Analyzing Qualitative Spatio-Temporal Calculi using Algebraic Geometry

Abstract

Qualitative spatial reasoning is based on calculi which comprise relations and operation tables that encode operations like relation composition. Designing a calculus involves determining these tables and analyzing reasoning properties—a demanding task that is susceptible to errors if performed manually. This paper is concerned with automating computation of operation tables and analysis of qualitative calculi over real-valued domains like the plane ². We present an approach to specify qualitative relations using polynomial equations that allows methods from algebraic geometry to be applied. This paper shows how reasoning with qualitative relations can be posed algebraically and demonstrates algebraic reasoning using Gröbner base analysis. We evaluate this approach and describe our implementation, which is freely available as part of the spatial reasoning toolbox SparQ.     

Qualitative Spatial Reasoning for Topological Map Learning

Abstract

In this paper we investigate the application of qualitative spatial reasoning methods for learning the topological map of an unknown environment. We develop a topological mapping framework that achieves robustness against ambiguity in the available information by tracking all possible graph hypotheses simultaneously. We then exploit spatial reasoning to reduce the space of possible hypotheses. The considered constraints are qualitative direction information and the assumption that the map is planar. We investigate the effects of absolute and relative direction information using two different spatial calculi and combine the approach with a real mapping system based on Voronoi graphs.     

The cognitive adequacy of Allen's interval calculus for qualitative spatial representation and reasoning

Qualitative spatial reasoning (QSR) is often claimed to be cognitively more plausible than conventional numerical approaches to spatial reasoning, because it copes with the indeterminacy of spatial data and allows inferences based on incomplete spatial knowledge. The paper reports experimental results concerning the cognitive adequacy of an important approach used in QSR, namely the spatial interpretation of the interval calculus introduced by Allen (1983). Knauff, Rauh and Schlieder (1995) distinguished between the conceptual and inferential cognitive adequacy of Allen's interval calculus. The former refers to the thirteen base relations as a representational system and the latter to the compositions of these relations as a tool for reasoning. The results of two memory experiments on conceptual adequacy show that people use ordinal information similar to the interval relations when representing and remembering spatial arrangements. Furthermore, symmetry transformations on the interval relations were found to be responsible for most of the errors, whereas conceptualneighborhood theory did not appear to correspond to cognitively relevant concepts. Inferential adequacy was investigated by two reasoning experiments and the results show that in inference tasks where the number of possible interval relations for the composition is more than one, subjects ignore numerous possibilities and interindividually prefer the same relations. Reorientations and transpositions operating on the relations seem to be important for reasoning performance as well, whereas conceptual neighborhood did not appear to affect the difficulty of reasoning tasks based on the interval relations.     

Reasoning about Depth and Motion from an Observer's Viewpoint

ABSTRACT

The goal of this paper is to present a logic-based formalism for representing knowledge about objects in space and their movements, and show how this knowledge could be built up from the viewpoint of an observer immersed in a dynamic world. In this paper space is represented using functions that extract attributes of depth, size and distance from snapshots of the world. These attributes compose a novel spatial reasoning system named Depth Profile Calculus (DPC). Transitions between qualitative relations involving these attributes are represented by an extension of this calculus called Dynamic Depth Profile Calculus (DDPC). We argue that knowledge about objects in the world could be built up via a process of abduction on DDPC relations.     

The Space within Fisherman's Folly: Playing with a Puzzle in Mereotopology

Abstract

In this paper we propose a spatial ontology for reasoning about holes, rigid objects and a string, taking a classical puzzle as a motivating example. In this ontology the domain is composed of spatial regions whereby a theory about holes is defined over a mereotopological basis. Within this theory we define a data structure, named chain, that facilitates a clear and efficient representation of the puzzle states and its solution.     

