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

We study the model checking problem, for fixed structures A, over positive equality-free first-order logic – a natural generalisation of the non-uniform quantified constraint satisfaction problem QCSP(A). We prove a complete complexity classification for this problem when A ranges over 1.) boolean structures and 2.) digraphs of size (less than or equal to) three. The former class displays dichotomy between Logspace and Pspace-complete, while the latter class displays tetrachotomy between Logspace, NP-complete, co-NP-complete and Pspace-complete.