# Temporal Constraint Satisfaction Problems in Fixed-Point Logic

Finite-domain constraint satisfaction problems are either solvable by Datalog, or not even expressible in fixed-point logic with counting. The border between the two regimes coincides with an important dichotomy in universal algebra; in particular, the border can be described by a strong height-one Maltsev condition. For infinite-domain CSPs the situation is more complicated even if the template structure of the CSP is model-theoretically tame. We prove that there is no Maltsev condition that characterises Datalog already for the CSPs of first-order reducts of (Q;<); such CSPs are called temporal CSPs and are of fundamental importance in infinite-domain constraint satisfaction. Our main result is a complete classification of temporal CSPs that can be solved in Datalog, and that can be solved in fixed point logic (with or without counting); the classification shows that many of the equivalent conditions in the finite fail to capture expressibility in any of these formalisms already for temporal CSPs.

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