On the Number of Quantifiers as a Complexity Measure

Journal of Logic and Computation, 2006

We present a second order logic of proportional quantifiers, SOLP, which is essentially a first order language extended with quantifiers that act upon second order variables of a given arity r, and count the fraction of elements in a subset of r-tuples of a model that satisfy a formula. Our logic is capable of expressing proportional versions of different problems of complexity up to NP-hard as, for example, the problem of deciding if at least a fraction 1/n of the set of vertices of a graph form a clique; and fragments within our logic capture complexity classes as NL and P, with auxiliary ordering relation. When restricted to monadic second order variables our logic of proportional quantifiers admits a semantic approximation based on almost linear orders, which is not as weak as other known logics with counting quantifiers (restricted to almost orders), for it does not has the bounded number of degrees property. Moreover, we show that in this almost ordered setting different fragments of this logic vary in their expressive power, and show the existence of an infinite hierarchy inside our monadic language. We extend our inexpressibility result over almost ordered structure to a fragment of SOLP, that in the presence of full order captures P. To obtain all our inexpressibility results we developed combinatorial games appropriate for these logics, whose application could go beyond the almost ordered models and hence are interesting by themselves.

Journal of the ACM, 1996

In this paper, a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given. This algorithm improves the complexity of the asymptotically fastest algorithm for this problem, known to this data. A new feature of this algorithm is that the role of the algebraic part (the dependence on the degrees of the imput polynomials) and the combinatorial part (the dependence on the number of polynomials) are sparated. Another new feature is that the degrees of the polynomials in the equivalent quantifier-free formula that is output, are independent of the number of input polynomials. As special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields, and also for solving the existential problem in the first order theory over real closed fields, are obtained.

Discrete Mathematics, 1984

This paper considers the definability of graph-properties by restricted second-order and first-order sentences. For example, it is shown that the class of Hamiltonian graphs cannot be defined by monadic second-order sentences (i.e.. if quantification over the subsets of vertices is allowed); any first-order sentence that defines Harniltonian graphs on n vertices must contain at least ½n quantifiers. The proofs use Frai'ss6--Ehrenfeucht games and ultraproducts.

Discrete Applied Mathematics, 2001

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is deÿnable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quantiÿcation. Such quantiÿcations are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this a ects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL deÿnable graph properties. Finally, our results are also applicable to SAT and ]SAT .

Lecture Notes in Computer Science, 2012

We study constraint satisfaction problems (CSPs) in the presence of counting quantifiers ∃ ≥j , asserting the existence of j distinct witnesses for the variable in question. As a continuation of our previous (CSR 2012) paper [11], we focus on the complexity of undirected graph templates. As our main contribution, we settle the two principal open questions proposed in [11]. Firstly, we complete the classification of clique templates by proving a full trichotomy for all possible combinations of counting quantifiers and clique sizes, placing each case either in P, NP-complete or Pspace-complete. This involves resolution of the cases in which we have the single quantifier ∃ ≥j on the clique K 2j. Secondly, we confirm a conjecture from [11], which proposes a full dichotomy for ∃ and ∃ ≥2 on all finite undirected graphs. The main thrust of this second result is the solution of the complexity for the infinite path which we prove is a polynomial-time solvable problem. By adapting the algorithm for the infinite path we are then able to solve the problem for finite paths, and then trees and forests. Thus as a corollary to this work, combining with the other cases from [11], we obtain a full dichotomy for ∃ and ∃ ≥2 quantifiers on finite graphs, each such problem being either in P or NP-hard. Finally, we persevere with the work of [11] in exploring cases in which there is dichotomy between P and Pspace-complete, in contrast with situations in which the intermediate NP-completeness may appear.