Algebraic foundations for qualitative calculi and networks
Robin Hirsch, Marcel Jackson, Tomasz Kowalski

TL;DR
This paper introduces a new algebraic framework for qualitative calculi and networks, showing computational advantages and limitations of qualitative representations compared to traditional relation algebra representations.
Contribution
It defines qualitative representations, explores their properties, and demonstrates their computational benefits and limitations, including finite representability and complexity results.
Findings
Finite qualitatively representable algebras have finite representations.
Network satisfaction over finite qualitative algebras is in NP.
Validity of equations over qualitative representations is co-NP-complete.
Abstract
A qualitative representation is like an ordinary representation of a relation algebra, but instead of requiring , as we do for ordinary representations, we only require that , for each 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. 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…
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