Promise Systems of Equations over Magmas with Identity and over Algebras in Congruence Modular Varieties

Abstract: We study the computational complexity of solving promise systems of equations over finite algebras. Given two algebras $\mathbf{A}$ and $\mathbf{B}$ with a homomorphism from $\mathbf{A}$ to $\mathbf{B}$, the promise system of equations problem is to determine if an input system of equations has a solution in $\mathbf{A}$ or not even in $\mathbf{B}$. We generalize the results of Larrauri, Mottet, and Živný [ACM ToCL'26] to obtain a $\mathbf{P}-\mathbf{NP}$-hard dichotomy result for promise systems of equations over a class of algebras which contains all monoids, and a dichotomy result for promise systems of equations over algebras in a congruence modular variety. We then consider the metaproblem for promise systems of equations over algebras in a congruence modular variety: given finite algebras $\mathbf{A}$ and $\mathbf{B}$ such that $\mathbf{A}$ is in a congruence modular variety, we show there is a quasi-polynomial time algorithm for determining whether or not the associated promise system of equations problem is in $\mathbf{P}$.