Algorithms and Complexity for Temporal and Spatial Formalisms

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. it is very unlikely that there exists a polynomial algorithm for the problem. Satis ability for the point-interval algebra de ned by Vilain [1982], where the only relations are those relating a point with an interval, is NP-complete [Meiri, 1991]. There exists a maximal tractable3 subclass of Allen's algebra, the ORDHorn subclass [Nebel and B urckert, 1995], i.e. satis ability is tractable for that subclass, and is tractable for no proper superclass of it. Furthermore, it is the unique maximal subclass containing all basic relations. There exists one maximal tractable subclass in the spatial algebra RCC-5 [Renz, 1996] and one in the spatial algebra RCC-8 [Renz, 1996; Renz and Nebel, 1997]. An important assumption that underlies most of the above results is that variables (taking for example values of temporal intervals or spatial regions) are allowed to be unrelated, and this also holds for the results of this thesis. Relaxing that assumption yields more tractable classes [Golumbic and Shamir, 1993]. An area that is related to temporal reasoning is the area of reasoning about action (see e.g. [Sandewall, 1994] for an introduction), which is the topic of paper V. In addition to treating relations between temporal entities, propositions being true at time points and actions which have been executed are also modelled in this eld, giving rise to the infamous frame problem (the problem of concisely expressing what does not change when it is known what does change) [McCarthy and Hayes, 1969]. However, emphasis of the research in this area has almost exclusively been on modelling, in contrast to the temporal reasoning eld. One attempt to address the computational issues of the area is that of Schwalb et al. [1994], but it cannot be considered satisfactory, since their \solution" is to sacri ce the completeness requirement of making inferences, but providing no other criterion for correctness4. Furthermore, they do not correctly deal with the frame problem. 3The expression \tractable problem" is often used in place of \polynomial-time problem". 4Note that any algorithm for entailment that yields no conclusions at all is sound. 41.2 Contributions This thesis contributes in line with the above results, rst by nding new maximal tractable subclasses for temporal and spatial reasoning, thus providing knowledge about the borderline between tractable and intractable reasoning, and second by exhibiting the rst tractable class for reasoning about action known in the literature (to my knowledge), and also providing some intractability results in this area. In addition, a new tractable class of intutionistic logic is presented; this result was obtained as a by-product of the results on spatial reasoning. 1.3 About the Thesis The structure of the thesis is simple: this introductory chapter describes the context of and summarises the eight papers, which in turn constitute the rest of the thesis. The papers can be read in any order, but I recommend that papers II and III be read before paper IV, and paper VII before paper VIII. All of the papers are intended to be reasonably self-contained, presupposing knowledge about discrete mathematics and some theory of computational complexity. (An good introduction to complexity theory is that of Papadimitriou [1994].) Due to this fact, some papers overlap slightly, but this should hopefully not be too disturbing. Concerning the amount of work put into the papers by the author, it should be noted that the order of the authors re ects this (if indeed it can be quanti ed), in that the rst author of a paper has done most of the research reported therein, and so on for the remaining list of authors. Of course, I am responsible for any errors in the papers. 1.4 Brief Summary of the Papers Paper I It is known in the literature that testing satis ability of a set of point variables over real numbers related with the relations <, >, , , = and 6=, that is, using the point algebra [Vilain, 1982], can be done in polynomial time [Ladkin and Maddux, 1994; van Beek and Cohen, 1990; Gerevini et al., 1993], and that there are several tractable special cases of checking satis ability using Allen's interval algebra [van Beek, 1989; van Beek, 1990; Golumbic and Shamir, 1993; Nebel and B urckert, 1995] 5 (the general case is NP-complete [Vilain and Kautz, 1986]). However, not much is known about the computational properties of the point-interval algebra [Meiri, 1991], where the only relations allowed are those between a point and an interval, except for the general satis ability problem being NP-complete [Meiri, 1991]. Paper I classi es every possible subset of the point-interval algebra into having an NP-complete or polynomial satis ability problem, respectively. It is proved that there are exactly ve maximal tractable subclasses. This represents a step towards obtaining tractable subclasses of Meiri's qualitative algebra, which includes the expressive power of the point algebra, the point-interval algebra and the Allen interval algebra in one formalism. An interesting feature of the method of proof employed, is that it requires computer support for testing approximately 2:44 105 cases. Paper II In Paper I, we could directly