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

Framework development for providing accessibility to qualitative spatial calculi

Dissertation submitted in partial fulfillment of the requirements for the Degree of Master of Science in Geospatial Technologies.Qualitative spatial reasoning deals with knowledge about an infinite spatial domain using a finite set of qualitative relations without using numerical computation. Qualitative knowledge is relative knowledge where we obtain the knowledge on the basis of comparison of features with in the object domain rather then using some external scales. Reasoning is an intellectual facility by which, conclusions are drawn from premises and is present in our everyday interaction with the geographical world. The kind of reasoning that human being relies on is based on commonsense knowledge in everyday situations. During the last decades a multitude of formal calculi over spatial relations have been proposed by focusing on different aspects of space like topology, orientation and distance. Qualitative spatial reasoning engines like SparQ and GQR represents space and reasoning about the space based on qualitative spatial relations and bring qualitative reasoning closer to the geographic applications. Their relations and certain operations defined in qualitative calculi use to infer new knowledge on different aspects of space. Today GIS does not support common-sense reasoning due to limitation for how to formalize spatial inferences. It is important to focus on common sense geographic reasoning, reasoning as it is performed by human. Human perceive and represents geographic information qualitatively, the integration of reasoner with spatial application enables GIS users to represent and extract geographic information qualitatively using human understandable query language. In this thesis, I designed and developed common API framework using platform independent software like XML and JAVA that used to integrate qualitative spatial reasoning engines (SparQ) with GIS application. SparQ is set of modules that structured to provides different reasoning services. SparQ supports command line instructions and it has a specific syntax as set of commands. The developed API provides interface between GIS application and reasoning engine. It establishes connection with reasoner over TCP/IP, takes XML format queries as input from GIS application and converts into SparQ module specific syntax. Similarly it extracts given result, converts it into defined XML format and passes it to GIS application over the same TCP/IP connection. The most challenging part of thesis was SparQ syntax analysis for inputs and their outputs. Each module in Sparq takes module specific query syntax and generates results in multiple syntaxes like; error, simple result and result with comments. Reasoner supports both binary and ternary calculi. The input query syntax for binary-calculi is different for ternary-calculi in the terms of constraint-networks. Based on analysis I, identified commonalities between input query syntaxes for both binary and ternary calculi and designed XML structures for them. Similarly I generalized SparQ results into five major categories and designed XML structures. For ternary-calculi, I considered constraint-reasoning module and their specific operations and designed XML structure for both of their inputs and outputs

Reasoning about topological and cardinal direction relations between 2-dimensional spatial objects

Increasing the expressiveness of qualitative spatial calculi is an essential step towards meeting the requirements of applications. This can be achieved by combining existing calculi in a way that we can express spatial information using relations from multiple calculi. The great challenge is to develop reasoning algorithms that are correct and complete when reasoning over the combined information. Previous work has mainly studied cases where the interaction between the combined calculi was small, or where one of the two calculi was very simple. In this paper we tackle the important combination of topological and directional information for extended spatial objects. We combine some of the best known calculi in qualitative spatial reasoning, the RCC8 algebra for representing topological information, and the Rectangle Algebra (RA) and the Cardinal Direction Calculus (CDC) for directional information. We consider two different interpretations of the RCC8 algebra, one uses a weak connectedness relation, the other uses a strong connectedness relation. In both interpretations, we show that reasoning with topological and directional information is decidable and remains in NP. Our computational complexity results unveil the significant differences between RA and CDC, and that between weak and strong RCC8 models. Take the combination of basic RCC8 and basic CDC constraints as an example: we show that the consistency problem is in P only when we use the strong RCC8 algebra and explicitly know the corresponding basic RA constraints

Reasoning about topological and cardinal direction relations between 2-dimensional spatial objects

Increasing the expressiveness of qualitative spatial calculi is an essential step towards meeting the requirements of applications. This can be achieved by combining existing calculi in a way that we can express spatial information using relations from multiple calculi. The great challenge is to develop reasoning algorithms that are correct and complete when reasoning over the combined information. Previous work has mainly studied cases where the interaction between the combined calculi was small, or where one of the two calculi was very simple. In this paper we tackle the important combination of topological and directional information for extended spatial objects. We combine some of the best known calculi in qualitative spatial reasoning, the RCC8 algebra for representing topological information, and the Rectangle Algebra (RA) and the Cardinal Direction Calculus (CDC) for directional information. We consider two different interpretations of the RCC8 algebra, one uses a weak connectedness relation, the other uses a strong connectedness relation. In both interpretations, we show that reasoning with topological and directional information is decidable and remains in NP. Our computational complexity results unveil the significant differences between RA and CDC, and that between weak and strong RCC8 models. Take the combination of basic RCC8 and basic CDC constraints as an example: we show that the consistency problem is in P only when we use the strong RCC8 algebra and explicitly know the corresponding basic RA constraints

Connecting qualitative spatial and temporal representations by propositional closure

This paper establishes new relationships between existing qualitative spatial and temporal representations. Qualitative spatial and temporal representation (QSTR) is concerned with abstractions of infinite spatial and temporal domains, which represent configurations of objects using a finite vocabulary of relations, also called a qualitative calculus. Classically, reasoning in QSTR is based on constraints. An important task is to identify decision procedures that are able to handle constraints from a single calculus or from several calculi. In particular the latter aspect is a longstanding challenge due to the multitude of calculi proposed. In this paper we consider propositional closures of qualitative constraints which enable progress with respect to the longstanding challenge. Propositional closure allows one to establish several translations between distinct calculi. This enables joint reasoning and provides new insights into computational complexity of individual calculi. We conclude that the study of propositional languages instead of previously considered purely relational languages is a viable research direction for QSTR leading to expressive formalisms and practical algorithms

