A technical note on AC-Unification. The number of minimal unifiers of the equation αx₁ + ⋯ + αx_p ≐_AC βy₁ + ⋯ + βy_q

Abstract

We show in this note that the equation αx₁ + #x22EF; +αx_p≐_ACβy₁ + α +βy_q where + is an AC operator and αx stands for x+...+x (α times), has exactly

$$\left( { - 1} \right)^{p + q} \sum\limits_{i = 0}^p {\sum\limits_{j = 0}^q {\left( { - 1} \right)^{1 + 1} \left( {\begin{array}{*{20}c} p \\ i \\ \end{array} } \right)\left( {\begin{array}{*{20}c} q \\ j \\ \end{array} } \right)} 2^{\left( {\alpha + \begin{array}{*{20}c} {j - 1} \\ \alpha \\ \end{array} } \right)\left( {\beta + \begin{array}{*{20}c} {i - 1} \\ \beta \\ \end{array} } \right)} } $$

minimal unifiers if gcd(α, β)=1.