Quantified Constraint Satisfaction Problems: From Relaxations to Explanations Alex Ferguson and Barry O’Sullivan Cork Constraint Computation Centre, University College Cork, Ireland and Centre for Telecommunications Value-Chain Research {a.ferguson|b.osullivan}@4c.ucc.ie Abstract The Quantified Constraint Satisfaction Problem (QCSP) is a generalisation of the classical CSP in which some of variables can be universally quanti- fied. In this paper, we extend two well-known con- cepts in classical constraint satisfaction to the quan- tified case: problem relaxation and explanation of inconsistency. We show that the generality of the QCSP allows for a number of different forms of re- laxation not available in classical CSP. We further present an algorithm for computing a generalisation of conflict-based explanations of inconsistency for the QCSP. 1 Introduction One of the disadvantages of the classical CSP framework is that it lacks sufficient expressive power for modelling par- ticular aspects of real world problems, for example, uncer- tainty or other forms of uncontrollability in the environment in which a set of constraints must be satisfied. In this paper we are concerned with the Quantified CSP, a generalisation of the classical CSP that allows some of the variables to be universally quantified [Chen, 2004]. The se- mantics of universal quantification over a variable is that the set of constraints must be satisfiable for any assignment to it. This is in contrast with classical CSP, where all variables are existentially quantified, i.e. any assignment to an existentially quantified variable that satisfies the constraints is satisfactory. While the classical CSP is known to be NP-complete in gen- eral, the QCSP is PSPACE-complete [Chen, 2004]. We con- sider the problem of relaxing an instance of the QCSP when it is, for example, unsatisfiable. We propose several novel forms of problem relaxation for the QCSP and present an algorithm for generating conflict-based explanations of inconsistency. Our motivation comes from problems in conformant plan- ning and supply-chain management. We are interested in using constraints to support the local decision-making pro- cesses of a company that must supply products to a set of cus- tomers, while managing a complex network of its own suppli- ers. Classical CSP is not sufficiently expressive to model this type of problem concisely, since some of the variables are not under the control of the decision-maker, but we can model the problem as a Quantified CSP. The parallels between ad- versarial games and QCSP are natural: we can model variables under our control using existential quantifiers and those vari- ables outside our control using universal quantifiers [Chen, 2004]. The remainder of this paper is organised as follows. In Section 2 we present a formal definition of the fundamen- tal concepts in constraint satisfaction and the Quantified CSP. We present several new forms of problem relaxation in Sec- tion 3 that have previously not been considered due to the literature’s focus on classical CSP. We show how these forms of relaxation can be captured using the notion of requirement relaxation. Section 4 presents an approach to generating ex- planations of conflict in Quantified CSP based on requirement relaxation. We show how an existing explanation generation algorithm for classical CSPs can be extended to the quanti- fied case. We review the most related work in Section 5. A number of concluding remarks are made in Section 6. 2 Preliminaries Definition 1 (Classical Constraint Satisfaction Problem) A constraint satisfaction problem (CSP) is a 3-tuple P ˆ = 〈X , D, C〉 where X is a finite set of variables X ˆ = {x 1 ,...,x n }, D is a set of finite domains D ˆ = {D(x 1 ),...,D(x n )} where the domain D(x i ) is the finite set of values that variable x i can take, and a set of constraints C ˆ = {c 1 ,...,c m }. Each constraint c i is defined by the ordered set var(c i ) of the variables it involves, and a set sol(c i ) of allowed combinations of values. An assignment of values to the variables in var(c i ) satisfies c i if it belongs to sol(c i ).A solution to a CSP is an assignment of a value from its domain to each variable such that every constraint in C is satisfied. A fundamental notion used in reasoning about classical CSPs is that of arc consistency [Mackworth, 1977]. Due to logical conjunction, any assignment to the variables that is locally inconsistent with a constraint is guaranteed not to be part of any consistent solution. In the classical CSP we can regard all variables as being existentially quantified: we wish to show that there exist as- signments for each of the variables that satisfy all constraints simultaneously. However, we can generalise further by also allowing variables to be universally quantified. IJCAI-07 74