5 research outputs found

    A spatio-temporalisation of ALC(D) and its translation into alternating automata augmented with spatial constraints

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    The aim of this work is to provide a family of qualitative theories for spatial change in general, and for motion of spatial scenes in particular. To achieve this, we consider a spatio-temporalisation MTALC(Dx), of the well-known ALC(D) family of Description Logics (DLs) with a concrete domain: the MTALC(Dx) concepts are interpreted over infinite k-ary Sigma-trees, with the nodes standing for time points, and Sigma including, additionally to its uses in classical k-ary Sigma-trees, the description of the snapshot of an n-object spatial scene of interest; the roles split into m+n immediate-successor (accessibility) relations, which are serial, irreflexive and antisymmetric, and of which m are general, not necessarily functional, the other n functional; the concrete domain Dx is generated by an RCC8-like spatial Relation Algebra (RA) x, and is used to guide the change by imposing spatial constraints on objects of the "followed" spatial scene, eventually at different time points of the input trees. In order to capture the expressiveness of most modal temporal logics encountered in the literature, we introduce weakly cyclic Terminological Boxes (TBoxes) of MTALC(Dx), whose axioms capture the decreasing property of modal temporal operators. We show the important result that satisfiability of an MTALC(Dx) concept with respect to a weakly cyclic TBox can be reduced to the emptiness problem of a Buchi weak alternating automaton augmented with spatial constraints. In another work, complementary to this one, also submitted to this conference, we thoroughly investigate Buchi automata augmented with spatial constraints, and provide, in particular, a translation of an alternating into a nondeterministic, and an effective decision procedure for the emptiness problem of the latter.Comment: See footnote 1 on the first page of the paper. arXiv admin note: substantial text overlap with arXiv:cs/0307040 and text overlap with arXiv:2002.1151

    Buchi automata augmented with spatial constraints: simulating an alternating with a nondeterministic and deciding the emptiness problem for the latter

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    The aim of this work is to thoroughly investigate Buchi automata augmented with spatial constraints. The input trees of such an automaton are infinite k-ary Sigma-trees, with the nodes standing for time points, and Sigma including, additionally to its uses in classical k-ary Sigma-trees, the description of the snapshot of an n-object spatial scene of interest. The constraints, from an RCC8-like spatial Relation Algebra (RA) x, are used to impose spatial constraints on objects of the spatial scene, eventually at different nodes of the input trees. We show that a Buchi alternating automaton augmented with spatial constraints can be simulated with a classical Buchi nondeterministic automaton of the same type, augmented with spatial constraints. We then provide a nondeterministic doubly depth-first polynomial space algorithm for the emptiness problem of the latter automaton. Our main motivation came from another work, also submitted to this conference, which defines a spatio-temporalisation of the well-known family ALC(D) of description logics with a concrete domain: together, the two works provide an effective solution to the satisfiability problem of a concept of the spatio-temporalisation with respect to a weakly cyclic TBox.Comment: See footnote 1 on the first page of the paper. arXiv admin note: substantial text overlap with arXiv:cs/030704

    Integrating cardinal direction relations and other orientation relations in Qualitative Spatial Reasoning

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    We propose a calculus integrating two calculi well-known in Qualitative Spatial Reasoning (QSR): Frank's projection-based cardinal direction calculus, and a coarser version of Freksa's relative orientation calculus. An original constraint propagation procedure is presented, which implements the interaction between the two integrated calculi. The importance of taking into account the interaction is shown with a real example providing an inconsistent knowledge base, whose inconsistency (a) cannot be detected by reasoning separately about each of the two components of the knowledge, just because, taken separately, each is consistent, but (b) is detected by the proposed algorithm, thanks to the interaction knowledge propagated from each of the two compnents to the other.Comment: Includes new material, such as a section on the use of the work in the concrete domain of the ALC(D) spatio-temporalisation defined in http://arXiv.org/abs/cs.AI/030704

    A ternary Relation Algebra of directed lines

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    We define a ternary Relation Algebra (RA) of relative position relations on two-dimensional directed lines (d-lines for short). A d-line has two degrees of freedom (DFs): a rotational DF (RDF), and a translational DF (TDF). The representation of the RDF of a d-line will be handled by an RA of 2D orientations, CYC_t, known in the literature. A second algebra, TA_t, which will handle the TDF of a d-line, will be defined. The two algebras, CYC_t and TA_t, will constitute, respectively, the translational and the rotational components of the RA, PA_t, of relative position relations on d-lines: the PA_t atoms will consist of those pairs of a TA_t atom and a CYC_t atom that are compatible. We present in detail the RA PA_t, with its converse table, its rotation table and its composition tables. We show that a (polynomial) constraint propagation algorithm, known in the literature, is complete for a subset of PA_t relations including almost all of the atomic relations. We will discuss the application scope of the RA, which includes incidence geometry, GIS (Geographic Information Systems), shape representation, localisation in (multi-)robot navigation, and the representation of motion prepositions in NLP (Natural Language Processing). We then compare the RA to existing ones, such as an algebra for reasoning about rectangles parallel to the axes of an (orthogonal) coordinate system, a ``spatial Odyssey'' of Allen's interval algebra, and an algebra for reasoning about 2D segments.Comment: 60 pages. Submitted. Technical report mentioned in "Report-no" below is an earlier version of the work, and its title differs slightly (Reasoning about relative position of directed lines as a ternary Relation Algebra (RA): presentation of the RA and of its use in the concrete domain of an ALC(D)-like description logic

    Bridging the gap between modal temporal logics and constraint-based QSR as an ALC(D) spatio-temporalisation with weakly cyclic TBoxes

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    The aim of this work is to provide a family of qualitative theories for spatial change in general, and for motion of spatial scenes in particular. To achieve this, we consider a spatio-temporalisation MTALC(D_x), of the well-known ALC(D) family of Description Logics (DLs) with a concrete domainan. In particular, the concrete domain D_x is generated by a qualitative spatial Relation Algebra (RA) x. We show the important result that satisfiability of an MTALC(D_x) concept with respect to a weakly cyclic TBox is decidable in nondeterministic exponential time, by reducing it to the emptiness problem of a weak alternating automaton augmented with spatial constraints, which we show to remain decidable, although the accepting condition of a run involves, additionally to the standard case, consistency of a CSP (Constraint Satisfaction Problem) potentially infinite. The result provides an effective tableaux-like satisfiability procedure which is discussed.Comment: 58 pages, 7 figures (I have only split each of Figures 1, 2 and 3, as it appears in the first version, into two figures, so that the number of figures is now 7 instead of the original 4 -it is expected that the reader will find the look of the paper better
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