Abstract
We investigate the decidability of unions of decidable equational theories. We focus on monadic theories, i.e., theories over signatures with unary symbols only. This allows us to make use of the equivalence between monoid amalgams and unions of monadic theories. We show that if the intersection theory is unitary, then the decidability of the union is guaranteed by the decidability of tensor products. We prove that if the intersection theory is a group or a group with zero, then the union is decidable. Finally, we show that even if the intersection theory is a 3-element monoid and is unitary, the union may be undecidable, but that it will always be decidable if the intersection is 2-element unitary. We also show that unions of regular theories, i.e., theories recognizable by finite automata, can be undecidable. However, we prove that they are decidable if the intersection theory is unitary.
This work has been partially supported by EU project SENSORIA (no. 016004).
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Pigozzi, D.: The join of equational theories. Coll. Math. 30(1), 15–25 (1974)
Baader, F., Tinelli, C.: Deciding the word problem in the union of equational theories. Inf. Comp. 178(2), 346–390 (2002)
Fiorentini, C., Ghilardi, S.: Combining word problems through rewriting in categories with products. Th. Comp. Sci. 294, 103–149 (2003)
Hoffman, P.: Union of equational theories: An algebraic approach. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 61–73. Springer, Heidelberg (2005)
Sapir, M.V.: Algorithmic problems for amalgams of finite semigroups. J. Algebra 229(2), 514–531 (2000)
Minsky, M.: Computation: Finite and Infinite Machines. Prentice-Hall, Englewood Cliffs (1967)
Birget, J.C., Margolis, S., Meakin, J.: On the word problem for tensor products and amalgams of monoids. Intnl. J. Alg. Comp. 9, 271–294 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hoffman, P. (2006). Unions of Equational Monadic Theories. In: Pfenning, F. (eds) Term Rewriting and Applications. RTA 2006. Lecture Notes in Computer Science, vol 4098. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11805618_7
Download citation
DOI: https://doi.org/10.1007/11805618_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-36834-2
Online ISBN: 978-3-540-36835-9
eBook Packages: Computer ScienceComputer Science (R0)