Classification transfer for qualitative reasoning problems

TL;DR

This paper classifies the computational complexity of various qualitative reasoning problems in temporal and spatial domains using logical and algebraic methods, providing new insights into tractability and open questions.

Contribution

It introduces a novel methodology using primitive positive interpretations to classify the complexity of first-order definable constraint languages in Allen's Interval Algebra and related formalisms.

Findings

Complete complexity classification for all first-order definable constraint languages in Allen's Interval Algebra.

Confirmed that ORD-Horn is maximally tractable among disjunctive binary relations in Rectangle Algebra.

Extended results to the r-dimensional Block Algebra, generalizing the classification.

Abstract

We study formalisms for temporal and spatial reasoning in the modern context of Constraint Satisfaction Problems (CSPs). We show how questions on the complexity of their subclasses can be solved using existing results via the powerful use of primitive positive (pp) interpretations and pp-homotopy. We demonstrate the methodology by giving a full complexity classification of all constraint languages that are first-order definable in Allen's Interval Algebra and contain the basic relations (s) and (f). In the case of the Rectangle Algebra we answer in the affirmative the old open question as to whether ORD-Horn is a maximally tractable subset among the (disjunctive, binary) relations. We then generalise our results for the Rectangle Algebra to the r-dimensional Block Algebra.