Algorithms and Complexity for Temporal and Spatial Formalisms

Abstract: The problem of computing with temporal information was early recognised within the area of artificial intelligence, most notably the temporal interval algebra by Allen has become a widely used formalism for representing and computing with qualitative knowledge about relations between temporal intervals. However, the computational properties of the algebra and related-formalisms are known to be bad: most problems (like satisfiability) are NP-hard. This thesis contributes to finding restrictions (as weak as possible) on Allen's algebra and related temporal formalisms (the point-interval algebra and extensions of Allen's algebra for metric time) for which the satisfiability problem can be computed in polynomial time. Another research area utilising temporal information is that of reasoning about action, which treats the problem of drawing conclusions based on the knowledge about actions having been performed at certain time points (this amounts to solving the infamous frame problem). One paper of this thesis attacks the computational side of this problem; one that has not been treated in the literature (research in the area has focused on modelling only). A nontrivial class of problems for which satisfiability is a polynomial-time problem is isolated, being able to express phenomena such as concurrency, conditional actions and continuous time.Similar to temporal reasoning is the field of spatial reasoning, where spatial instead of temporal objects are the field of study. In two papers, the formalism RCC-5 for spatial reasoning, very similar to Allen's algebra, is analysed with respect to tractable subclasses, using techniques from temporal reasoning.Finally, as a spin-off effect from the papers on spatial reasoning, a technique employed therein is used for finding a class of intuitionistic logic for which computing inference is tractable.