Randomness in complexity theory and logics

Abstract

This thesis is comprised of two main parts whose common theme is the question of how powerful randomness as a computational resource is. In the first part (chapter 2) we deal with random structures such as graphs or families of functions and explain how these can possess – with high probability – properties than can be exploited by computer algorithms. Though it may seem counterintuitive at first, it can be very hard to deterministically construct a structure (such as a graph) possessing some desirable property such as good expansion which a random structure has with high probability. We review some cases where such deterministic constructions have indeed been obtained, and add two new results of this kind: We derandomise a randomised reduction due to Alekhnovich and Razborov by constructing certain unbalanced bipartite expander graphs, and we give a reduction from a problem concerning bipartite graphs to the problem of computing the minmax-value in threeplayer games. The latter reduction had been conceived by Hansen and Verbin in a randomised form, the derandomisation is a contribution of this thesis. In the second part (chapters 3 and 4), we study the expressive power of various logics when they are enriched by random relation symbols. Our goal is to apply techniques from descriptive complexity theory to the study of randomised complexity classes, and indeed we show that our randomised logics do capture complexity classes under study in complexity theory. Using strong results on the expressive power of first-order logic and the computational power of bounded-depth circuits, we give both positive and negative derandomisation results for our logics. On the negative side, we show that randomised first-order logic gains expressive power over standard first-order logic even on structures with a built-in addition relation. Furthermore, it is not contained in monadic second-order logic on ordered structures, nor in infinitary counting logic on arbitrary structures. On the positive side, we show that randomised first-order logic can be derandomised on structures with a unary vocabulary and is contained in monadic second-order logic on additive structures. The definition of randomised logics, as well as our results concerning their expressive power, are contributions of this thesis.