attack the problem of nding the complete set of tractable subalgebras of the point-interval algebra. However, this is possible only due to size of the algebra: it contains only 32 relations, making the total number of potential subclasses 232 4:3 109, which would be theoretically possible to enumerate on a computer. Allen's algebra, on the other hand, contains 8192 relations, yielding 28192 102466 subclasses, clearly being impossible to enumerate on today's computers. Therefore, a less systematic approach to nding tractable subclasses could be excused. For the Allen algebra, only one maximal tractable class was previously known, theORD-Horn subclass by Nebel and B urckert [1995], containing 868 relations. This algebra extends all known tractable classes, except for one four-element set found by Golumbic and Shamir [1993]. Paper II expands our knowledge of tractable and maximal tractable subclasses, by de ning twentyone new tractable subclasses whose satis ability problems are solvable in linear time, all of which are considerably larger than the ORD-Horn class, nine of them being maximal tractable, and the remaining ones nonmaximal (these are extended in paper III, but to the price of using an O(n2) time algorithm). Each algebra contains exactly three basic relations, and four of them contain the relation ( ), \before or after", which is needed for expressing the important notion of sequentiality. Also the tractable class by Golumbic and Shamir is subsumed by these algebras. 6Paper III This paper combines two directions in temporal reasoning: the search for tractable subalgebras of Allen's algebra, and providing temporal formalisms with the possibility of expressing not only qualitative temporal information, but also metric information (such as statements like 3 4 t 0:6s 3_x 6= 17). Eight new subclasses of Allen's algebra are de ned, and they are all proved to be maximal tractable. It is proved that all nonmaximal algebras of paper II are included in these algebras. Six of the algebras contain exactly ve basic relations each, and two of them contain three basic relations each. Furthermore, we use the new linear-programming approach to temporal reasoning by Jonsson and Backstrom [1996] in order to provide these algebras with metric temporal information: four of the algebras can have metric information on interval starting points and the remaining ones on interval ending points. It is also proved that we cannot have both, without losing tractability, except for the already known case of Horn DLRs [Jonsson and Backstr om, 1996], which provides the ORD-Horn algebra with metric temporal information. Paper IV This paper sets papers II and III in perspective, by proving a partial classi cation of tractability in Allen's algebra, in contrast to the complete classi cation of paper I. Given the results of papers II and III, an important question is whether the algebras found are just a few out of a multitude of algebras of comparable expressivity. Using intensive computer support (equivalent to 40 CPU weeks on a Sun SPARC 10 station), we show that the answer to this question has to be answered negatively, in the following precise way: any tractable subclass of Allen's algebra that is not a subset of either the ORD-Horn class or one of the algebras presented in papers II and III, cannot contain more than three basic relations, which have to be , b and the converse of b, where b is one of d, o, s or f. Furthermore, there is no remaining tractable subclass containing the relation ( ), required for expression the property of sequentiality. Thus any remaining tractable subclass is necessarily of weak expressivity concerning basic relations and sequentiality. Note that this result also entails a complete classi cation of tractable algebras containing the basic relation . 7 Paper V As mentioned in the introduction, the motivation for investigating the problem of reasoning about time in separation has been that the concept of time is used in a large number of arti cial intelligence applications. Paper V is the only temporal reasoning paper in this thesis which does not consist solely of a temporal component: here we can compute with propositions which are true at time points. The paper achieves two goals in this: rst, a propositional temporal logic is developed, and second, this logic is extended to model the so-called common-sense law of inertia. Logics of this kind are mostly referred to as \logics of action of change", and in modelling power this logic has to be compared to such logics. However, the emphasis of this paper is not the traditional one in this eld, that of modelling, but that of analysing computational complexity: no analysis of that kind has yet been done in the literature for such logics. It is proved that both in the basic temporal logic and in the extension for reasoning about action, the satis ability problems are NP-complete, but perhaps more important, nontrivial tractable classes are found in both, capable of modelling phenomena like concurrent actions, continuous time and conditional results of actions. The expressivity of the temporal component is equal to that of Horn DLRs [Jonsson and Backstr om, 1996], which means that qualitative as well as metric time can be used. One interesting (although not surprising) conclusion to draw