On A Semi-Automatic Method for Generating Composition Tables

Originating from Allen’s Interval Algebra, composition-based reasoning has been widely acknowledged as the most popular reasoning technique in qualitative spatial and temporal reasoning. Given a qualitative calculus (i.e. a relation model), the first thing we should do is to establish its composition table (CT). In the past three decades, such work is usually done manually. This is undesirable and error-prone, given that the calculus may contain tens or hundreds of basic relations. Computing the correct CT has been identified by Tony Cohn as a challenge for computer scientists in 1995. This paper addresses this problem and introduces a semi-automatic method to compute the CT by randomly generating triples of elements. For several important qualitative calculi, our method can establish the correct CT in a reasonable short time. This is illustrated by applications to the Interval Algebra, the Region Connection Calculus RCC-8, the INDU calculus, and the Oriented Point Relation Algebras. Our method can also be used to generate CTs for customised qualitative calculi defined on restricted domains.

💡 Research Summary

The paper tackles a long‑standing bottleneck in qualitative spatial and temporal reasoning: the construction of composition tables (CTs) for qualitative calculi. A composition table records, for every pair of basic relations α and β, the set of basic relations γ that can result from their weak composition (the smallest relation containing the ordinary composition). Traditionally, CTs have been built manually by checking the consistency of the network {x α y, y β z, x γ z} for each triple of basic relations. When a calculus contains dozens or even hundreds of basic relations, this manual process becomes infeasible and error‑prone—a challenge highlighted by Tony Cohn in 1995.

The authors propose a semi‑automatic, Monte‑Carlo‑style algorithm that generates CT entries by randomly sampling triples of domain elements. The key observation is that any three objects a, b, c in the universe U of a calculus automatically yield a valid composition triad (c‑triad) (ρ(a,b), ρ(a,c), ρ(b,c)), where ρ(x,y) denotes the basic relation holding between x and y. By enumerating all six permutations of (a,b,c) (including inverses), each sampled triple contributes up to six c‑triads. The algorithm (Algorithm 1) repeatedly draws random triples from a finite subdomain D ⊂ U, records any newly discovered c‑triads in a dynamic table, and stops when a termination condition Ψ is met (e.g., a fixed number of iterations, or no new triad after a large number of loops).

A crucial theoretical requirement is that the chosen subdomain D be 3‑complete: every consistent basic network of three variables must have a solution inside D. If D is 3‑complete, the random sampling will eventually generate all possible c‑triads, and the resulting table equals the true CT. The paper discusses how to verify 3‑completeness experimentally (by testing supersets of D) and notes that a formal proof is needed for absolute certainty.

The method is evaluated on four well‑known calculi:

1. Interval Algebra (IA) – 13 basic relations, 409 c‑triads. Using integer‑bounded intervals D_M = {