hapredwa.pro/54.php Formalization of conceptual spatial models bridges the gap between conceptual models and their successful implementation for the following reasons: 1 it helps to understand the underlying theory and structure, i. An example illustrates an application of the hierarchical spatial reasoning to model wayfinding. It shows how a particular type of hierarchy is used for spatial reasoning section 4.
We discuss the formalization method and tool briefly, as sufficient background information is available in the cited literature e. In section 5 we give conclusions and suggest directions for future work. Hierarchy is one of the most common forms of organizing and structuring complex systems where a system is subdivided in smaller subsystems, and further subdivision of subsystems can be recursively repeated as long as the subdivision makes sense Koestler Such a hierarchical structure can be represented as a tree whose leaves represent the system components.
A hierarchy consists of levels each representing a subset of objects selected from the initial set according to some criteria. The number of levels determines the depth of the hierarchy. The number of objects on each level determines its span, and in turn, the span of the hierarchy tree. Each of these components has local and global properties. Local properties are properties of a level per se , determining objects which belong to a particular hierarchical level. A local property depends on a context in which a hierarchy is introduced e.
They describe general structure of a hierarchy such as:. Koestler further suggests a rough classification of hierarchies into structural and functional hierarchies. The first emphasizes the spatial aspect of a system a structure of the spatial domain whereas the second focuses more on a process as another aspect. These classes tend to overlap even though the aspects are complementary aspects of an indivisible spatio-temporal process. Therefore, it is often useful and necessary to focus on just one of these aspects in order to successfully solve a task at hand. Hierarchical Spatial Reasoning HRS is a method of spatial problem solving that uses hierarchy to infer spatial information and draw conclusions Car Hierarchy, as an abstraction mechanism, is used to decrease the cognitive load in two major ways: to reduce the complexity of the problem domain by cutting down the problem space, or it can be used as a divide and conquer method where a problem is divided in smaller parts and then solved.
Both strategies result in an efficient problem solving because only the necessary data is processed, usually in a shorter period of time.
One of the outcomes of this research is the requirements for hierarchical spatial reasoning, which state how to perform such a reasoning. The following must be given Car and Frank :. For example, Shapiro, Weixman and Nir proposed a graph with an imposed hierarchical structure, and an adequate, simple heuristic shortest path search.
Therefore, the last two requirements have been added to the set, and can be understood as quality control of hierarchical reasoning. A number of cognitive studies have shown the evidence that people may reason hierarchically in geographic space Hirtle and Jonides ; McNamara, J. Hardy and Hirtle People often use approximate reasoning either to compensate for missing information or to simplify reasoning by using only information that is needed Fotheringam To account for such reasoning, hierarchical algorithms, which approximate the strict results, are also of interest.
Thus, we can say that hierarchical spatial reasoning follows patterns of human cognition, as it computes approximations. The rising question then is What is a good enough solution? This is similar to the economics term satisficing which is used when something is "good enough". The idea in economics is that people might look around for alternatives until they are merely satisfied, which is not quite the same as looking harder for the "optimal" alternative Simon Thus, we can talk about satisficing versus optimizing solutions.
An example addressing this issue is found in Timpf and Frank showing that HSR can compute increasingly better results and stops the computation when the good enough result is achieved. Considering the classification of hierarchies suggested by Koestler, we distinguish 2 types of spatial hierarchies Table 1 : functional spatial hierarchies are based on a task-subtask division and are useful for modeling spatial processes e. Hierarchization methods. At least two different methods to derive a spatial hierarchy emerged from analysis of hierarchies for spatial reasoning:.
In both cases operations have been defined, describing criteria for selecting spatial objects into subsets, each of which represents different hierarchical levels. Structural Hierarchies. In a structural hierarchy spatial objects are usually organized in levels such that each level contains the same type of objects interacting in the same way among themselves. That is, ontology is the same for each level. The levels differ only in the degree of detail.
It is meaningful to consider hierarchies of geometric primitives, i. Typical examples for such hierarchies are quadtrees Samet , HTINs De Floriani and Puppo , triacons or quaternary triangular meshes Dutton , graph-subgraph subdivisions, or fractals Batty and Longley Structural hierarchy often refers to the granularity of the representation.
Hernandez , for example, discusses a model for qualitative spatial reasoning using levels with different granularity for projection and orientation e. Cohn considers the problem of representing the shape of a region, qualitatively, within a logical theory of space.
He uses just two primitive notions, that of two regions connecting, and the convex hull of a region, to distinguish various concave shapes. By applying the technique recursively to the inside of a region, Cohn achieves a hierarchical representation at varying levels of granularity. Glasgow uses hierarchization in spatial planning. In these examples reasoning becomes simpler on levels with fewer details.