from this paper is that sometimes a separation of temporal and non-temporal components is not the best way to attack the computational problems involving time: we obtain the tractable cases by coding the logics into Horn DLRs, and not by treating truth-value propositions and temporal propositions separately. It is not at all clear that a separated approach could handle this as e ciently. Paper VI In this paper, we apply methods from results in temporal reasoning on a related eld, that of spatial reasoning. Also in this eld, formalisms like Allen's algebra, the point-interval algebra and the point algebra have been invented to model qualitative spatial phenomena. In this paper, we take an existing such formalism from the RCC approach to spatial reasoning [Randell and Cohn, 1989; Randell et al., 1992b], in particular the RCC-5 algebra [Bennett, 1994], and classify every subset of it (there are 32 relations in it, so the e ort is comparable to that of paper I) into having a tractable 8or NP-complete satis ability problem. Only one maximal tractable subclass of this algebra was known previously [Renz, 1996], which in turn built on earlier results by Nebel [1995]. In the paper we nd the remaining three maximal tractable classes. Paper VII This paper relates the temporal and spatial reasoning areas: generalising the method of proof employed by Jonsson and Backstr om [1996] for proving that satis ability for Horn-DLRs is tractable, we prove that for a corresponding Horn class of reasoning about nonempty sets, satis ability checking can be done in polynomial time. Then we apply this result to nding a simpler algorithm for the previously known maximal tractable class of the RCC-5 algebra for spatial reasoning [Renz, 1996]. The existing polynomial algorithm involves complicated transformations using modal logic, whereas our algorithm is based on simple set-theoretical manipulations. In addition, by approaching the problem from the more abstract set-theoretical level, it is possible to express statements about spatial relationships that are not expressible in the RCC-5 algebra alone. Furthermore, some topological properties of a tractable subclass of the RCC-5 algebra are investigated, showing that satis ability of that subclass is equivalent to satis ability in three-dimensional space. This complements the results by Grigni et al. [1995], which show that satis ability in two-dimensional space for a restricted version of the same subclass is NP-complete. Paper VIII This paper is somewhat more loosely related to the central topic of the thesis, that of reasoning about temporal and spatial relations, than the rest of the papers. It connects mainly to paper VII, by providing a new tractable class of tractable reasoning about sets. The di erence from paper VII is that we investigate the case when empty sets are also allowed as set variables, and it turns out that the Horn class found in paper VII has an NP-complete satis ability problem under this assumption. A new Horn-like class (being a subset of the class of paper VII) is further de ned, which is proved to have a tractable satis ability problem also when empty sets are allowed. The technique for proving tractability is very similar to that of paper VII, and was originally employed by Jonsson and Backstr om [1996]. Using a connection between topological spaces and intuitionistic logic es9 tablished by Tarski [1956], we use the tractable class to obtain a subclass of intuitionistic logic for which inference is tractable. This result is somewhat related to spatial reasoning: in a more expressive approach to spatial reasoning than the RCC-5 algebra, the RCC-8 algebra [Bennett, 1994], a di erent tractable class of intuitionistic logic was used in order to nd a tractable class [Nebel, 1995] (extended by Renz and Nebel [1997]), and it is possible that this class can be used in a similar way, to nd new tractable subclasses of the RCC-8 algebra. 2 The Papers in Context This section describes a broader perspective of the papers than the one given in Section 1.4. 2.1 Papers I{IV: Temporal Reasoning Outside the area of Arti cial intelligence, much work has been done concerning the modelling of temporal phenomena using various kinds of temporal logics (see e.g. Gabbay et al. [1994] for an overview). However, the computational issues dealt with are usually on the level of establishing decidability or axiomatisability, due to the expressiveness of the logics involved. Therefore, since the emphasis of this thesis is on nding polynomial-time algorithms for reasoning about time, these logics merit less attention here. Within the area of Arti cial Intelligence, early attempts at computing with temporal information are those of Bruce [1972] and Kahn and Gorry [1977]. The former seems to be the rst one de ning what constitute the basic relations of Allen's algebra. Both of these present an implemented system for reasoning about temporal information, but there is no evidence of the existence of correctness proofs for their algorithms (which are not even reported), and at least for the latter, the intended semantics for the formalism employed seems to be unclear. Allen [1983] seems to be one of the rst to take formalisation, computation and correctness seriously, by giving a precise de nition of his algebra of interval