Algebraic foundations for qualitative calculi and networks

Abstract

Binary Constraint Problems have traditionally been considered as Network Satisfaction Problems over some relation algebra. A constraint network is satisfiable if its nodes can be mapped into some representation of the relation algebra in such a way that the constraints are preserved. A qualitative representation ϕ is like an ordinary representation, but instead of requiring that (a;b)ϕ is the composition aϕ∘bϕ of the relations aϕ and bϕ, as we do for ordinary representations, we only require that cϕ⊇aϕ∘bϕ⇔c≥a;b, for each c in the algebra. A constraint network is qualitatively satisfiable if its nodes can be mapped to elements of a qualitative representation, preserving the constraints. If a constraint network is satisfiable then it is clearly qualitatively satisfiable, but the converse can fail, as we show. However, for a wide range of relation algebras including the point algebra, the Allen Interval Algebra, RCC8 and many others, a network is satisfiable if and only if it is qualitatively satisfiable.Unlike ordinary composition, the weak composition arising from qualitative representations need not be associative, so we can generalise by considering network satisfaction problems over non-associative algebras. We prove that computationally, qualitative representations have many advantages over ordinary representations: whereas many finite relation algebras have only infinite representations, every finite qualitatively representable algebra has a finite qualitative representation; the representability problem for (the atom structures of) finite non-associative algebras is NP-complete; the network satisfaction problem over a finite qualitatively representable algebra is always in NP; the validity of equations over qualitative representations is co-NP-complete. On the other hand we prove that there is no finite axiomatisation of the class of qualitatively representable algebras.