Relation algebras and their application in temporal and spatial reasoning

Abstract Qualitative temporal and spatial reasoning is in many cases based on binary relations such as before, after, starts, contains, contact, part of, and others derived from these by relational operators. The calculus of relation algebras is an equational formalism; it tells us which relations must exist, given several basic operations, such as Boolean operations on relations, relational composition and converse. Each equation in the calculus corresponds to a theorem, and, for a situation where there are only nitely many relations, one can construct a composition table which can serve as a look up table for the relations involved. Since the calculus handles relations, no knowledge about the concrete geometrical objects is necessary. In this sense, relational calculus is pointless. Relation algebras were introduced into temporal reasoning by Allen [1] and into spatial reasoning by Egenhofer and Sharm

Region Connection Calculus: Composition Tables and Constraint Satisfaction Problems

Qualitative spatial reasoning (QSR) is an important field of AI that deals with qualitative aspects of spatial entities. Regions and their relationships are described in qualitative terms instead of numerical values. This approach models human based reasoning about such entities closer than other approaches. Any relationships between regions that we encounter in our daily life situations are normally formulated in natural language. For example, one can outline one's room plan to an expert by indicating which rooms should be connected to each other. Mereotopology as an area of QSR combines mereology, topology and algebraic methods. As mereotopology plays an important role in region based theories of space, our focus is on one of the most widely referenced formalisms for QSR, the region connection calculus (RCC). RCC is a first order theory based on a primitive connectedness relation, which is a binary symmetric relation satisfying some additional properties. By using this relation we can define a set of basic binary relations which have the property of being jointly exhaustive and pairwise disjoint (JEPD), which means that between any two spatial entities exactly one of the basic relations hold. Basic reasoning can now be done by using the composition operation on relations whose results are stored in a composition table. Relation algebras (RAs) have become a main entity for spatial reasoning in the area of QSR. These algebras are based on equational reasoning which can be used to derive further relations between regions in a certain situation. Any of those algebras describe the relation between regions up to a certain degree of detail. In this thesis we will use the method of splitting atoms in a RA in order to reproduce known algebras such as RCC15 and RCC25 systematically and to generate new algebras, and hence a more detailed description of regions, beyond RCC25

Spatial reasoning with RCC8 and connectedness constraints in Euclidean spaces

The language RCC8 is a widely-studied formalism for describing topological arrangements of spatial regions. The variables of this language range over the collection of non-empty, regular closed sets of n-dimensional Euclidean space, here denoted RC+(R^n), and its non-logical primitives allow us to specify how the interiors, exteriors and boundaries of these sets intersect. The key question is the satisfiability problem: given a finite set of atomic RCC8-constraints in m variables, determine whether there exists an m-tuple of elements of RC+(R^n) satisfying them. These problems are known to coincide for all n ≥ 1, so that RCC8-satisfiability is independent of dimension. This common satisfiability problem is NLogSpace-complete. Unfortunately, RCC8 lacks the means to say that a spatial region comprises a ‘single piece’, and the present article investigates what happens when this facility is added. We consider two extensions of RCC8: RCC8c, in which we can state that a region is connected, and RCC8c0, in which we can instead state that a region has a connected interior. The satisfiability problems for both these languages are easily seen to depend on the dimension n, for n ≤ 3. Furthermore, in the case of RCC8c0, we show that there exist finite sets of constraints that are satisfiable over RC+(R^2), but only by ‘wild’ regions having no possible physical meaning. This prompts us to consider interpretations over the more restrictive domain of non-empty, regular closed, polyhedral sets, RCP+(R^n). We show that (a) the satisfiability problems for RCC8c (equivalently, RCC8c0) over RC+(R) and RCP+(R) are distinct and both NP-complete; (b) the satisfiability problems for RCC8c over RC+(R^2) and RCP+(R^2) are identical and NP-complete; (c) the satisfiability problems for RCC8c0 over RC+(R^2) and RCP+(R^2) are distinct, and the latter is NP-complete. Decidability of the satisfiability problem for RCC8c0 over RC+(R^2) is open. For n ≥ 3, RCC8c and RCC8c0 are not interestingly different from RCC8. We finish by answering the following question: given that a set of RCC8c- or RCC8c0-constraints is satisfiable over RC+(R^n) or RCP+(R^n), how complex is the simplest satisfying assignment? In particular, we exhibit, for both languages, a sequence of constraints Φ_n, satisfiable over RCP+(R^2), such that the size of Φ_n grows polynomially in n, while the smallest configuration of polygons satisfying Φ_n cuts the plane into a number of pieces that grows exponentially. We further show that, over RC+(R^2), RCC8c again requires exponentially large satisfying diagrams, while RCC8c0 can force regions in satisfying configurations to have infinitely many components