relations, providing an algorithm for computing the closure of an interval network (that is, nding the strongest possible implied relation between two interval variables), and showing incompleteness of the proposed algorithm. In Allen's paper, the reason for emphasising the algebraic operations (making it an algebra) on interval relations (composition, intersection 10 and converse of relations) was primarily for the operation of his particular closure algorithm. Interestingly, the property of being an algebra was later exploited for a completely di erent purpose, pioneered by Nebel and B urckert [1995] and frequently used in papers I, II, III, IV and VI of this thesis: it turns out that the satis ability problem for interval networks using only a restricted set S of interval relations is NP-complete if and only if the satis ability problem using the set S0 of interval relations is, where S0 is the least set of relations containing S and being closed under the algebraic operations (this is a di erent kind of closure). Soon, the computational properties of Allen's algebra were investigated by Vilain and Kautz [1986]: the satis ability problem is NP-complete, and so is the problem of computing the closure of an interval network. This means that it is very unlikely that there exist polynomial-time (let alone e cient) algorithms for these problems. Essentially three lines of attack have been pursued in response to this fact: Not solving the intended problem, by designing the algorithms as being incomplete and hoping that they will work for most cases anyway. Allen's original algorithm [Allen, 1983] is an example of this approach. Trying to design fast algorithms for the full algebra, concentrating on the performance on certain randomly-generated instances and on benchmark problems which are naturally occurring in, for instance, molecular biology [Vald es-P erez, 1987; Ladkin and Reinefeld, 1992; van Beek and Manchak, 1996; Mitra and Loganantharaj, 1996; Nebel, 1996]. Of course, these algorithms cannot solve the full problem e ciently, but it appears that improvement is made towards solving instances which are of somewhat practical size. However, the di culty of empirically analysing algorithms (see e.g. [Johnson, 1996]) will probably require more investigation. Restricting the set of relations, proving that the problem for the restricted set of relations is a polynomial-time one. This is a more theoretical approach than the former, since no experimental data is required for establishing results. It can also be argued that it might be appropriate to investigate what e cient algorithms exist before a general-purpose superpolynomial algorithm is employed. The papers of this thesis follow the third approach, which was indeed the rst one to be pursued: Vilain and Kautz [1986] found that when the point 11 algebra [Vilain, 1982] is used (that is, relating time point variables with the relations <, , =, , > and 6=), satis ability can be solved in polynomial time (the algorithm was improved by Gerevini et al. [1993] to a linear-time one), and expressing interval relations in terms of these yields a fragment of Allen's algebra for which computing satis ability is polynomial; the pointisable subalgebra. However, it was not clear whether this subalgebra could be extended further, until Nebel and B urckert [1995] de ned a superset thereof, the ORD-Horn algebra, which furthermore was shown to be maximal tractable, that is, impossible to extend without losing tractability, and also that no other subalgebras than those contained in the ORD-Horn algebra can both be tractable and contain all the basic relations. Also Golumbic and Shamir [1993] de ned several tractable subclasses of Allen's algebra (and provided NP-completeness results) using graph-theoretic methods. One of their tractable classes is not included in the ORD-Horn class (there are also other tractable classes, but these cannot express the relation \unrelated", as discussed above). When time intervals or points are related using Allen's algebra or the point-algebra, only qualitative information can be expressed, that is, we cannot express, for instance, that a time point comes ve seconds after some other time point. In order to remedy this, several di erent formalisms allowing such expressiveness have been developed and analysed [Dechter et al., 1991; Meiri, 1991; Kautz and Ladkin, 1991; Koubarakis, 1992; Lassez and McAloon, 1992]. By the main result of Jonsson and Backstr om [1996], all tractable classes de ned in these papers (except for two by Meiri, involving nonstandard entities such as discrete domains and multi-intervals) are uni ed into a single tractable formalism based on linear programming, that of Horn-DLRs, which also includes the ORD-Horn algebra (this result was independently found by Koubarakis [1996], but using a less transparent algorithm). The contributions of papers I{IV fall mainly within this category of results. Related to papers I and IV is also the complete classi cation by Pe'er and Shamir [1997] of tractable subclasses of a subset of Allen's algebra (using the same kind of macro relations as Golumbic and Shamir [1993]) with a nonstandard semantics where all intervals are required to have the same length. We should also mention that much research is being done on nding special-purpose data structures for making reasoning practically e cient even though a polynomial-time algorithm is already known (note that the requirement of polynomial-time algorithms is not a su cient condition for practical usefulness), mostly for the point-algebra [Ghallab and Alaoui, 1989; 12 Dean and McDermott, 1987; Dorn, 1992; Schrag et al., 1992; Delgrande and Gupta, 1996]. 