On the Complexity of Unification and Disunification in Commutative Idempotent Semigroups

We analyze the computational complexity of elementary uni-cation and disuniication problems for the equational theory ACI of commutative idempotent semigroups. From earlier work, it was known that the decision problem for elementary ACI-uniication is solvable in polynomial time. We show that this problem is inherently sequential by establishing that it is complete for polynomial time (P-complete) via logarithmic-space reductions. We also investigate the decision problem and the counting problem for elementary ACI-matching and observe that the former is solvable in logarithmic space, but the latter is #P-complete. After this, we analyze the computational complexity of the decision problem for elementary ground ACI-disuniication. Finally, we study the computational complexity of a restricted version of elementary ACI-matching, which arises naturally as a set-term matching problem in the context of the logic data language LDL. In both cases, we delineate the boundary between polynomial-time solvability and NP-hardness by taking into account two parameters, the number of free constants and the number of disequations or equations.