Localization,Exploration, and Navigation Based on Qualitative Angle Information

Abstract

Three of the major problems in building autonomous mobile robots are localization, exploration, and navigation. This paper investigates how well different qualitative methods based on angle information, most of them originally invented for representation of spatial knowledge, are suited for addressing these problems. It combines results from discrete and computational geometry with methods from qualitative spatial reasoning, gaining some new insights on the complexity of robot navigation. It turns out that essentially only with panoramas (special roundviews) the qualitative localization problem can be solved in a satisfactory manner. The exploration problem (qualitative map building), remains difficult for all considered approaches.     

Approximate qualitative spatial reasoning

Qualitative relations between spatial regions play an importantrole in the representation and manipulation of spatial knowledge.The RCC5 and RCC8 systems of relations,used in the Region-Connection Calculus, are of fundamentalimportance. These two systems deal with ideal regions havingprecisely determined location. However,in many practical examples of spatial reasoning,regions are represented by finite approximations rather than known precisely.Approximations may be given by describing how a regionrelates to cells forming a partition of the space underconsideration. Although the RCC5 and RCC8 systems have beengeneralized to ``egg-yolk' regions, in order to modelcertain types of vagueness, their extension to regionsapproximated in this way has not been discussed before.This paper presents two methods, the syntactic and the semantic, by which the RCC5 and RCC8 systemsmay be defined for approximate regions. The syntactic uses algebraicoperations on approximate regions which generalize operations on preciseregions. The semantic method makes use of the set of preciseregions which could be the intended interpretation of anapproximate region. Relationships between these two methods arediscussed in detail.alternative to navigation training with a map.     

The Psychological Validity of Qualitative Spatial Reasoning in One Dimension

One of the central questions of spatial reasoning research is whether the underlying processes are inherently visual, spatial, or logical. We applied the dual task interference paradigm to spatial reasoning problems in one dimension, using Allen's interval calculus, in order to make progress towards resolving this argument. Our results indicate that spatial reasoning with interval relations is largely based on the construction and inspection of qualitative spatial representations, or mental models, while no evidence for logical proofs of derivations or the involvement of visual representations and processes was found.     

On Embedding a Qualitative Representation in a Two-Dimensional Plane

Abstract

This paper discusses embedding in a two-dimensional plane a symbolic representation for spatial data using the simple objects, points (P), lines (L), circuits (C), and areas (A). We have proposed PLCA as a new framework for a qualitative spatial reasoning. In a PLCA expression, the entire figure is represented in a form in which all the objects are related. We investigate the conditions for two-dimensional realizability of a PLCA expression, and derive the relation that the numbers of objects in a PLCA expression should have. In this process, we use the well-known Euler's formula. We also give an algorithm for drawing the figure of the PLCA expression that satisfies this condition in a two-dimensional plane and prove its correctness. The algorithm generates a quantitative expression from qualitative expression.     

Expert Algorithmic and Imagistic Problem Solving Strategies in Advanced Chemistry

Abstract

Visualization and imagistic reasoning appear central to expert practice in science; however, expert use of these strategies on authentic tasks has not been examined in detail. This study documents how science experts use both algorithms and imagistic reasoning to solve problems. Using protocol analysis, we report expert chemists' preferential use of algorithms for solving spatial problems and imagistic reasoning for deducing spatial transformations. We observed experts employ algorithms to solve the majority of spatial tasks while reserving imagistic strategies to solve a class of tasks that required translating between representations. Strategy used varied widely among experts and tasks.     

Qualitative Spatial Reasoning in Interpreting Text and Narrative

Abstract

Simple natural language texts and narratives often raise problems in commonsense spatial knowledge and reasoning of surprising logical complexity and geometric richness. In this article, I consider a dozen short texts—five taken from literature, the remainder contrived as illustrations—and discuss the spatial reasoning involved in understanding them. I conclude by summarizing their common features, and by tentatively drawing some morals for research in this area.     

