Abstract
We study two-way tree automata modulo equational theories. We deal with the theories of Abelian groups (ACUM), idempotent commutative monoids (ACUI), and the theory of exclusive-or (ACUX), as well as some variants including the theory of commutative monoids (ACU). We show that the one-way automata for all these theories are closed under union and intersection, and emptiness is decidable. For two-way automata the situation is more complex. In all these theories except ACUI, we show that two-way automata can be effectively reduced to one-way automata, provided some care is taken in the definition of the so-called push clauses. (The ACUI case is open.) In particular, the two-way automata modulo these theories are closed under union and intersection, and emptiness is decidable. We also note that alternating variants have undecidable emptiness problem for most theories, contrarily to the non-equational case where alternation is essentially harmless.
Partially supported by the ACI “cryptologie” PSI-Robuste, ACI VERNAM, the RNTL project EVA and the ACI jeunes chercheurs “Sécurité informatique, protocoles cryptographiques et détection d’intrusions”.
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
T. Colcombet. Rewriting in the partial algebra of typed terms modulo AC. In A. Kucera and R. Mayr, editors, Electronic Notes in Theoretical Computer Science, volume 68. Elsevier Science Publishers, 2002.
H. Comon, M. Dauchet, R. Gilleron, F. Jacquemard, D. Lugiez, S. Tison, and M. Tommasi. Tree automata techniques and applications. http://www.grappa.univ-lille3.fr/tata, 1997.
T. Frühwirth, E. Shapiro, M. Y. Vardi, and E. Yardeni. Logic programs as types for logic programs. In LICS’91, 1991.
F. Gécseg and M. Steinby. Tree languages. In G. Rozenberg and A. Salomaa, editors, Handbook of Formal Languages, volume 3, pages 1–68. Springer Verlag, 1997.
S. Ginsburg and E. H. Spanier. Semigroups, Presburger formulas and languages. Pacific Journal of Mathematics, 16(2):285–296, 1966.
J. Goubault-Larrecq. A method for automatic cryptographic protocol verification. In FMPPTA’2000, 15th IPDPS Workshops, pages 977–984. Springer-Verlag LNCS 1800, 2000.
J. Goubault-Larrecq and K. N. Verma. Alternating two-way AC-tree automata. In preparation.
D. Lugiez. A good class of tree automata. Application to inductive theorem proving. In ICALP’98, pages 409–420. Springer-Verlag LNCS 1443, 1998.
D. Lugiez. Counting and equality constraints for multitree automata. In FOSSACS’03. Springer-Verlag LNCS, 2003.
M. L. Minsky. Recursive unsolvability of Post’s problem of “tag” and other topics in the theory of Turing machines. Annals of Mathematics, Second Series, 74(3):437–455, 1961.
D. Monniaux. Abstracting cryptographic protocols with tree automata. In SAS’99, pages 149–163. Springer-Verlag LNCS 1694, 1999.
H. Ohsaki. Beyond regularity: Equational tree automata for associative and commutative theories. In CSL’01, pages 539–553. Springer-Verlag LNCS 2142, 2001.
H. Ohsaki and T. Takai. Decidability and closure properties of equational tree languages. In RTA’02, pages 114–128. Springer-Verlag LNCS 2378, 2002.
R. J. Parikh. On context-free languages. Journal of the ACM, 13(4):570–581, 1966.
M. Y. Vardi. Reasoning about the past with two-way automata. In ICALP’98, pages 628–641. Springer Verlag LNCS 1443, 1998.
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
Verma, K.N. (2003). Two-Way Equational Tree Automata for AC-Like Theories: Decidability and Closure Properties. In: Nieuwenhuis, R. (eds) Rewriting Techniques and Applications. RTA 2003. Lecture Notes in Computer Science, vol 2706. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44881-0_14
Download citation
DOI: https://doi.org/10.1007/3-540-44881-0_14
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40254-1
Online ISBN: 978-3-540-44881-5
eBook Packages: Springer Book Archive