Research on HSR involves also formal description of functions or operations that form a hierarchy, that is, methods to transform a nonhierarchical structure into a hierarchical one see section 3. For example, Frank and Timpf examine the idea of the intelligent zoom which implies that more details about objects become visible with the increasing scale and restricted display view. To achieve this zoom operation ability, they employ a hierarchical tree-like structure to store scale-dependent representations of objects from cartographic maps.
Timpf further investigates how to link map objects at different levels of detail that represent the same real-world objects and proposes a hierarchical data structure that supports and describes the behavior of map objects at multiple levels. The hierarchical structure is a composition of three different types of hierarchies defined according to the functions of generalization, aggregation and filtering.
The formal specification is available as the Gofer code. Another example comes from cognitive linguistics and is given by Habel ; ; Habel proposes a hierarchical system of granularity levels as an adequate means of mental representations of space and time. He suggests that different levels of granularity of spatial memory are connected by specific operations such as embedding , which allow for building up increasingly refined representations. For example, mental images in spatial memory are connected through embedding, which allows for changing of their scale.
In the case of time, the refinement operation splits the events. In these examples a hierarchical system has proven useful as a finite representation of a spatio-temporal continuum. Formal logic is used to formally describe the system. Functional hierarchies are based on task decomposition. A subtask represents one level of hierarchy. Usually each of the subtasks requires different types of spatial objects with different interaction among them. Consequently, each level is expected to have different ontology.
In case of functional hierarchies, a hierarchization method usually starts with the more general task and ends at the specific task level. This corresponds to the top-down hierarchization method. A multilevel highway navigation Timpf et al. Its conceptual model consists of the Planning level, Instructional level, and Driving level each describing a subtask in planning and navigating a journey. The most general level is the planning task that involves a highway network consisting of highways, places and interchanges. The next lower level provides sufficient instruction information for navigating such as entrances and exits.
The lowest level describes the objects and actions necessary to drive a vehicle according to the instructions. The full formal specification is available in Timpf Each of these levels has different ontology, describing different levels of information needed to move around.
For example, places, paths and regions on the topological level, linked by topological relations such as connectivity, order and containment, create a topological map of space. The same types of objects on the metrical level, linked by metric relations such as distance, direction, or shape, provide metric information for movement such as amount of rotation and travel distance. Transitions of objects and actions from one level to objects and actions on another level are described by a set of rules.
In this section we give an example in which a method of hierarchical spatial reasoning has been applied to model wayfinding in a hierarchically structured network Car and Frank ; Car First we describe the design steps, and the give a brief overview of the modeling and formalization methods used. The modeling method used here is known as conceptual modeling. Conceptual modeling is modeling of real-world situations on a higher level of abstraction, before a detailed logical and physical design takes place Brodie, Mylopoulos and Schmidt Conceptual models provide the description of space that is closer to human conceptualization and its semantics.
They communicate the formalized ideas of space and as such enhance communication between domain expert, system designer and end-user. Research on conceptual modeling, particularly developing suitable methods for specification of large, complex models requires further efforts. A formalization method introduced here uses object orientation as a design method, formal specifications, and a functional programming language as a specification and prototyping Car and Frank The object-oriented approach has been accepted as appropriate to modeling complex data structures and hierarchically organized knowledge domains.
Software engineering proposes object orientation as a design method Khoshafian and Abnous because there is a strong correspondence between the idea to model both the structure of an object and its behavior and the way humans build classes of similar objects according to their experience Lakoff The main concern of the introduction of the object-oriented method for modeling spatial data in GIS applications is the attempt to capture more semantics than the existing data models do Egenhofer and Frank ; Worboys In a relational data model, for example, all data is represented in tabular form, whereas the object-oriented model considers complex objects as individual units with certain behavior see, for example, Herring ; Mainguenaud Formal specifications support object-oriented design because of their ability to describe the structure of objects and operations applicable to them.
The formal specifications method combines the advantages of data abstraction with an axiom-based mathematical method to describe semantics Guttag, Horowitz and Musser Denotational semantics Stoy show how to use mathematical methods to describe semantics, which is a direct link to functional programming Frank and Kuhn Formal specifications and functional programming languages are based on similar mathematical theories and use similar syntax.
Functional programming languages Bird and Wadler allow to formally check specifications, and to observe if the specifications capture the intended behavior. The executable code is treated as a prototype. A prototype is expected to show if the implementation is possible and if a formal model behaves as expected.