Mereological inheritance

In recent investigations of Spatial Reasoning, SpatialKnowledge Representation, and Geographic Information Systems, asignificant effort has been spent by many scholars of the areaabout the problem of representing properties of spatial objects bymeans of logical theories.An analogous effort has been the problem of analyzingthe qualitative relations which can be established between twospace regions. These investigations have led to a generalframework of the field known by the term ``mereo-topology'.Though both the above mentioned approaches have proved to besuccessful in the investigation of formal and practically relevantaspects of spatial objects, no attempt has been carried out in the direction of integrating the approaches and looking at the relationshipbetween a general logical theory of space and mereo-topology from an analytical point of view, in particular for exploiting thecombinatorial behaviour of such an integrated model.This paper intends to fill the gap and analyze the behaviour ofspatial formulae of a logical theory of space as objects which canbe classified based on the behaviour they exhibit with respect tothe parts and supertparts of the regions where they are true. Wename these categories of behaviourinheritance modalities.     

基于模糊SDG模型的航天器故障诊断方法研究

基于符号有向图(SDG)的故障诊断具有良好的完备性、易于解释性,但是其不足为 分辨 率差,因此,提出了基于模糊SDG模型和模糊推理相结合的半定量故障诊断方法。将节点 变 量变为模糊变量使节点承载了更多的定量信息,节点间的定性关系通过模糊关系矩阵来表达 ,通过模糊推理判断相容支路找出故障源候选集合,通过部件故障概率和传播故障权重对候 选故障源进行故障可能性的排序。最后应用该方法建立了某卫星一次电源系统的诊断模型, 并进行了故障诊断仿真,结果表明该方法有效地提高了诊断的分辨率,适用于航天器在轨 故障诊断。     

Space,Language and Ontology: A Response to Davis

Abstract

Ernest Davis' article “Qualititative Spatial Reasoning in Interpreting Text and Narrative” discusses challenges that the interpretation of natural language appears to raise for the formalization of commonsense spatial reasoning. Davis finds these to be of “surprising logical complexity,” but also “erratic” in that they do not show a logical structuring of the problem space that could guide productive research. In this response I argue that much of the apparent lack of structure Davis laments is due to the very style of formal modeling he pursues. By augmenting logical considerations with substantial input from other disciplines and by adopting a heterogeneous and modular approach to formalization, I suggest that the problem space is by no means as ill-structured as Davis presents it.     

Spatio-Temporal Ontologies and Attention

ABSTRACT

This paper presents a new proposal for the design of spatio-temporal ontologies which has its origin in cognitively motivated spatial semantics. It is shown that selective attention not only plays a central role in the characterization of spatial relations but that the representation of attentional aspects also leads to the possibility of defining an ontological upper structure which systematically covers both the spatial and the temporal domain.     

Spatial Thinking Across the College Curriculum: A Report on a Specialist Meeting

This report presents findings from a specialist meeting of spatially-minded researchers and administrators from education and industry to consider prospects for introducing courses and curricula on spatial thinking in higher education. More than 40 participants explored the rationale for expanding student exposure to concepts, tools, and applications of spatial reasoning across a range of science, engineering, and humanities disciplines. The focus was on what we know and what we need to know to make the case for space, underscoring basic research on what is meant by spatial thinking and on variations in the spatial reasoning skills required in different domains of knowledge. The need for rigorous assessments of learning outcomes associated with different approaches to teaching spatial thinking was emphasized.     

A Neuro-Cognitive Theory of Deductive Relational Reasoning with Mental Models and Visual Images

Abstract

Many neuro-imaging studies have provided evidence that the parietal cortex plays a key role in reasoning based on mental models, which are supposed to be of abstract spatial nature. However, these studies have also shown concurrent activation in vision-related cortical areas which have often been interpreted as evidence for the role of visual mental imagery in reasoning. The aim of the paper is to resolve the inconsistencies in the previous literature on reasoning and imagery and to develop a neurally and cognitively plausible theory of human relational reasoning. The main assumption is that visual brain areas are only involved if the problem information is easy to visualize and when this information must be processed and maintained in visual working memory. A regular reasoning process, however, does not involve visual images but more abstract spatial representations—spatial mental models—held in parietal cortices. Only these spatial representations are crucial for the genuine reasoning processes.