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 PTIME. 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 Cinf, 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. Similarly, we present a query on ordered structures which is definable in BPFO but not in monadic second-order logic, and a query on additive structures which is definable in BPFO but not in FO. The latter of these queries shows that certain uniform variants of AC0 (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.

💡 Research Summary

The paper introduces a systematic study of probabilistic complexity classes from the perspective of descriptive complexity theory. For any logic L, the authors define a probabilistic extension BP L, mirroring the way BPP is obtained from PTIME. The construction adds random bits to the evaluation of L‑formulas, allowing the definition of bounded‑error probabilistic logics (BP L), one‑sided error variants (R L, co‑RL) and an unbounded‑error version (P L). The central question is whether the expressive power of a probabilistic logic can be captured by a deterministic logic, i.e., whether BP L is “derandomisable” within some deterministic logic L′.

The main technical contributions are negative derandomisation results. First, the authors prove that BP FO (probabilistic first‑order logic) is strictly more expressive than C^∞_ω, the finite‑variable infinitary logic with counting. They construct a query that can be defined by a BP FO‑formula but not by any C^∞_ω‑formula. This separation immediately implies that many standard logics of finite model theory—transitive‑closure logic (TC), deterministic TC, fixed‑point logics (LFP, IFP, PFP) and all their counting extensions—cannot be derandomised.

Second, they examine structures equipped with built‑in relations. On ordered structures, they exhibit a query definable in BP FO but not in monadic second‑order logic (MSO), showing that even with a linear order the probabilistic logic exceeds MSO. Third, on additive structures (structures with built‑in addition and multiplication), they present a query definable in BP FO but not in FO. This yields a concrete separation between FO