We provide a characterization of the resolution width introduced in the context of propositional proof complexity in terms of the existential pebble game introduced in the context of finite model theory. The characterization is tight and purely combinatorial. Our first application of this result is a surprising proof that the minimum space of refuting a 3-CNF formula is always bounded from below by the minimum width of refuting it (minus 3). This solves a well-known open problem. The second application is the unification of several width lower bound arguments, and a new width lower bound for the Dense Linear Order Principle. Since we also show that this principle has resolution refutations of polynomial size, this provides yet another example showing that the size-width relationship is tight.

this paper is to highlight some of these aspects of the model theory of nite structures, where the nite and in nite interact fruitfully, in order to dispel the perhaps too common perception that ( rst-order) model theory has little to say about nite structures

We prove a lower bound of ω(n k/4) on the size of constantdepth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound of ω(n k/89d2) due to Beame where d is the circuit depth. Our lower bound has the advantage that it does not depend on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeoff. Our k-clique lower bound derives from a stronger result of independent interest. Suppose fn: {0, 1} (n2) − → {0, 1} is a sequence of functions computed by constant-depth circuits of size O(n t). Let G be an Erdős-Rényi random graph with vertex set {1,..., n} and independent edge probabilities n −α where α ≤ 1 2t−1. Let A be a uniform random k-element subset of {1,..., n} (where k is any constant independent of n) and let KA denote the clique supported on A. We prove that fn(G) = fn(G ∪ KA) asymptotically almost surely. These results resolve a long-standing open question in finite model theory (going back at least to Immerman in 1982). The m-variable fragment of first-order logic, denoted by FO m, consists of the first-order sentences which involve at most m variables. Our results imply that the bounded variable hierarchy FO 1 ⊂ FO 2 ⊂ · · · ⊂ FO m ⊂ · · · is strict in terms of expressive power on finite ordered graphs. It was previously unknown that FO 3 is less expressive than full first-order logic on finite ordered graphs.

Abstract. The present paper gives a classification of the expressive power of two-variable least fixed-point logics. The main results are: 1. The two-variable fragment of monadic least fixed-point logic with parameters is as expressive as full monadic least fixed-point logic (on binary structures). 2. The two-variable fragment of monadic least fixed-point logic without parameters is as expressive as the two-variable fragment of binary least fixed-point logic without parameters. 3. The two-variable fragment of binary least fixed-point logic with parameters is strictly more expressive than the two-variable fragment of monadic least fixed-point logic with parameters (even on finite strings). 1.

Introduction Investigation of the measure-theoretic structure of complexity classes began with the development of resource-bounded measure in 1991 [56]. Since that time, a growing body of research by more than forty scientists around the world has shown resource-bounded measure to be a powerful tool that sheds new light on many aspects of computational complexity. Recent survey papers by Lutz [60], Ambos-Spies and Mayordomo [3], and Buhrman and Torenvliet [22] describe many of the achievements of this line of inquiry. In this column, we give a more recent snapshot of resource-bounded measure, focusing not so much on what has been achieved to date as on what we hope will be achieved in the near future. Section 2 below gives a brief, nontechnical overview of resource-bounded measure in terms of its motivation and principal ideas. Sections 3, 4, and 5 describe twelve specific open problems in the area. We have used the following three criteria in choosing these problems. 1. Their