Abstract
We study deterministic computability over sets with atoms. We characterize those alphabets for which Turing machines with atoms determinize. To this end, the determinization problem is expressed as a Constraint Satisfaction Problem, and a characterization is obtained from deep results in CSP theory. As an application to Descriptive Complexity Theory, within a substantial class of relational structures including Cai-Furer-Immerman graphs, we precisely characterize those subclasses where the logic IFP+C captures order-invariant polynomial time computation.
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