle Scholar - 6.B. Courcelle. The monadic second-order logic of graphs VI: On several representations of graphs by relational structures.
*Discrete Applied Mathematics*, 54:117–149, 1994.MATHCrossRefMathSciNetGoogle Scholar - 7.V. Diekert. Makaninś algorithm. In M. Lothaire, editor,
*Algebraic Combinatorics on Words*, pages 342–390. Cambridge University Press, 2001.Google Scholar - 8.V. Diekert and G. Rozenberg, editors.
*The Book of Traces*. World Scientific, 1995.Google Scholar - 9.R. Diestel.
*Graph Theory, Second Edition*. Springer, 2000.Google Scholar - 10.M. J. Dunwoody. Cutting up graphs.
*Combinatorica*, 2(1):15–23, 1981.CrossRefMathSciNetGoogle Scholar - 11.C. C. Elgot and M. O. Rabin. Decidability and undecidability of extensions of second (first) order theory of (generalized) successor.
*Journal of Symbolic Logic*, 31(2):169–181, 1966.MATHCrossRefGoogle Scholar - 12.S. Feferman and R. L. Vaught. The first order properties of products of algebraic systems.
*Fundamenta Mathematicae*, 47:57–103, 1959.MATHMathSciNetGoogle Scholar - 13.J. Ferrante and C. Racko..
*The Computational Complexity of Logical Theories*, number 718 of*Lecture Notes in Mathematics.*Springer, 1979.Google Scholar - 14.H. Gaifman. On local and nonlocal properties. In J. Stern, editor,
*Logic Colloquium’ 81*, pages 105–135. North Holland, 1982.Google Scholar - 15.E. R. Green.
*Graph Products of Groups*. PhD thesis, The University of Leeds, 1990.Google Scholar - 16.B. Khoussainov and A. Nerode. Automatic presentations of structures. In
*LCC: International Workshop on Logic and Computational Complexity*, number 960 in Lecture Notes in Computer Science, pages 367–392, 1994.Google Scholar - 17.D. Kuske and M. Lohrey. Decidable theories of graphs, factorized unfoldings and cayley-graphs. Technical Report2002/37, University of Leicester, MCS, 2002.Google Scholar
- 18.G. S. Makanin. The problem of solvability of equations in a free semigroup.
*Math. Sbornik*, 103:147–236, 1977. In Russian; English translation in: Math. USSR Sbornik 32, 1977.MathSciNetGoogle Scholar - 19.G. S. Makanin. Equations in a free group.
*Izv. Akad. Nauk SSR*, Ser. Math. 46:1199–1273, 1983. In Russian; English translation in*Math. USSR Izvestija 21, 1983*.MathSciNetGoogle Scholar - 20.D. E. Muller and P. E. Schupp. Groups, the theory of ends, and context-free languages.
*Journal of Computer and System Sciences*, 26:295–310, 1983.MATHCrossRefMathSciNetGoogle Scholar - 21.D. E. Muller and P. E. Schupp. The theory of ends, pushdown automata, and second-order logic.
*Theoretical Computer Science*, 37(1):51–75, 1985.MATHCrossRefMathSciNetGoogle Scholar - 22.P. Narendran and F. Otto. Some results on equational unification. In M. E. Stickel, editor,
*Proceedings of the 10th International Conference on Automated Deduction (CADE 90)*,*Kaiserslautern (Germany)*, number 449 in Lecture Notes in Computer Science, pages 276–291. Springer, 1990.Google Scholar - 23.D. Seese. Tree-partite graphs and the complexity of algorithms. In L. Budach, editor,
*Proceedings of Fundamentals of Computation Theory (FCT’85), Cottbus (GDR)*, number 199 in Lecture Notes in Computer Science, pages 412–421Google Scholar - 24.D. Seese. The structure of models of decidable monadic theories of graphs.
*Annals of Pure and Applied Logic*, 53:169–195, 1991.MATHCrossRefMathSciNetGoogle Scholar - 25.G. Sénizergues. Semi-groups acting on context-free graphs. In F. M. auf der Heide and B. Monien, editors,
*Proceedings of the 28th International Colloquium on Automata, Languages and Programming (ICALP 96)*,*Paderborn (Germany)*, number 1099 in Lecture Notes in Computer Science, pages 206–218. Springer, 1996.Google Scholar - 26.S. Shelah. The monadic theory of order.
*Annals of Mathematics, II. Series*, 102:379–419, 1975.Google Scholar - 27.J. Stupp. The lattice-model is recursive in the original model. The Hebrew University, Jerusalem, 1975.Google Scholar
- 28.W. Thomas. A short introduction to infinite automata. In W. Kuich, G. Rozenberg, and A. Salomaa, editors,
*Proceedings of the 5th International Conference on Developments in Language Theory (DLT 2001)*,*Vienna (Austria)*, number 2295in Lecture Notes in Computer Science, pages 130–144. Springer, 2001.Google Scholar - 29.C. Thomassen and W. Woess. Vertex-transitive graphs and accessibility.
*Journal of Combinatorial Theory, Series B*, 58:248–268, 1993.MATHCrossRefMathSciNetGoogle Scholar - 30.A. Veloso da Costa. Graph products of monoids.
*Semigroup Forum*, 63(2):247–277, 2001.MATHMathSciNetGoogle Scholar - 31.I. Walukiewicz. Monadic second-order logic on tree-like structures.
*Theoretical Computer Science*, 275(1–2):311–346, 2002.MATHCrossRefMathSciNetGoogle Scholar