Randomisation and Derandomisation in Descriptive Complexity Theory

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 Cω ∞ω , 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 AC (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.