Closures and dichotomies for quantified constraints

Quantified constraint satisfaction is the generalization of constraint satisfaction that allows for both universal and existential quantifiers over constrained variables, instead of just existential quantifiers. We study quantified constraint satisfaction problems CSP(Q,S), where Q denotes a pattern of quantifier alternation ending in exists or the set of all possible alternations of quantifiers, and S is a set of relations constraining the combinations of values that the variables may take. These problems belong to the corresponding level of the polynomial hierarchy or in PSPACE, depending on whether Q is a fixed pattern of quantifier alternation or the set of all possible alternations of quantifiers. We also introduce and study the quantified constraint satisfaction problems CSP′(Q,S) in which the universally quantified variables are restricted to range over given subsets of the domain. We first show that CSP(Q,S) and CSP′(Q,S) are polynomial-time equivalent to the problem of evaluating certain syntactically restricted monadic second-order formulas on finite structures. After this, we establish three broad sufficient conditions for polynomial-time solvability of CSP′(Q,S) that are based on closure functions; these results generalize and extend earlier results by other researchers about polynomial-time solvability of CSP(Q,S). Our study culminates with a dichotomy theorem for the complexity of list CSP′(Q,S), that is, CSP′(Q,S) where the relations of S include every subset of the domain of S. Specifically, list CSP′(Q,S) is either solvable in polynomial-time or complete for the corresponding level of the polynomial hierarchy, if Q is a fixed pattern of quantifier alternation (or PSPACE-complete if Q is the set of all possible alternations of quantifiers). The proofs are based on a more general unique sink property formulation.