Existentially Restricted Quantified Constraint Satisfaction

The quantified constraint satisfaction problem (QCSP) is a powerful framework for modelling computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified variables, is restricted. Our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model, of inclusion in the complexity class coNP

Existentially restricted quantified constraint satisfaction

AbstractThe quantified constraint satisfaction problem (QCSP) is a framework for modelling PSPACE computational problems. The general intractability of the QCSP has motivated the pursuit of restricted cases that avoid its maximal complexity. In this paper, we introduce and study a new model for investigating QCSP complexity in which the types of constraints given by the existentially quantified variables, is restricted. Our primary technical contribution is the development and application of a general technology for proving positive results on parameterizations of the model, of inclusion in the complexity class coNP

Constraint Satisfaction with Counting Quantifiers

We initiate the study of constraint satisfaction problems (CSPs) in the presence of counting quantifiers, which may be seen as variants of CSPs in the mould of quantified CSPs (QCSPs). We show that a single counting quantifier strictly between exists^1:=exists and exists^n:=forall (the domain being of size n) already affords the maximal possible complexity of QCSPs (which have both exists and forall), being Pspace-complete for a suitably chosen template. Next, we focus on the complexity of subsets of counting quantifiers on clique and cycle templates. For cycles we give a full trichotomy -- all such problems are in L, NP-complete or Pspace-complete. For cliques we come close to a similar trichotomy, but one case remains outstanding. Afterwards, we consider the generalisation of CSPs in which we augment the extant quantifier exists^1:=exists with the quantifier exists^j (j not 1). Such a CSP is already NP-hard on non-bipartite graph templates. We explore the situation of this generalised CSP on bipartite templates, giving various conditions for both tractability and hardness -- culminating in a classification theorem for general graphs. Finally, we use counting quantifiers to solve the complexity of a concrete QCSP whose complexity was previously open

Quantified Constraints in Twenty Seventeen

I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions

Decomposing quantified conjunctive (or disjunctive) formulas

Model checking---deciding if a logical sentence holds on a structure---is a basic computational task that is well known to be intractable in general. For first-order logic on finite structures, it is PSPACE-complete, and the natural evaluation algorithm exhibits exponential dependence on the formula. We study model checking on the quantified conjunctive fragment of first-order logic, namely, prenex sentences having a purely conjunctive quantifier-free part. Following a number of works, we associate a graph to the quantifier-free part; each sentence then induces a prefixed graph, a quantifier prefix paired with a graph on its variables. We give a comprehensive classification of the sets of prefixed graphs on which model checking is tractable based on a novel generalization of treewidth that generalizes and places into a unified framework a number of existing results

Logic Column 17: A Rendezvous of Logic, Complexity, and Algebra

This article surveys recent advances in applying algebraic techniques to constraint satisfaction problems.Comment: 30 page