The functional language Gofer Jones is used as the specification language. In Gofer it is possible to describe abstract behavior of objects, to construct specific objects, and describes how they achieve the intended behavior. The code produced by a functional programming language is clearly readable, and allows for the insight to be gained from reading or writing specifications.
Gofer enables specification and prototyping that are integrated in a single and easy to use environment. This is a contribution to the research on the development of suitable methods for specifications of large, complex systems. The conceptual modeling method is used to model hierarchical wayfinding. The conceptual model of hierarchical wayfinding is based on the proposed theory of hierarchical spatial reasoning.
The model describes the hierarchical structure of a road network and explains how the reasoning process progresses on such a structure. The tools used are graph theory, object orientation, algebraic specifications and functional programming. According to the requirements of the hierarchical spatial reasoning, the conceptual modeling process consists of the following major steps: First, a model for the nonhierarchical case is designed.
It includes a model of a road network at only one level of representation and a shortest path algorithm. Second, a hierarchical structure is introduced and a set of rules is given stating how the algorithm determines the shortest path in such a structure.
Keywords social robot consciousness-emotion interaction machine consciousness signal-detection theory. This finding suggests that the same neural mechanisms that underlie the projection of future events may also be sensitive to hierarchical structure in those events. From actions to goals and vice-versa: Theoretical analysis and models of the ideomotor principle and TOTE. Resource management and scalability of the XCSF learning system. Rothkopf, V.
In the third step results achieved by the hierarchical algorithm will be analyzed and compared to the results achieved by the nonhierarchical algorithm. In the last step the performance of the hierarchical algorithm is analyzed and compared to the performance of the nonhierarchical algorithm. The nonhierarchical case. We start with specifying a generic graph based on which a bidirectional, weighted, and hierarchical graph are then built. As an application of it, we give a specification of a road network for the nonhierarchical and hierarchical case.
Such graphs can then be used for the shortest path determination. The following assumptions are made about the graph in order to preserve the simplicity of the model:. A road network can be seen a special case of a bidirectional graph, which in turn, is a special case of a generic graph. To derive a formal model of such a road network, the generic graph has to be specified first. A road network is also a weighted graph, which requires introduction of weights.
The complete set of objects is collected and brought together into an ontology. An ontology is an abstracted, idealized model of reality containing only those objects, relations among them, and rules that govern them, which are of interest in a particular reasoning system being designed Davis These objects, i. This algorithm determines the lengths of the shortest paths between the given node and any other node in a graph. This is the optimal running time for a fully dense graph i. Improvements of the running time can be achieved through different implementations Ahuja, Thomas L.
Magnanti and Orlin : e. The hierarchical case. The ontology for the nonhierarchical case is enriched by hierarchical levels.
A bottom-up operation based on the road classification criterion, extracts a subgraph from the basic graph creating a structural hierarchy. The lowest level contains the entire network. Thus, the ontology contains the objects node, edge, graph and level. A set of rules states which part of the graph to use, when to switch between the levels, and how to combine single-level, partial results into a complete solution.
The underlying idea of the hierarchical algorithm is the stepwise reduction of the initial graph causing the flat algorithm to operate on a subgraph. The improvement of the running time is, therefore, achieved by reducing the size of the graph, and this is enabled through the hierarchical structure of the network. We have shown that the strategy of hierarchical subdivision of graphs improves the performance of the standard, nonhierarchical shortest path algorithm.
The hierarchical algorithm is formally described, which allows to see how such algorithms are structured in general i. From the formal description further properties of the hierarchical wayfinding can be derived e. The impact of this research on hierarchy and its use for modeling is quite large. The applied methods and results provide better insight into the theory of HSR. The major contribution to the theory is in combining advances in human knowledge and representation, and hierarchical reasoning with classical wayfinding algorithms.
The theory is expected to be useful in cases where large datasets or incomplete spatial information needs processing. In-car navigation is such a case. The method of formalization used here has proven useful, particularly to define objects in a conceptual model i. Gofer, as a specification tool, produced executable specification, which adds a new aspect to the formalization for the GIS purposes.
Gofer-specifications were relatively easy to write and read, which was helpful in checking syntax. Executable code enabled immediate testing, i. Addiction beyond pharmacological effects: The role of environment complexity and bounded rationality. Neural Networks. Fiore, VG. Giglia, G. Changing perspective on perception physiology: Can you really see what is happening? EuroMediterranean Biomedical Journal. Cognitive Neuroscience. Lee, K.
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Addiction beyond pharmacological effects: the role of environment complexity and bounded rationality.
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