Expressibility Of Constraint Satisfaction Problems In Extensions Of First-Order Logic

A fundamental open problem in finite model theory and, specifically, descriptive complexity is the question whether there exists a logic which characterizes solvability of algorithmic decision polynomial time on the class of finite relational structures. A prominent candidate is the logic of Choiceless Polynomial Time (CPT) (optionally counting), a strict extension of the first-order logic. The expressibility problem for properties of finite relational structures in CPT (or CPT+C) is intrinsically hard, but CPT can be replaced by a more standard model-theoretic construction.

We show that constraint satisfaction problems over Maltsev templates can be expressed in the logic PIL+H. Furthermore, we show that the algorithm for Maltsev templates also solves said instances of constraint satisfaction problems over finite, idempotent Taylor algebras. This proves that for every finite, idempotent algebra A, the problem CSP(A) is either NP-complete or its solvability can be defined in the logic PIL+H.