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We study the model checking problem, for fixed structures A, over positiveequality-free first-order logic -- a natural generalisation of the non-uniformquantified constraint satisfaction problem QCSP(A). We prove a completecomplexity classification for this problem when A ranges over 1.) booleanstructures and 2.) digraphs of size (less than or equal to) three. The formerclass displays dichotomy between Logspace and Pspace-complete, while the latterclass displays tetrachotomy between Logspace, NP-complete, co-NP-complete andPspace-complete.

Model Checking Positive Equality-free FO: Boolean Structures and Digraphs of Size Three