Combining word problems through rewriting in categories with products
Theoretical Computer Science - Category theory and computer science
Computation: finite and infinite machines
Computation: finite and infinite machines
Union of equational theories: an algebraic approach
RTA'05 Proceedings of the 16th international conference on Term Rewriting and Applications
Reachability in unions of commutative rewriting systems is decidable
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
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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.