2.2 Paper V: Reasoning about Action Early in the history of Arti cial Intelligence, McCarthy [1958] introduced logic as a means for representing knowledge. Soon, several problems were discovered with this approach; except for the ine ciencies involved with running a theorem prover for computing consequences of facts [Green, 1969], the representation issues themselves provided some problems, notably the infamous frame problem, identi ed by McCarthy and Hayes [1969]. The frame problem is how to represent what does not change in the world, when it is known that certain actions a ecting the world have been performed, and maybe some additional observations are to be taken into account. Since the emphasis of this thesis is not on modelling but on computing, it serves no purpose to describe the development of these modelling formalisms, but it is interesting to note that after several papers proposing solutions to the frame problem, Hanks and McDermott [1986] pointed out that none of these was correct, using the Yale shooting problem (thereafter a prototypical example for reasoning about action) as a diagnostic example. As a reaction to this, some researchers fortunately started to be more systematic [Sandewall, 1994; Gelfond and Lifschitz, 1993; Kartha and Lifschitz, 1994; Lin and Shoham, 1991], by starting to de ne clearly what problem was to be solved, and under which assumptions the approach would hold as a correct solution to the frame problem. This so-called systematic approach (as opposed to the previously used example-based approach) is followed in paper V, of course, by stating mathematically what problem is to be solved and proving that this is indeed done. In this area, the focus of interest has generally been that of modelling rather than computation, probably by the impact of the complications surrounding the frame problem, and by the fact that formalisms employed are almost always at least as expressive as rst order logic, where computing entailment is known to be undecidable [Boolos and Je rey, 1989]. To our knowledge, no complexity analyses of such formalisms have yet appeared in the literature5. 5Liberatore [1997] has performed a complexity analysis of the language A (which is less expressive than that used in the paper) for reasoning about action [Gelfond and Lifschitz, 1993]; the publication is however unrefereed. 13 2.3 Papers VI-VII: Spatial Reasoning The problem of spatial reasoning is slightly more involved than that of temporal reasoning: whereas the choice of ontological entities in temporal reasoning is relatively obvious (points or intervals; sometimes perhaps multiintervals or sets of time points [Meiri, 1991; Ladkin, 1986a; Ladkin, 1986b]), this is not the case for spatial reasoning. For instance, we have the following aspects to decide on: How many dimensions (two or three6) are to be used? Is the form or orientation of objects to be represented? How can (qualitative) spatial relations be naturally chosen? For a discussion of such issues, see Freksa and Rohrig [1993]. In this thesis we shall only consider spatial reasoning in its form closest related to temporal reasoning formalisms like Allen's algebra, and not the more general (and more imprecise) kind of reasoning as, for instance, na ve physics [Hayes, 1979; Hayes, 1985], which concerns representing knowledge about physical processes without using standard physics. The so-called RCC approach by Randell, Cohn and Cui [Randell and Cohn, 1989; Randell et al., 1992b; Bennett, 1994] is a rst-order formalism for expressing knowledge about spatial relationships, with two limited subformalisms, named RCC-5 and RCC-8. These formalisms are de ned as algebras like the Allen interval algebra, containing ve and eight basic relations, respectively, but of di erent expressive power. Essentially, RCC-5 expresses relations between nonempty sets, whereas RCC-8 instead has sets in a topological space as its ontological primitive, thus distinguishing the border of a region from the rest of it. Until recently, the computational properties of this approach were largely unknown; existing methods were either not known to be complete [Egenhofer, 1991; Hern andez, 1994] or very ine cient [Randell et al., 1992b; Bennett, 1994]. Even computing a transitivity table posed a computational problem [Randell et al., 1992a]! Nebel [1995] presented the rst results on the computational complexity of these formalisms, showing that consistency-checking of the basic relations of RCC-8 can be done in polynomial time, and Grigni et al. [1995] classi ed the complexity of several cases when models are required to be two-dimensional. Renz [1996] extended Nebel's result, providing one maximal tractable subclass of RCC-8 (see also Renz and Nebel [1997]) and also one of RCC-5. Paper VI identi es the remaining tractable subclasses of RCC-5, and paper VII presents an algorithm for reasoning about arbitrary nonempty sets, yielding as a side-e ect a new, simpler algorithm for the maximal tractable RCC-5 subclass of Renz [1996]. 