Qualitative reasoning with directional relations

AbstractQualitative spatial reasoning (QSR) pursues a symbolic approach to reasoning about a spatial domain. Qualitative calculi are defined to capture domain properties in relation operations, granting a relation algebraic approach to reasoning. QSR has two primary goals: providing a symbolic model for human common-sense level of reasoning and providing efficient means for reasoning. In this paper, we dismantle the hope for efficient reasoning about directional information in infinite spatial domains by showing that it is inherently hard to decide consistency of a set of constraints that represents positions in the plane by specifying directions from reference objects. We assume that these reference objects are not fixed but only constrained through directional relations themselves. Known QSR reasoning methods fail to handle this information

The Effect of Verbalizer, Spatial and Object Visualizer Cognitive Style to Students’ Algebraic Thinking: Insights from Empirical Study Focusing on Middle School Students

Everyone has a preferred approach and different habits in organizing and representing the information they received. It is known as cognitive style. The theory of verbalizer and visualizer cognitive style developed into three dimensions; verbalizer, object visualizer and spatial visualizer. Algebraic thinking consists of four components, namely generalized arithmetic (GA), functional thinking (FT), modeling languages (M), and algebraic proof (AP). Previous research has shown that spatial visualizers have a more significant relationship with numbers sense and algebraic reasoning compared to verbalizers and object visualizers. However, there is no empirical research that shows the effect of three-dimensional cognitive style on the 4 components of algebra, especially for students in junior high school. The subjects of this study were 149 8th grade students in Gresik and Sidoarjo regencies, Indonesia. This study uses Structural Equation Modeling (SEM) data analysis with AMOS 24. The instrument used is an adaptation of the Object- Spatial Imagery and Verbal Questionnaire (OSIVQ) Blazhenkova & Kozhevnikov (2009) and an algebraic thinking test instrument. The results of this study indicate that three-dimensional cognitive style affects the components of algebraic thinking, especially in mathematical modeling (M). In addition, functional thinking (FT) has the greatest contribution to algebraic thinking skills compared to GA, M and AP

Qualitative Spatial Reasoning about Relative Orientation --- A Question of Consistency ---

Abstract. After the emergence of Allen s Interval Algebra Qualitative Spatial Reasoning has evolved into a fruitful field of research in artificial intelligence with possible applications in geographic information systems (GIS) and robot navigation Qualitative Spatial Reasoning abstracts from the detailed metric description of space using rich mathematical theories and restricts its language to a finite, often rather small, set of relations that fulfill certain properties. This approach is often deemed to be cognitively adequate . A major question in qualitative spatial reasoning is whether a description of a spatial situation given as a constraint network is consistent. The problem becomes a hard one since the domain of space (often R2 ) is infinite. In contrast many of the interesting problems for constraint satisfaction have a finite domain on which backtracking methods can be used. But because of the infinity of its domains these methods are generally not applicable to Qualitative Spatial Reasoning. Anyhow the method of path consistency or rather its generalization algebraic closure turned out to be helpful to a certain degree for many qualitative spatial calculi. The problem regarding this method is that it depends on the existence of a composition table, and calculating this table is not an easy task. For example the dipole calculus (operating on oriented dipoles) DRAf has 72 base relations and binary composition, hence its composition table has 5184 entries. Finding all these entries by hand is a hard, long and error-prone task. Finding them using a computer is also not easy, since the semantics of DRAf in the Euclidean Plane, its natural domain, rely on non-linear inequalities. This is not a special problem of the DRAf calculus. In fact, all calculi dealing with relative orientation share the property of having semantics based on non-linear inequalities in the Euclidean plane. This not only makes it hard to find a composition table, it also makes it particularly hard to decide consistency for these calculi. As shown in [79] algebraic closure is always just an approximation to consistency for these calculi, but it is the only method that works fast. Methods like Gröbner reasoning can decide consistency for these calculi but only for small constraint networks. Still finding a composition table for DRAf is a fruitful task, since we can use it analyze the properties of composition based reasoning for such a calculus and it is a starting point for the investigation of the quality of the approximation of consistency for this calculus. We utilize a new approach for calculating the composition table for DRAf using condensed semantics, i.e. the domain of the calculus is compressed in such a way that only finitely many possible configurations need to be investigated. In fact, only the configurations need to be investigated that turn out to represent special characteristics for the placement of three lines in the plane. This method turns out to be highly efficient for calculating the composition table of the calculus. Another method of obtaining a composition table is borrowing it via a suitable morphism. Hence, we investigate morphisms between qualitative spatial calculi. Having the composition table is not the end but rather the beginning of the problem. With that table we can compute algebraically closed refinements of constraint networks, but how meaningful is this process? We know that all constraint networks for which such a refinement does not exist are inconsistent, but what about the rest? In fact, they may be consistent or not. If they are all consistent, then we can be happy, since algebraic closure would decide consistency for the calculus at hand. We investigate LR, DRAf and DRAfp and show that for all these calculi algebraic closure does not decide consistency. In fact, for the LR calculus algebraic closure is an extremely bad approximation of consistency. For this calculus we introduce a new method for the approximation of consistency based on triangles, that performs far better than algebraic closure. A major weak spot of the field of Qualitative Spatial Reasoning is the area of applications. It is hard to refute the accusation of qualitative spatial calculi having only few applications so far. As a step into the direction of scrutinizing the applicability of these calculi, we examine the performance of DRA and OPRA in the issue of describing and navigating street networks based on local observations. Especially for OPRA we investigate a factorization of the base relations that is deemed cognitively adequate . Whenever possible we use real-world data in these investigations obtained from OpenStreetMap

