The problem of computing with temporal information was early recognised within the area of arti cial 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 satis ability) are NPhard. This thesis contributes to nding 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 satis ability 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 satis ability 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 eld of spatial reasoning, where spatial instead of temporal objects are the eld 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-o e ect from the papers on spatial reasoning, a technique employed therein is used for nding a class of intuitionistic logic for which computing inference is tractable. Acknowledgements First, I would like to thank my supervisor Christer B ackstr om for letting me do this research, and for pointing out to me the existence of the area of temporal reasoning. Next, I have had an enjoyable cooperation with those co-authoring papers of this thesis. Without them, this thesis would have been (if at all) completely di erent: Peter Jonsson, Marcus Bjareland and Christer Backstrom. Many people (other than those above) have contributed to a stimulating research environment with facts, comments and discussions of various kinds, particularly Simin Nadjm-Tehrani, Anders Henriksson, Erik Sandewall, Patrick Doherty, Magnus Andersson, Lars Karlsson, and also anonymous reviewers of papers of the thesis. Thanks also to Lise-Lott Andersson and Lillemor Wallgren for making administrative issues non-problems, the TUS group for keeping the computers running, and Ivan Rankin for improving my English. List of Papers The thesis includes the following eight papers. I. Peter Jonsson, Thomas Drakengren and Christer Backstr om. The Computational Complexity of Relating Points with Intervals on the Real Line. The paper is a revised and extended version of another paper: Peter Jonsson, Thomas Drakengren and Christer Backstr om. A Complete Classi cation of Tractability in the Point-Interval Algebra. In J. Doyle and L. Aiello, editors, Proceedings of the 5th International Conference on Principles of Knowledge Representation and Reasoning, KR '96, pages 352{363, Cambridge, MA, USA October 1996. Morgan Kaufmann. II. Thomas Drakengren and Peter Jonsson, 1997. Twenty-one Large Tractable Subclasses of Allen's Algebra. Arti cial Intelligence, 93(1{2):297{319. The paper is an extended version of another paper: Thomas Drakengren and Peter Jonsson. Maximal Tractable Subclasses of Allen's Interval Algebra: Preliminary Report. In Proceedings of the 13th (US) National Conference on Arti cial Intelligence, AAAI '96, Portland, OR, USA, August 1996. III. Thomas Drakengren and Peter Jonsson, 1997. Eight Maximal Tractable Subclasses of Allen's Algebra with Metric Time. Journal of Arti cial Intelligence Research, 7:25{45. IV. Thomas Drakengren and Peter Jonsson, 1997. Towards a Complete Classi cation of Tractability in Allen's Algebra. In Proceedings of the Fifteenth International Joint Conference on Arti cial Intelligence, IJCAI '97, Nagoya, Japan, August 1997. V. Thomas Drakengren and Marcus Bjareland. Reasoning about Action in Polynomial Time. The paper is a version of the following paper, but with full proofs: Thomas Drakengren and Marcus Bjareland, 1997. Reasoning about Action in Polynomial Time. In Proceedings of the Fifteenth International Joint Conference on Arti cial Intelligence, IJCAI '97, Nagoya, Japan, August 1997. VI. Peter Jonsson and Thomas Drakengren, 1997. RCC-5 and its Tractable Subclasses. Journal of Arti cial Intelligence Research, 6:211{221. VII. Peter Jonsson and Thomas Drakengren. Qualitative Reasoning about Sets Applied to Spatial Reasoning. VIII. Thomas Drakengren and Peter Jonsson. Reasoning about Set Constraints Applied to Tractable Inference in Intuitionistic Logic. 1 1 Thesis Overview 1.1 Topic The topic of this thesis is the study of computational complexity of temporal and spatial reasoning1. Much of the work within the area of Arti cial Intelligence (and elsewhere) has been recognised as including the notions of time and space, naturally, since these are fundamental aspects of the real world. It is clear that capabilities of modelling and computing with facts about time and space will be needed for any reasonably exible system interacting with its environment. The areas of temporal and spatial reasoning in AI have developed from this recognition. In this thesis, the focus is on temporal reasoning, and the results on spatial reasoning can largely be considered as by-products of the research on temporal reasoning. Vila [1994] has written an overview of the general uses of temporal reasoning in AI. The thesis contributes to the problem of computing with temporal and spatial information, as opposed to the task of modelling temporal and spatial phenomena. The modelling problem consists of deciding what are the ideal primitive concepts and what is the appropriate formalism to use for a given application. Once it is decided how to model a class of phenomena, methods can be devised for computing facts given information expressed in the chosen formalism. Attacking the problem of computation certainly requires a choice of modelling formalism, and in order for the solutions to the computational problems to be relevant, here formalisms that are well established and uncontroversial have been used, at least for the temporal case, where the basis is Allen's interval algebra [Allen, 1983], both in its original form and extended with constructs for expressing metric temporal information. For the spatial case, however, the modelling language used is the spatial algebra, RCC-5 [Bennett, 1994], taken from the RCC approach to spatial reasoning [Randell and Cohn, 1989; Randell et al., 1992b], about which there is still some controversy whether or when it correctly models spatial phenomena [Lemon, 1996]. 1Instead of temporal reasoning, sometimes the expression temporal constraint reasoning is used to distinguish the eld from that of reasoning about action, which deals with the problem of inferring what propositions hold at what time points when it is known that certain actions have been performed (see e.g. [Sandewall, 1994] and paper V of this thesis). I think that this use is unfortunate, since in line with this, any reasoning task having a temporal component could be called temporal reasoning. 2 There are fundamental computational problems associated with temporal and spatial formalisms, including the following: Given some information, checking whether some temporal or spatial relation is entailed from it, that is, being a logical consequence thereof. Checking whether some information is satis able, that is, checking whether there exists a model that satis es the information. Cleary, there is a connection between the two problems, and indeed, they are polynomially equivalent2 for Allen's algebra [Golumbic and Shamir, 1993], and an analogous proof would easily establish the same result for the RCC-5 algebra. Another reasoning problem that is not as \logical" as the above problems is that of nding a polynomially-sized representation from which models can be extracted in polynomial time [Golumbic and Shamir, 1993]. This thesis is mainly concerned with the satis ability problem. However, it is important to note that although the above two problems are equivalent for the full algebras, this does not necessarily hold for restricted subsets of the algebras (see e.g. the end of the discussion section in paper II). It was early recognised that computing even with moderately expressive temporal formalisms such as Allen's interval algebra is di cult (the same holds for spatial formalisms); in particular, the polynomial-time algorithm originally proposed by Allen [1983] was designed to be incomplete in order to make the computation reasonably fast. However, this is a very dangerous way of attacking computational obstacles, unless the algorithm is accompanied with some kind of guarantee, like a class for which it is correct. This was not the case for Allen's algorithm. The computational problems inspired several researchers to apply methods from the eld of complexity theory to temporal reasoning (and recently, also to spatial reasoning), since the eld addresses precisely the correctness of problems related to how di cult they are to compute. Results obtained from this approach include the following: It is a polynomial-time problem to check satis ability (and entailment) for the point algebra, i.e. where point variables are related by the relations <, >, , , = and 6= [Ladkin and Maddux, 1994]. 2That is, one problem can be solved in polynomial time using at most a polynomial number of calls to the other, and vice versa. 3 Satis ability for Allen's algebra is NP-complete [Vilain and Kautz, 1986], i.e.