Randomisation and Derandomisation in Descriptive Complexity Theory

TL;DR

This paper explores the logical foundations of probabilistic complexity classes, demonstrating limitations of derandomisation in finite model theory and establishing that certain probabilistic logics capture BPP.

Contribution

It introduces the logic BPL for probabilistic classes, proves non-derandomisability of key queries in various logics, and shows BPIFP+C captures BPP on unordered structures.

Findings

Certain queries definable in BPFO are not in Cinf, showing limits of derandomisation.

Some standard logics cannot be derandomised, like transitive closure and fixed-point logic.

BPIFP+C captures BPP even on unordered structures.

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 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.