Autornated reasoning about sets has received a considerable amount of interest in the literature. Techniques for such reasoning have been used in, for instance, analyses of programming languages, terminological logics and spatial reasoning. In this paper, we identify a new class of set constraints where checking satisfiability is tractable. (i.e. polynomial-time). We show how to use this tractability result for constructing a new tractable fragment of intuitionistic logic. Furthermore, we prove NP-completeness of several other cases of reasoning about sets.

This paper combines two important directions of research in temporal resoning: that of finding maximal tractable subclasses of Allen's interval algebra, and that of reasoning with metric temporal information. Eight new maximal tractable subclasses of Allen's interval algebra are presented, some of them subsuming previously reported tractable algebras. The algebras allow for metric temporal constraints on interval starting or ending points, using the recent framework of Horn DLRs. Two of the algebras can express the notion of sequentiality between intervals, being the first such algebras admitting both qualitative and metric time.

We investigate the computational properties of the spatial algebra RCC-5 which is a restricted version of the RCC framework for spatial reasoning. The satisfiability problem for RCC-5 is known to be NP-complete but not much is known about its approximately four billion subclasses. We provide a complete classification of satisfiability for all these subclasses into polynomial and NP-complete respectively. In the process, we identify all maximal tractable subalgebras which are four in total.