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

Circuit Complexity of Properties of Graphs with Constant Planar Cutwidth

We study the complexity of several of the classical graph decision problems in the setting of bounded cutwidth and how imposing planarity affects the complexity. We show that for 2-coloring, for bipartite perfect matching, and for several variants of disjoint paths the straightforward NC1 upper bound may be improved to AC0[2], ACC0, and AC0 respectively for bounded planar cutwidth graphs. We obtain our upper bounds using the characterization of these circuit classes in tems of finite monoids due to Barrington and Thérien. On the other hand we show that 3-coloring and Hamilton cycle remain hard for NC1 under projection reductions, analogous to the NP-completeness for general planar graphs. We also show that 2-coloring and (non-bipartite) perfect matching are hard under projection reductions for certain subclasses of AC0[2]. In particular this shows that our bounds for 2-coloring are quite close.

Invariant Definability and P/poly

. We look at various uniform and non-uniform complexity classes within P=poly and its variations L=poly, NL=poly, NP=poly and PSpace=poly, and look for analogues of the Ajtai-Immerman theorem which characterizes AC0 as the non-uniformly First Order Definable classes of finite structures. We have previously observed that the AjtaiImmerman theorem can be rephrased in terms of invariant definability: A class of finite structures is FOL invariantly definable iff it is in AC0 . Invariant definability is a notion closely related to but different from implicit definability and \Delta-definability. Its exact relationship to these other notions of definability has been determined in [Mak97]. Our first results are a slight generalization of similar results due to Molzan and can be stated as follows: let C be one of L; NL;P, NP, PSpace and L be a logic which captures C on ordered structures. Then the non-uniform L-invariantly definable classes of (not necessarily ordered) finite structures are..

Electronic Colloquium on Computational Complexity, Report No. 130 (2006) One-input-face MPCVP is Hard for L, but in

Abstract. A monotone planar circuit (MPC) is a Boolean circuit that can be embedded in a plane, and that has only AND and OR gates. Yang showed that the one-input-face monotone planar circuit value problem (MPCVP) is in NC 2, and Limaye et. al. improved the bound to LogCFL. Barrington et. al. showed that evaluating monotone upward stratified circuits, a restricted version of the one-input-face MPCVP, is in LogDCFL. In this paper, we prove that the unrestricted one-input-face MPCVP is also in LogDCFL. We also show this problem to be L-hard under quantifier free projections. Key Words: L, LogDCFL, monotone planar circuits.

Uniform circuits, & Boolean proof nets

The relationship between Boolean proof nets of multiplicative linear logic (APN) and Boolean circuits has been studied [Ter04] in a non-uniform setting. We refine the results taking care of uniformity: the relationship can be expressed in term of the (Turing) polynomial hierarchy. We give a proofs-as-programs correspondence between proof nets and deterministic as well as non-deterministic Boolean circuits with a uniform depth-preserving simulation of each other. The Boolean proof nets class m&BN(poly) is built on multiplicative and additive linear logic with a polynomial amount of additive connectives as the nondeterministic circuit class NNC(poly) is with non-deterministic variables. We obtain uniform-APN = NC and m&BN(poly) = NNC(poly) = NP