Let D(G) be the smallest quantifier depth of a first order formula which is true for a graph G but false for any other non-isomorphic graph. This can be viewed as a measure for the descriptive complexity of G in first order logic. We show that almost surely D(G) = Θ( ln n ln ln n ), where G is a random tree of order n or the giant component of a random graph G(n, c n ) with constant c > 1. Thes...

In this paper we give a characterization of both Boolean and arithmetic circuit classes of logarithmic depth in the vein of descriptive complexity theory, i.e., the Boolean classes NC1, SAC1 and AC1 as well as their arithmetic counterparts #NC1, #SAC1 and #AC1. We build on Immerman’s characterization of constant-depth polynomial-size circuits by formulae of first-order logic, i.e., AC0 = FO, an...

The talk includes: ¯ a brief survey of the literature on descriptive complexity results; this journey takes us from regular to context-sensitive languages, ¯ a brief survey of the complexity of (realistic) linguistic formalisms; this journey takes us from context-sensitive grammars to fragments of richer grammars, ¯ a discussion of generative and model-theoretic means to arrive at more adequate...

This paper deals with descriptive complexity of picture languages of any dimension by syntactical fragments of existential second-order logic. • We uniformly generalize to any dimension the characterization by Giammarresi et al. [GRST96] of the class of recognizable picture languages in existential monadic second-order logic. • We state several logical characterizations of the class of picture ...

We present a functional characterization of deterministic logspace-computable predicates based on a variant (although not a subsystem) of propositional linear logic, which we call parsimonious logic. The resulting calculus is simply-typed and contains no primitive besides those provided by the underlying logical system, which makes it one of the simplest higher-order languages capturing logspac...

The first order logic Ring(0,+, ∗, <) for finite residue class rings with order is presented, and extensions of this logic with generalized quantifiers are given. It is shown that this logic and its extensions capture DLOGTIME-uniform circuit complexity classes ranging from AC to TC. Separability results are obtained for the hierarchy of these logics when order is not present, and for Ring(0,+,...

We discuss the definability of finite graphs in first-order logic with two relation symbols for adjacency and equality of vertices. The logical depth D(G) of a graph G is equal to the minimum quantifier depth of a sentence defining G up to isomorphism. The logical width W (G) is the minimum number of variables occurring in such a sentence. The logical length L(G) is the length of a shortest def...

Slowdown factors determine the extent of slowdown a computing system can experience based on functional and performance requirements. Dynamic Voltage Scaling (DVS) of a processor based on slowdown factors can lead to considerable energy savings. For the Earliest Deadline First (EDF) scheduling, the problem of DVS in the presence of task synchronization has not yet been addressed. We compute slo...