6One-dimensional spatial reasoning seems to collapse to temporal reasoning. 14 Furthermore, we cast some light on the semantical foundations of RCC-5 by showing that for a particular subclass, satis ability is equivalent to satis ability in R3 (see Lemon [1996] for a criticism of the semantics of various spatial logics). 3 Paper VIII: Intuitionistic Logic Intuitionistic logic represents an attempt to formalise the ideas arising from Brouwer's criticism of the foundations of mathematics [Heyting, 1971; Dragalin, 1988]. Not much is known about the computational properties thereof, except for inference being PSPACE-complete in the propositional case [Statman, 1979] (there also exists an O(nlogn)-space procedure [Hudelmaier, 1993]), and the tractable class used by Nebel [1995] to nd a tractable class of the RCC-8 algebra for spatial reasoning. In paper VIII, we improve on this by nding a new tractable subclass of intuitionistic logic, using a technique for reasoning about sets similar to that of paper VII. Furthermore, the results on reasoning about sets are interesting on their own, as are the corresponding results of paper VII. 15 References [AAAI '86, 1986] American Association for Arti cial Intelligence. Proceedings of the 5th (US) National Conference on Arti cial Intelligence (AAAI '86), Philadelphia, PA, USA, August 1986. Morgan Kaufmann. [AAAI '91, 1991] American Association for Arti cial Intelligence. Proceedings of the 9th (US) National Conference on Arti cial Intelligence (AAAI '91), Anaheim, CA, USA, July 1991. AAAI Press/MIT Press. [AAAI '96, 1996] American Association for Arti cial Intelligence. Proceedings of the 13th (US) National Conference on Arti cial Intelligence (AAAI '96), Portland, OR, USA, August 1996. [Allen et al., 1990] James Allen, James Hendler, and Austin Tate, editors. Readings in Planning. Morgan Kaufmann, San Mateo, CA, 1990. [Allen, 1983] James F. Allen. Maintaining knowledge about temporal intervals. Communications of the ACM, 26(11):832{843, 1983. [Bennett, 1994] B. Bennett. Spatial reasoning with propositional logics. In Doyle et al. [1994], pages 165{176. [Boolos and Je rey, 1989] George S. Boolos and Richard C. Je rey. Computability and Logic. Cambridge University Press, 3rd edition, 1989. [Bruce, 1972] Bertram C. Bruce. A model for temporal references and its application in a question answering program. Arti cial Intelligence, 3:1{ 25, 1972. [Dean and McDermott, 1987] Thomas L. Dean and Drew V. McDermott. Temporal data base management. Arti cial Intelligence, 32:1{55, 1987. [Dechter et al., 1991] Rina Dechter, Itay Meiri, and Judea Pearl. Temporal constraint networks. Arti cial Intelligence, 49:61{95, 1991. [Delgrande and Gupta, 1996] James P. Delgrande and Arvind Gupta. A representation for e cient temporal reasoning. In AAAI '96 [1996], pages 381{388. [Dorn, 1992] J urgen Dorn. Temporal reasoning in sequence graphs. In Proceedings of the 10th (US) National Conference on Arti cial Intelligence (AAAI '92), pages 735{740, San Jos e, CA, USA, July 1992. American Association for Arti cial Intelligence. 16 [Doyle and Aiello, 1996] J. Doyle and L. Aiello, editors. Proceedings of the 5th International Conference on Principles of Knowledge Representation and Reasoning (KR '96), Cambridge, MA, USA, October 1996. Morgan Kaufmann. [Doyle et al., 1994] J. Doyle, E. Sandewall, and P. Torasso, editors. Proceedings of the 4th International Conference on Principles of Knowledge Representation and Reasoning (KR '94), Bonn, Germany, May 1994. Morgan Kaufmann. [Dragalin, 1988] A. G. Dragalin. Mathematical Intuitionism: Introduction to Proof Theory, volume 67 of Translations of Mathematical Monographs. American Mathematical Society, 1988. [Drakengren and Bjareland, 1997] Thomas Drakengren and Marcus Bjareland. Reasoning about action in polynomial time. In Pollack [1997]. [Drakengren and Jonsson, 1996] Thomas Drakengren and Peter Jonsson. Maximal tractable subclasses of Allen's interval algebra: Preliminary report. In AAAI '96 [1996], pages 389{394. [Drakengren and Jonsson, 1997a] Thomas Drakengren and Peter Jonsson. Eight maximal tractable subclasses of Allen's algebra with metric time. Journal of Arti cial Intelligence Research, 7:25{45, 1997. [Drakengren and Jonsson, 1997b] Thomas Drakengren and Peter Jonsson. Towards a complete classi cation of tractability in Allen's algebra. In Pollack [1997]. [Drakengren and Jonsson, 1997c] Thomas Drakengren and Peter Jonsson. Twenty-one large tractable subclasses of Allen's algebra. Arti cial Intelligence, 93(1{2):297{319, 1997. [Egenhofer, 1991] M. J. Egenhofer. Reasoning about binary topological relations. In O. Gunther and H. J. Schek, editors, Advances in Spatial Databases, pages 143{160. Springer-Verlag, 1991. [Freksa and Rohrig, 1993] Christian Freksa and Ralf Rohrig. Dimensions of qualitative spatial reasoning. In P. Carret and M. G. Singh, editors, Qualitative Reasoning in Decision Technologies, pages 483{492, Barcelona, June 1993. 