Canonization for Two Variables and Puzzles on the Square

Abstract We consider infinitary logic with only two variable symbols, both with and without counting quantifiers, i.e. L 2 ≔ L ∞ ω 2 and C 2 ≔ L ∞ ω 2 (∃ ⩾ m ) mϵω . The main result is that finite relational structures admit Ptime canonization with respect to L 2 and C 2 : there are polynomial time com putable functors mapping finite relational structures to unique representatives of their equivalence class with respect to indistinguishability in either of these logics. In fact we exhibit Ptime in verses to the natural Ptime invariants that characterize structures up to L 2 - or C 2 -equivalence, respectively. As a corollary we obtain recursive presentations of the classes of all P time boolean queries that are closed with respect to L 2 - or C 2 -equivalence.

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