Qualitative constraint satisfaction problems : algorithms, computational complexity, and extended framework

University of Technology, Sydney. Faculty of Engineering and Information Technology.Qualitative Spatial and Temporal Reasoning (QSTR) is a subfield of artificial intelligence that represents and reasons with spatial/temporal knowledge in a qualitative way. In the past three decades, researchers have proposed dozens of relational models (known as qualitative calculi), including, among others, Point Algebra (PA) and Interval Algebra (IA) for temporal knowledge, Cardinal Relation Algebra (CRA) and Cardinal Direction Calculus (CDC) for directional spatial knowledge, and the Region Connection Calculus RCC-5/RCC-8 for topological spatial knowledge. Relations are used in qualitative calculi for representing spatial/temporal information (e.g. Germany is to the east of France) and constraints (e.g. the to-be-established landfill should be disjoint from any lake). The reasoning tasks in QSTR are formalised via the qualitative constraint satisfaction problem (QCSP). As the central reasoning problem in QCSP, the consistency problem (which decides the consistency of a number of constraints in certain qualitative calculi) has been extensively investigated in the literature. For PA, IA, CRA, and RCC-5/RCC-8, the consistency problem can be solved by composition-based reasoning. For CDC, however, composition-based reasoning is incomplete, and the consistency problem in CDC remains challenging. Previous works in QCSP assume that qualitative constraints only concern completely unknown entities. Therefore, constraints about landmarks (i.e., fixed entities) cannot be properly expressed. This has significantly restricted the usefulness of QSTR in real-world applications. The main contributions of this thesis are as follows. (i) The composition-based method is one of the most important reasoning methods in QSTR. This thesis designs a semi-automatic algorithm for generating composition tables for general qualitative calculi. This provides a partial answer to the challenge proposed by Cohn in 1995. (ii) Schockaert et al. (2008) extend the RCC models interpreted in Euclidean topologies to the fuzzy context and show that composition-based reasoning is sufficient to solve fuzzy QCSP, where 31 composition rules are used. This thesis first shows that only six of the 31 composition rules are necessary, and then introduces a method which consistently fuzzifies any classical RCC models. This thesis also proposes a polynomial algorithm for realizing solutions of consistent fuzzy RCC constraints. (iii) Composition-based reasoning is incomplete for solving QCSP over the CDC. This thesis provides a cubic algorithm which for the first time solves the consistency problem of complete basic CDC networks, and further shows that the problem becomes NP-complete if the networks are allowed to be incomplete. This draws a sharp boundary between the tractable and intractable subclasses of the CDC. (iv) This thesis proposes a more general and more expressive QCSP framework, in which a variable is allowed to be a landmark (i.e., a fixed object), or to be chosen among several landmarks. The computational complexity of the consistency problems in the new framework is then investigated, covering all qualitative calculi mentioned above. For basic networks, the consistency problem remains tractable for Point Algebra, but becomes NP-complete for all the remaining qualitative calculi. A special case in which a variable is either a landmark or is totally unknown has also been studied. (v) A qualitative network is minimal if it cannot be refined without changing its solution set. Unlike the assumptions in the literature, this thesis shows that computing a solution of minimal networks is NP-complete for (partially ordered) PA, CRA, IA, and RCC-5/RCC-8. As a by-product, it has also been proved that determining the minimality of networks in these qualitative calculi is NP-complete

On Distributive Subalgebras of Qualitative Spatial and Temporal Calculi

Qualitative calculi play a central role in representing and reasoning about qualitative spatial and temporal knowledge. This paper studies distributive subalgebras of qualitative calculi, which are subalgebras in which (weak) composition distributives over nonempty intersections. It has been proven for RCC5 and RCC8 that path consistent constraint network over a distributive subalgebra is always minimal and globally consistent (in the sense of strong $n$-consistency) in a qualitative sense. The well-known subclass of convex interval relations provides one such an example of distributive subalgebras. This paper first gives a characterisation of distributive subalgebras, which states that the intersection of a set of $n\geq 3$ relations in the subalgebra is nonempty if and only if the intersection of every two of these relations is nonempty. We further compute and generate all maximal distributive subalgebras for Point Algebra, Interval Algebra, RCC5 and RCC8, Cardinal Relation Algebra, and Rectangle Algebra. Lastly, we establish two nice properties which will play an important role in efficient reasoning with constraint networks involving a large number of variables.Comment: Adding proof of Theorem 2 to appendi

Qualitative constraint satisfaction problems: An extended framework with landmarks

Dealing with spatial and temporal knowledge is an indispensable part of almost all aspects of human activity. The qualitative approach to spatial and temporal reasoning, known as Qualitative Spatial and Temporal Reasoning (QSTR), typically represents spatial/temporal knowledge in terms of qualitative relations (e.g., to the east of, after), and reasons with spatial/temporal knowledge by solving qualitative constraints. When formulating qualitative constraint satisfaction problems (CSPs), it is usually assumed that each variable could be "here, there and everywhere".1 Practical applications such as urban planning, however, often require a variable to take its value from a certain finite domain, i.e. it is required to be 'here or there, but not everywhere'. Entities in such a finite domain often act as reference objects and are called "landmarks" in this paper. The paper extends the classical framework of qualitative CSPs by allowing variables to take values from finite domains. The computational complexity of the consistency problem in this extended framework is examined for the five most important qualitative calculi, viz. Point Algebra, Interval Algebra, Cardinal Relation Algebra, RCC5, and RCC8. We show that all these consistency problems remain in NP and provide, under practical assumptions, efficient algorithms for solving basic constraints involving landmarks for all these calculi. © 2013 Elsevier B.V