17 [Gabbay et al., 1994] Dov M. Gabbay, Ian Hodgkinson, and Mark Reynolds. Temporal Logic: Mathematical Foundations and Computational Aspects. Clarendon Press, Oxford, 1994. [Gelfond and Lifschitz, 1993] Michael Gelfond and Vladimir Lifschitz. Representing action and change by logic programs. Journal of Logic Programming, 17:301{321, 1993. [Gerevini et al., 1993] Alfonso Gerevini, Lenhart Schubert, and Stephanie Schae er. Temporal reasoning in Timegraph I{II. SIGART Bulletin, 4(3):21{25, 1993. [Ghallab and Alaoui, 1989] Malik Ghallab and Amine Mounir Alaoui. Managing e ciently temporal relations through indexed spanning trees. In Sridharan [1989], pages 1297{1303. [Golumbic and Shamir, 1993] Martin Charles Golumbic and Ron Shamir. Complexity and algorithms for reasoning about time: A graph-theoretic approach. Journal of the ACM, 40(5):1108{1133, 1993. [Green, 1969] Cordell Green. Application of theorem proving to planning. In Donald E. Walker and Lewis M. Norton, editors, Proceedings of the 1st International Joint Conference on Arti cial Intelligence (IJCAI '69), pages 219{239, Washington, DC, USA, May 1969. William Kaufmann. Reprinted in Allen et al [1990], pages 67{87. [Grigni et al., 1995] M. Grigni, D. Papadias, and C. Papadimitriou. Topological inference. In Chris Mellish, editor, Proceedings of the 14th International Joint Conference on Arti cial Intelligence (IJCAI '95), pages 901{906, Montr eal, PQ, Canada, August 1995. Morgan Kaufmann. [Hanks and McDermott, 1986] Steve Hanks and Drew McDermott. Default reasoning, nonmonotonic logics, and the frame problem. In AAAI '86 [1986], pages 328{333. [Hayes, 1979] Patrick J. Hayes. The na ve physics manifesto. In D. Michie, editor, Expert Systems in the Micro Electronic Age, pages 242{270. Edinburgh University Press, 1979. [Hayes, 1985] Patrick J. Hayes. The second naive physics manifesto. In Jerry R. Hobbs and Robert C. Moore, editors, Formal Theories of the Commonsense World, pages 1{36. Ablex, 1985. 18 [Hern andez, 1994] D. Hern andez. Qualitative representation of spatial knowledge. In Lecture Notes in Arti cial Intelligence, volume 804. Springer-Verlag, Berlin, Heidelberg, New York, 1994. [Heyting, 1971] Arend Heyting. Intuitionism: An Introduction. NorthHolland, 3rd edition, 1971. [Hudelmaier, 1993] J. Hudelmaier. An O(nlogn)-space decision procedure for intuitionistic propositional logic. Journal of Logic and Computation, 3(1):63{75, 1993. [Johnson, 1996] David S. Johnson. A theoretician's guide to the experimental analysis of algorithms. AT&T Labs Research. Available from http://www.research.att.com/ dsj/papers/exper.ps, 1996. [Jonsson and Backstrom, 1996] Peter Jonsson and Christer Backstr om. A linear-programming approach to temporal reasoning. In AAAI '96 [1996], pages 1235{1240. [Jonsson and Drakengren, 1997] Peter Jonsson and Thomas Drakengren. A complete classi cation of tractability in RCC-5. Journal of Arti cial Intelligence Research, 6:211{221, 1997. [Jonsson et al., 1996] Peter Jonsson, Thomas Drakengren, and Christer Backstrom. Tractable subclasses of the point-interval algebra: A complete classi cation. In Doyle and Aiello [1996], pages 352{363. [Kahn and Gorry, 1977] K. Kahn and G. Gorry. Mechanizing temporal knowledge. Arti cial Intelligence, 9:87{108, 1977. [Kartha and Lifschitz, 1994] G. Neelakantan Kartha and Vladimir Lifschitz. Actions with indirect e ects (preliminary report). In Doyle et al. [1994]. [Kautz and Ladkin, 1991] Henry Kautz and Peter Ladkin. Integrating metric and temporal qualitative temporal reasoning. In AAAI '91 [1991], pages 241{246. [Koubarakis, 1992] Manolis Koubarakis. Dense time and temporal constraints with 6=. In Swartout and Nebel [1992], pages 24{35. [Koubarakis, 1996] M. Koubarakis. Tractable disjunctions of linear constraints. In Proceedings of the 2nd International Conference on Principles and Practice for Constraint Programming, pages 297{307, Cambridge, MA, August 1996. 19 [Ladkin and Maddux, 1994] Peter B. Ladkin and Roger Maddux. On binary constraint networks. Journal of the ACM, 41(3):435{469, May 1994. [Ladkin and Reinefeld, 1992] Peter B. Ladkin and Alexander Reinefeld. Effective solution of qualitative interval constraint problems. Arti cial Intelligence, 57:105{124, 1992. [Ladkin, 1986a] Peter Ladkin. Primitives and units for time speci cation. In AAAI '86 [1986]. [Ladkin, 1986b] Peter Ladkin. Time representation: A taxonomy of interval relations. In AAAI '86 [1986]. [Lassez and McAloon, 1992] Jean-Louis Lassez and Ken McAloon. A canonical form for generalized linear constraints. Journal of Symbolic Computation, 13:1{14, 1992. [Lemon, 1996] O. Lemon. Semantical foundations of spatial logics. In Doyle and Aiello [1996], pages 212{219. [Liberatore, 1997] P. Liberatore. The complexity of the language A. Link oping Electronic Articles in Computer and Information Science, 2(2), July 1997. URL: http://www.ep.liu.se/ea/cis/1997/006. [Lin and Shoham, 1991] Fangzhen Lin and Yoav Shoham. Provably correct theories of actions: Preliminary report. In AAAI '91 [1991]. [McCarthy and Hayes, 1969] J. McCarthy and P.J. Hayes. Some philosophical problems from the standpoint of Arti cial Intelligence. Machine Intelligence, 4:463{502, 1969. [McCarthy, 1958] John McCarthy. Programs with common sense. In Mechanisation of Thought Processes, Proceedings of the Symposium of the National Physics Laboratory, pages 77{84, London, U.K., 1958. Her Majesty's Stationery O ce. [Meiri, 1991] Itay Meiri. Combining qualitative and quantitative constraints in temporal reasoning. In AAAI '91 [1991], pages 260{267. [Mitra and Loganantharaj, 1996] Debasis Mitra and Rasiah Loganantharaj. Experimenting with a temporal constraint propagation algorithm. Applied Intelligence, 6:39{48, 1996. 