From research to practice: The case of mathematical reasoning

Mathematical proficiency is a key goal of the Australian Mathematics curriculum. However, international assessments of mathematical literacy suggest that mathematical reasoning and problem solving are areas of difficulty for Australian students. Given the efficacy of teaching informed by quality assessment data, a recent study focused on the development of evidence-based Learning Progressions for Algebraic, Spatial and Statistical Reasoning that can be used to identify where students are in their learning and where they need to go to next. Importantly, they can also be used to generate targeted teaching advice and activities to help teachers progress student learning. This paper explores the processes involved in taking the research to practice

Algebraic Properties of Qualitative Spatio-Temporal Calculi

Qualitative spatial and temporal reasoning is based on so-called qualitative calculi. Algebraic properties of these calculi have several implications on reasoning algorithms. But what exactly is a qualitative calculus? And to which extent do the qualitative calculi proposed meet these demands? The literature provides various answers to the first question but only few facts about the second. In this paper we identify the minimal requirements to binary spatio-temporal calculi and we discuss the relevance of the according axioms for representation and reasoning. We also analyze existing qualitative calculi and provide a classification involving different notions of a relation algebra.Comment: COSIT 2013 paper including supplementary materia

Answer Set Programming Modulo `Space-Time'

We present ASP Modulo `Space-Time', a declarative representational and computational framework to perform commonsense reasoning about regions with both spatial and temporal components. Supported are capabilities for mixed qualitative-quantitative reasoning, consistency checking, and inferring compositions of space-time relations; these capabilities combine and synergise for applications in a range of AI application areas where the processing and interpretation of spatio-temporal data is crucial. The framework and resulting system is the only general KR-based method for declaratively reasoning about the dynamics of `space-time' regions as first-class objects. We present an empirical evaluation (with scalability and robustness results), and include diverse application examples involving interpretation and control tasks

Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning

Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of models. To cope with the lack of explicit set of states in institutions, the proposed abstract logical dual operators are defined in an extension of institutions, the stratified institutions, which take into account the notion of open sentences, the satisfaction of which is parametrized by sets of states. A hint on the potential interest of the proposed framework for spatial reasoning is also provided.Comment: 36 page