A Fixed-Point Logic with Symmetric Choice

Gire and Hoang introduce a fixed-point logic with a ‘symmetric’ choice operator that makes a nondeterministic choice from a definable set of tuples at each stage in the inductive construction of a relation, as long as the set of tuples is an automorphism class of the structure. We present a clean definition of the syntax and semantics of this logic and investigate its expressive power. We extend the logic of Gire and Hoang with parameterized and nested fixed points and first-order combinations of fixed points. We show that the ability to supply parameters to fixed points strictly increases the power of the logic. Our logic can express the graph isomorphism problem and we show that, on almost all structures, it captures P GI , the class of problems decidable in polynomial time by a deterministic Turing machine with an oracle for graph isomorphism.

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