We define matchings, and show that they capture the essence of context-freeness. More precisely, we show that the class of context-free languages coincides with the class of those sets of strings which can be defined by sentences of the form 9 b', where ' is first order, b is a binary predicate symbol, and the range of the second order quantifier is restricted to the class of matchings. Several variations and extensions are discussed.

Inexpressibility results in Finite Model Theory are often proved by showing that Duplicator, one of the two players of an Ehrenfeucht game, has a winning strategy on certain structures. In this article a new method is introduced that allows, under certain conditions, the extension of a winning strategy of Duplicator on some small parts of two finite structures to a global winning strategy. As applications of this technique it is shown that (*) Graph Connectivity is not expressible in existential monadic second-order logic (MonNP), even in the presence of a built-in linear order, (*) Graph Connectivity is not expressible in MonNP even in the presence of arbitrary built-in relations of degree n^o(1), and (*) the presence of a built-in linear order gives MonNP more expressive power than the presence of a built-in successor relation.

Most proofs showing limitations of expressive power of first-order logic rely on Ehrenfeucht-Fraisse games. Playing the game often involves a nontrivial combinatorial argument, so it was proposed to find easier tools for proving expressivity bounds. Most of those known for first-order logic are based on its "locality", that is defined in different ways. In this paper we characterize the relationship between those notions of locality. We note that Gaifman's locality theorem gives rise to two notions: one deals with sentences and one with open formulae. We prove that the former implies Hanf's notion of locality, which in turn implies Gaifman's locality for open formulae. Each of these implies the bounded degree property, which is one of the easiest tools for proving expressivity bounds. These results apply beyond the first-order case. We use them to derive expressivity bounds for first-order logic with unary quantifiers and counting. Finally, we apply these results to relational database...

It has been conjectured [FSV93] that an existential secondoder formula, in which the second-order quantification is restricted to unary relations (i.e. a Monadic NP formula), cannot express Graph Connectivity even in the presence of arbitrary built-in relations. In this paper it is shown that Graph Connectivity cannot be expressed by Monadic NP formulas in the presence of arbitrary built-in relations of degree n^o(1). The result is obtained by using a simplified version of a method introduced in [Sch94] that allows the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to a global winning strategy.

We show that in monadic second-order logic over finite directed graphs, a strict hierarchy of expressiveness is obtained by increasing the (second-order) quantifier alternation depth of formulas. Thus, the "monadic analogue" of the polynomial hierarchy is found to be strict, which solves a problem of Fagin. The proof is based on automata theoretic concepts (rather than Ehrenfeucht-Frasse games) and starts from a restricted class of graph-like structures, namely finite two-dimensional grids. We investigate monadic second-order definable sets of grids where the width of grids is a function of the height. In this context, the infiniteness of the quantifier alternation hierarchy is witnessed by n-fold exponential functions for increasing n. It is notable that these witness sets of the monadic hierarchy all belong to the complexity class NP, the first level of the polynomial hierarchy. 1 Introduction The subject of this paper is monadic second-order logic over graphs. In this logic, one ca...

We present a powerful and versatile new sufficient condition for the second player (the

A query is local if...

The key tool in proving inexpressibility results in finite-model theory is EhrenfeuchtFra iss'e games. This paper surveys various game-theoretic techniques and tools that lead to simpler proofs of inexpressibility results. The focus is on first-order logic and monadic NP. Appeared in: Descriptive Complexity and Finite Models, DIMACS Series in Descrete Mathematics and Theoretical Computer Science, American Mathematical Society, Volume 1, 1997, pp. 1--32. 1 Introduction The computational complexity of a problem is the amount of resources, such as time or space, required by a machine that solves the problem. Complexity theory traditionally has focused on the computational complexity of problems. A more recent branch of complexity theory focuses on the descriptive complexity of problems, which is the complexity of describing problems in some logical formalism [Imm89]. One of the exciting developments in complexity theory is the discovery of a very intimate connection between computation...

Binary NP consists of all sets of finite structures which are expressible in existential second order logic with second order quantification restricted to relations of arity 2. We look at semantical restrictions of binary NP, where the second order quantifiers range only over certain classes of relations. We consider mainly three types of classes of relations: unary functions, order relations and graphs with degree bounds. We show that many of these restrictions have the same expressive power and establish a 4-level strict hierarchy, represented by sets, permutations, unary functions and arbitrary binary relations, respectively.

This paper studies expressivity bounds for extensions of first-order logic with counting and unary quantifiers in the presence of relations of large degree. There are several motivations for this work. First, it is known that first-order logic with counting quantifiers captures uniform TC 0 over ordered structures. Thus, proving expressivity bounds for first-order with counting can be seen as an attempt to show TC 0 $ DLOG using techniques of descriptive complexity. Second, the presence of auxiliary built-in relations (e.g., order, successor) is known to make a big impact on expressivity results in finite-model theory and database theory (where logics with counting and unary quantifiers have recently been used to model query languages with aggregation). For those logics, our goal is to extend techniques from "pure" setting to that of auxiliary relations. Until now, all known results on the limitations of expressive power of the counting and unary quantifier extensions of first-order...