Order-Invariant CFO Contains Local-Order-Invariant FO

Abstract

Order-invariant cluster first-order logic (CFO) and local-order-invariant first-order logic are two fragments of order-invariant first-order logic, both motivated by the study of invariant logics on classes of bounded degree. Our main result is that order-invariant CFO contains local-order-invariant FO on arbitrary classes of finite structures, settling the conjecture of Ghasemi and Grange that order-invariant CFO contains plain FO. By combining this with a collapse result of Ghasemi and Grange, we obtain the equivalence of FO, local-order-invariant FO, and order-invariant CFO on classes of finite colored graphs of bounded degree. The proof makes use of Gaifman normal form, which allows our translation from local-order-invariant FO to order-invariant CFO to be effective. Combined with the fixed-parameter tractability of model-checking for order-invariant CFO on bounded-degree colored graph classes, this shows that model-checking for local-order-invariant FO on bounded-degree colored graph classes is fixed-parameter tractable. To generalize our collapse and model-checking results beyond colored graphs, we use a method based on first-order interpretations which also yields a collapse of order-invariant FO to FO when restricted to graph classes of bounded shrub-depth. Finally, we give an application to the database-motivated frozen-witness logic of Abiteboul, Simon, and Vianu by showing that local-order-invariant FO contains deterministic frozen-witness FO, and thus that deterministic frozen-witness FO collapses to FO on classes of finite structures of bounded degree.