20 [Nebel and B urckert, 1995] Bernhard Nebel and Hans-J urgen B urckert. Reasoning about temporal relations: A maximal tractable subclass of Allen's interval algebra. Journal of the ACM, 42(1):43{66, 1995. [Nebel, 1995] Bernhard Nebel. Computational properties of qualitative spatial reasoning: First results. In I. Wachsmuth, C-R. Rollinger, and W. Brauer, editors, KI-95: Advances in Arti cial Intelligence, pages 233{ 244, Bielefeld, Germany, September 1995. Springer-Verlag. [Nebel, 1996] Bernhard Nebel. Solving hard qualitative temporal reasoning problems: Evaluating the e ciency of using the ORD-Horn class. In Wolfgang Wahlster, editor, Proceedings of the 12th European Conference on Arti cial Intelligence (ECAI '96), Budapest, Hungary, August 1996. Wiley. [Papadimitriou, 1994] Christos H. Papadimitriou. Computational Complexity. Addison Wesley, Reading, MA, 1994. [Pe'er and Shamir, 1997] Itsik Pe'er and Ron Shamir. Satis ability problems on intervals and unit intervals. Theoretical Computer Science, 175:349{372, 1997. [Pollack, 1997] Martha E. Pollack, editor. Proceedings of the 15th International Joint Conference on Arti cial Intelligence (IJCAI '97), Nagoya, Japan, August 1997. Morgan Kaufmann. [Randell and Cohn, 1989] D. A. Randell and A. G. Cohn. Modelling topological and metrical properties of physical processes. In Ronald J. Brachman, Hector J. Levesque, and Raymond Reiter, editors, Proceedings of the 1st International Conference on Principles of Knowledge Representation and Reasoning (KR '89), pages 55{66, Toronto, ON, Canada, May 1989. Morgan Kaufmann. [Randell et al., 1992a] D. A. Randell, Z. Cui, and A. G. Cohn. Computing transitivity tables: A challenge for automated theorem provers. In Proceedings of the 11th International Conference on Automated Deduction, Saratoga Springs, NY, USA, 1992. Springer-Verlag. [Randell et al., 1992b] D. A. Randell, Z. Cui, and A. G. Cohn. A spatial logic based on regions and connection. In Swartout and Nebel [1992], pages 165{176. 21 [Renz and Nebel, 1997] Jochen Renz and Bernhard Nebel. On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connected calculus. In Pollack [1997]. [Renz, 1996] Jochen Renz. Qualitatives raumliches Schlie en: Berechnungseigenschaften und e ziente Algorithmen. Master's thesis, Fakultat f ur Informatik, Universitat Ulm, 1996. Available from http://www.informatik.uni-freiburg.de/ sppraum. [Sandewall, 1994] Erik Sandewall. Features and Fluents. Oxford University Press, 1994. [Schrag et al., 1992] Robert Schrag, Mark Boddy, and Jim Carcio ni. Managing disjunciton for practical temporal reasoning. In Swartout and Nebel [1992], pages 36{46. [Schwalb et al., 1994] E. Schwalb, K. Kask, and R. Dechter. Temporal reasoning with constraints on uents and events. In Proceedings of the 12th (US) National Conference on Arti cial Intelligence (AAAI '94), Seattle, WA, USA, July{August 1994. American Association for Arti cial Intelligence. [Sridharan, 1989] N. S. Sridharan, editor. Proceedings of the 11th International Joint Conference on Arti cial Intelligence (IJCAI '89), Detroit, MI, USA, August 1989. Morgan Kaufmann. [Statman, 1979] R. Statman. Intuitionistic logic is polynomial-space complete. Theoretical Computer Science, 9(1):67{72, 1979. [Swartout and Nebel, 1992] Bill Swartout and Bernhard Nebel, editors. Proceedings of the 3rd International Conference on Principles of Knowledge Representation and Reasoning (KR '92), Cambridge, MA, USA, October 1992. Morgan Kaufmann. [Tarski, 1956] Alfred Tarski. Sentential calculus and topology. In Logic, Semantics, Metamathematics, chapter 17. Oxford Clarendon Press, 1956. [Vald es-P erez, 1987] Ra ul Vald es-P erez. The satis ability of temporal constraint networks. In Proceedings of the 6th (US) National Conference on Arti cial Intelligence (AAAI '87), Seattle, WA, USA, July 1987. American Association for Arti cial Intelligence. 22 [van Beek and Cohen, 1990] Peter van Beek and Robin Cohen. Exact and approximate reasoning about temporal relations. Computational Intelligence, 6(3):132{144, 1990. [van Beek and Manchak, 1996] Peter van Beek and Dennis W. Manchak. The design and experimental analysis of algorithms for temporal reasoning. Journal of Arti cial Intelligence Research, 4:1{18, 1996. [van Beek, 1989] Peter van Beek. Approximation algorithms for temporal reasoning. In Sridharan [1989], pages 1291{1296. [van Beek, 1990] Peter van Beek. Reasoning about qualitative temporal information. In Proceedings of the 8th (US) National Conference on Arti cial Intelligence (AAAI '90), pages 728{734, Boston, MA, USA, August 1990. American Association for Arti cial Intelligence, MIT Press. [Vila, 1994] Llu s Vila. A survey on temporal reasoning in arti cial intelligence. AI Communications, 7(1):4{28, 1994. [Vilain and Kautz, 1986] M. Vilain and H. Kautz. Constraint propagation algorithms for temporal reasoning. In AAAI '86 [1986], pages 373{381. [Vilain, 1982] Marc B. Vilain. A system for reasoning about time. In Proceedings of the 2nd (US) National Conference on Arti cial Intelligence (AAAI '82), pages 197{201, Pittsburgh, PA, USA, August 1982. American Association for Arti cial Intelligence.