ASOGU with a single second-order variable and no first-order variables is undecidable, shown by reduction from Hilbert's tenth problem, and the result holds even under power associativity.
On the limits of second-order unification
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One is all you need: Second-order Unification without First-order Variables
ASOGU with a single second-order variable and no first-order variables is undecidable, shown by reduction from Hilbert's tenth problem, and the result holds even under power associativity.