# On the complexity of unification and disunification in commutative idempotent semigroups

## Abstract

We analyze the computational complexity of elementary unification and disunification problems for the equational theory ACI of commutative idempotent semigroups. From earlier work, it was known that the decision problem for elementary ACI-unification 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-disunification. 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.

## Keywords

Polynomial Time Decision Problem Equational Theory Truth Assignment Horn Clause## Preview

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