What is Spatial Logic?

Aiello, Marco; Pratt-Hartmann, Ian; Benthem, Johan van

doi:10.1007/978-1-4020-5587-4_1

Cited by 79 publications

(141 citation statements)

References 4 publications

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“…The binding force of Boolean connectives also as usual, i.e., negation (¬) has a stronger binding force than conjunction (∧) or disjunction (∨) or implication (→) or bi-implication (↔). And spatial operators ݅ (݅ ∈ [1,9]) has the same binding force as negation (¬). Therefore, if ‫ܮ‬ ‫ܣܫ‬ does not involve other symbols like ݅ (݅ ∈ [1,9]), then ܵ ‫ܣܫ‬ has nothing to do only to describe general proposition independent space location.…”

Section: Syntaxmentioning

confidence: 99%

“…And spatial operators ݅ (݅ ∈ [1,9]) has the same binding force as negation (¬). Therefore, if ‫ܮ‬ ‫ܣܫ‬ does not involve other symbols like ݅ (݅ ∈ [1,9]), then ܵ ‫ܣܫ‬ has nothing to do only to describe general proposition independent space location. The ‫ܮ‬ ‫ܣܫ‬ not only describes the relationship between image points, but also can express the proposition of relationship between any two points in the image.…”

Section: Syntaxmentioning

confidence: 99%

“…By the current conventional methods, the image analysis based on computer is usually done with the following procedures: input high quality image, image segmentation, feature extraction, converting features into quantitative expression, image description, image understanding, and so on [7,8]. However, in this paper, we don't take a usual measure to analyse image but spatial logic that is a special technique in topology and modal logic [9,10]. We establish a corresponding semantic model and axiomatic system (notation: ܵ ‫ܣܫ‬ ), and then realize the image analysis in formal reasoning method.…”

Section: Introductionmentioning

confidence: 99%

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A reasoning system for image analysis

Gao

Zhou

2016

MATEC Web Conf.

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Abstract.For image analysis in computer, the traditional approach is extracting and transcoding features after image segmentation. However, in this paper, we present a different way to analyze image. We adopt spatial logic technology to establish a reasoning system with corresponding semantic model, and prove its soundness and completeness, and then realize the image analysis in formal way. And it can be applied in artificial intelligence. This is a new attempt and also a challenging approach.

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Section: Syntaxmentioning

confidence: 99%

Section: Syntaxmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A reasoning system for image analysis

Gao

Zhou

2016

MATEC Web Conf.

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“…Knowledge representation and reasoning about space may be formally interpreted within diverse frameworks such as: (a) geometric reasoning & constructive (solid) geometry [20]; (b) relational algebraic semantics of 'qualitative spatial calculi' [25]; and (c) by axiomatically constructed formal systems of mereotopology and mereogeometry [1]. Independent of formal semantics, commonsense spatio-linguistic abstractions offer a humancentred and cognitively adequate mechanism for logic-based automated reasoning about spatio-temporal information [4].…”

Section: Introductionmentioning

confidence: 99%

Encoding Relative Orientation and Mereotopology Relations with Geometric Constraints in CLP(QS)

Schultz

Bhatt

2015

Proceedings of the LQMR 2015 Workshop

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Abstract-One approach for encoding the semantics of spatial relations within a declarative programming framework is by systems of polynomial constraints. However, solving such constraints is computationally intractable in general (i.e. the theory of realclosed fields), and thus far the proposed symbolic algebraic methods do not scale to real world problems. In this paper we address this intractability by investigating the use of constructive geometric constraint solving (in combination with numerical optimisation) within the context of constraint logic programming over qualitative spaces, CLP(QS). We present novel geometric encodings for relative orientation and mereotopology relations and show that our encodings are significantly more robust than other proposed approaches for directly encoding inequalities, due to our encodings being based on a standard, well known set of relations encoded as quadratic equations. Our encodings are general (i.e. not implementation specific) and can thus be directly employed in all standard geometric constraint solvers, particularly solvers that are prominent within the Computer Aided Design and Manufacturing communities. We empirically evaluate our approach on a range of benchmark problems from the Qualitative Spatial Reasoning community, and show that our method outperforms other symbolic algebraic approaches to spatial reasoning by orders of magnitude on those benchmark problems (such as Cylindrical Algebraic Decomposition).

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“…Recently the modal logics of space have began to draw considerable interest from logicians and computer scientists. See, e.g., [1]. Much of the interest seems to stem from the perceived use of modal logics for qualitative reasoning about spatial relations between objects, and the potential applications in computer science and knowledge representation.…”

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confidence: 99%

The modal logic of affine planes is not finitely axiomatisable

Hodkinson

Hussain

2008

J. symb. log.

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Abstract. We consider a modal language for affine planes, with two sorts of formulas (for points and lines) and three modal boxes. To evaluate formulas, we regard an affine plane as a Kripke frame with two sorts (points and lines) and three modal accessibility relations, namely the point-line and line-point incidence relations and the parallelism relation between lines. We show that the modal logic of affine planes in this language is not finitely axiomatisable. §1. Introduction. Recently the modal logics of space have began to draw considerable interest from logicians and computer scientists. See, e.g., [1]. Much of the interest seems to stem from the perceived use of modal logics for qualitative reasoning about spatial relations between objects, and the potential applications in computer science and knowledge representation.In this paper, we are concerned with the modal logics of projective and affine planes. In [2], geometries of points and lines were viewed as Kripke frames, the domain of each frame being the point-line incidence relation itself (i.e., the set of pairs (s, l ) where s is a point on a line l ). A completeness theorem for 'incidence geometries' was proved, using a non-orthodox 'irreflexivity' inference rule, and extensions to projective and affine geometries were considered.In [13], Venema viewed projective planes in a somewhat more straightforward way, as Kripke frames with two sorts (points and lines), and two modal accessibility relations (incidence between points and lines and between lines and points). He formulated a corresponding modal language with two sorts of formulas -point formulas (evaluated at points) and line formulas (at lines). He then presented a finite set of axioms, essentially expressing that the two accessibility relations are the converses of each other; every point lies on at least one line; any two points lie on at least one common line; and the duals of these two properties obtained by exchanging points and lines. The inference rules were orthodox: modus ponens, (well-sorted) substitution, and universal generalisation for each of the two sorts. Venema proved that the system is (strongly) sound and complete for projective planes. He also proved that the problem of determining whether a given formula is

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What is Spatial Logic?

Cited by 79 publications

References 4 publications

A reasoning system for image analysis

A reasoning system for image analysis

Encoding Relative Orientation and Mereotopology Relations with Geometric Constraints in CLP(QS)

The modal logic of affine planes is not finitely axiomatisable

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