Matching disparate geospatial datasets and validating matches using spatial logic

In recent years, the emergence and development of crowd-sourced geospatial data has provided challenges and opportunities to national mapping agencies as well as commercial mapping organisations. Crowd-sourced data involves non-specialists in data collection, sharing and maintenance. Compared to authoritative geospatial data, which is collected by surveyors or other geodata professionals, crowd-sourced data is less accurate and less structured, but often provides richer user-based information and reflects real world changes more quickly at a much lower cost. In order to maximize the synergistic use of authoritative and crowd-sourced geospatial data, this research investigates the problem of how to establish and validate correspondences (matches) between spatial features from disparate geospatial datasets. To reason about and validate matches between spatial features, a series of new qualitative spatial logics was developed. Their soundness, completeness, decidability and complexity theorems were proved for models based on a metric space. A software tool `MatchMaps' was developed, which generates matches using location and lexical information, and verifies consistency of matches using reasoning in description logic and qualitative spatial logic. MatchMaps was evaluated by the author and experts from Ordnance Survey, the national mapping agency of Great Britain. In experiments, it achieved high precision and recall, as well as reduced human effort. The methodology developed and implemented in MatchMaps has a wider application than matching authoritative and crowd-sourced data and could be applied wherever it is necessary to match two geospatial datasets of vector data

Matching disparate geospatial datasets and validating matches using spatial logic

In recent years, the emergence and development of crowd-sourced geospatial data has provided challenges and opportunities to national mapping agencies as well as commercial mapping organisations. Crowd-sourced data involves non-specialists in data collection, sharing and maintenance. Compared to authoritative geospatial data, which is collected by surveyors or other geodata professionals, crowd-sourced data is less accurate and less structured, but often provides richer user-based information and reflects real world changes more quickly at a much lower cost. In order to maximize the synergistic use of authoritative and crowd-sourced geospatial data, this research investigates the problem of how to establish and validate correspondences (matches) between spatial features from disparate geospatial datasets. To reason about and validate matches between spatial features, a series of new qualitative spatial logics was developed. Their soundness, completeness, decidability and complexity theorems were proved for models based on a metric space. A software tool `MatchMaps' was developed, which generates matches using location and lexical information, and verifies consistency of matches using reasoning in description logic and qualitative spatial logic. MatchMaps was evaluated by the author and experts from Ordnance Survey, the national mapping agency of Great Britain. In experiments, it achieved high precision and recall, as well as reduced human effort. The methodology developed and implemented in MatchMaps has a wider application than matching authoritative and crowd-sourced data and could be applied wherever it is necessary to match two geospatial datasets of vector data

Extensionality of the RCC8 Composition Table

This paper is mainly concerned with the RCC8 composition table entailed by the Region Connection Calculus (RCC), a well-known formalism for Qualitative Spatial Reasoning. This table has been independently generated by Egenhofer in the context of Geographic Information Systems. It has been known for some time that the table is not extensional for each RCC model. This paper however shows that the Egenhofer model is indeed an extensional one for the RCC8 composition table. Moreover this model is the maximal extensional one for the RCC8 composition table in a sense

Deriving extensional spatial composition tables

Spatial composition tables are fundamental tools for the realisation of qualitative spatial reasoning techniques. Studying the properties of these tables in relation to the spatial calculi they are based on is essential for understanding the applicability of these calculi and how they can be extended and generalised. An extensional interpretation of a spatial composition table is an important property that has been studied in the literature and is used to determine the validity of the table for the models it is proposed for. It provides means for consistency checking of ground sets of relations and for addressing spatial constraint satisfaction problems. Furthermore, two general conditions that can be used to test for extensionality of spatial composition tables are proposed and applied to the RCC8 composition table to verify the allowable models in this calculus