This paper surveys two related lines of research: ffl Logical characterizations of (non-deterministic) linear time complexity classes, and ffl non-expressibility results concerning sublogics of existential second-order logic. Starting from Fagin's fundamental work there has been steady progress in both fields with the effect that the weakest logics that are used in characterizations of linear time complexity classes are closely related to the strongest logics for which inexpressibility proofs for concrete problems have been obtained. The paper sketches these developments and highlights their connections as well as the obstacles that prevent us from closing the remaining gap between both kinds of logics. 1 Introduction The theory of computational complexity is quite successful in classifying computational problems with respect to their intrinsic consumption of ressources. Unfortunately it is until now much less successful in proving that the complexity classes that are used for these...

This paper first surveys the near-total lack of superlinear lower bounds in complexity theory, for “natural” computational problems with respect to many models of computation. We note that the dividing line between models where such bounds are known and those where none are known comes when the model allows non-local communication with memory at unit cost. We study a model that imposes a “fair cost ” for non-local communication, and obtain modest superlinear lower bounds for some problems via a Kolmogorov-complexity argument. Then we look to the larger picture of what it will take to prove really striking lower bounds, and pull from ours and others’ work a concept of information vicinity that may offer new tools and modes of analysis to a young field that rather lacks them.

Monadic least fixed point logic MLFP is a natural logic whose expressiveness lies between that of first-order logic FO and monadic second-order logic MSO. In this paper we take a closer look at the expressive power of MLFP. Our results are (1) MLFP can describe graph properties beyond any fixed level of the monadic secondorder quantifier alternation hierarchy. (2) On strings with built-in addition, MLFP can describe at least all languages that belong to the linear time complexity class DLIN. (3) Settling the question whether addition-invariant MLFP? = addition-invariant MSO on finite strings or, equivalently, settling the question whether MLFP? = MSO on finite strings with addition would solve open problems in complexity theory: “= ” would imply that PH = PTIME whereas “�= ” would imply that DLIN � = LINH. Apart from this we give a self-contained proof of the previously known result that MLFP is strictly less expressive than MSO on the class of finite graphs.

In this paper an algebraic characterization of the class DLIN of functions that can be computed in linear time by a deterministic RAM using only numbers of linear size is given. This class was introduced by Grandjean, who showed that it is robust and contains most computational problems that are usually considered to be solvable in deterministic linear time. The characterization is in terms of a recursion scheme for unary functions. A variation of this recursion scheme characterizes DLINEAR, the class which allows polynomially large numbers. A second variation defines a class that still contains DTIME(n), the class of functions that are computable in linear time on a Turing machine. From these algebraic characterizations, logical characterizations of DLIN and DLINEAR as well as complete problems (under DTIME(n) reductions) are derived. 1 Introduction Although deterministic linear time is a frequently used notion in the theory of algorithms it still does not have a universally accept...

. The paper gives a logical characterisation of the class NTIME(n) of problems that can be solved on a nondeterministic Turing machine in linear time. It is shown that a set L of strings is in this class if and only if there is a formula of the form 9f1 \Delta \Delta 9fk9R1 \Delta \Delta 9Rm8x' that is true exactly for all strings in L. In this formula the f i are unary function symbols, the R i are unary relation symbols and ' is a quantifier-free formula. Furthermore, the quantification of functions is restricted to noncrossing, decreasing functions and in ' no equations in which different functions occur are allowed. There are a number of variations of this statement, e.g., it holds also for k = 3. From these results we derive an Ehrenfeucht game characterisation of NTIME(n). 1 Introduction Since Fagin's seminal result that NP is the class of problems that can be described by an existential second--order (ESO) formula [6] there have been several characterisations of subclasses of N...