Randomisation and Derandomisation in Descriptive Complexity Theory

Abstract We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the complexity class BPP, bounded error probabilistic polynomial time, is defined from P. Our main focus lies on questions of derandomisation, and we prove that there is a query which is definable in BPFO, the probabilistic version of first-order logic, but not in C \(^\omega_{\infty{\omega}}\), finite variable infinitary logic with counting. This implies that many of the standard logics of finite model theory, like transitive closure logic and fixed-point logic, both with and without counting, cannot be derandomised. We prove similar results for ordered structures and structures with an addition relation, showing that certain uniform variants of A C 0 (bounded-depth polynomial sized circuits) cannot be derandomised. These results are in contrast to the general belief that most standard complexity classes can be derandomised. Finally, we note that BPIFP+C, the probabilistic version of fixed-point logic with counting, captures the complexity class BPP, even on unordered structures.

Authors

• Kord Eickmeyer
• Martin Grohe

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