Spatial and Temporal Structures in Cognitive Processes *
|
|
- Abel Conley
- 6 years ago
- Views:
Transcription
1 In: Foundations of Computer Science, C Freksa, M Jantzen, R Valk (Eds.), Lecture Notes in Computer Science 1337, , Berlin: Springer-Verlag. Spatial and Temporal Structures in Cognitive Processes * Christian Freksa # University of Hamburg Abstract. The structures of space and time are identified as essential for the realization of cognitive systems. It is suggested that the omnipresence of space and time may have been responsible for neglecting these dimensions in knowledge processing in the past. The evolving interest in space and time in cognitive science and some of the current conceptions of space and time are briefly reviewed. It is argued that space and time not only structure cognitive representations and processes but also provide useful information for knowledge processing. Various ways of structuring space and time are discussed and the merits of different languages for describing space and time are addressed. In particular, qualitative and quantitative descriptions are related to local and global reference frames and crisp qualities are related to fuzzy quantities. The importance of selecting an appropriate level of interpretation for a given description is stressed. Examples of interpreting spatial and temporal object descriptions in various ways are presented. The Ubiquity of Space and Time in Cognitive Systems Space and time are everywhere particularly in cognitive systems and around them. This situation as trivial as it sounds may be responsible for the fact that the relevance of space and time to cognition has been neglected in modeling cognitive representations and processes for a long time. Perception, the origin of all cognition, takes place in spatially extended regions and requires time to be carried out; memory requires spatial extension and the processes of storage and retrieval require some time; processing perceived or recorded information takes place in space and requires time; actions carried out on the basis of computation require both space and time. From an information processing perspective, something that is everywhere tends to be not very interesting: at first glance it appears unspecific and therefore not informative. Thus, it is not surprising that many knowledge representation approaches in artificial intelligence abstracted from time and space while truth and falsity were of central interest in these approaches 1. Locations in particular memory locations were considered equivalent to one another and the times of occurrence of events were considered arbitrary and therefore not relevant in many models of the world. * # 1 Support from the Deutsche Forschungsgemeinschaft is gratefully acknowledged. freksa@informatik.uni-hamburg.de FB 18, Vogt-Kölln-Str. 30, Hamburg, Germany. See also the debates on the role of logic in AI in Computational Intelligence, vol 3, no 3, August 1987 pp and in KI, vol 6, no 3, September 1992.
2 But at second glance, the situation looks quite different: each location in space and each moment in time can be considered unique, and therefore very informative. However unique entities are not interesting from an information processing point of view, as they are unpredictable; they are too specific to be useful for generalization. Fortunately, at third glance we observe that space and time have rather regular structures. And structure means predictability. In other words, space and time bear the potential of being interpreted in very specific ways due to the specificity of their parts and of being interpreted in more general ways due to their regular structures; thus they behave in a predictable way and can be exploited by information processes in general and by cognitive processes in particular. The structures of space and time serve as reference frames for our understanding of the world. Animals and people go to familiar places for security and sovereignty; they exploit the periodicity of events to predict new situations. We describe nonperceivable abstract dimensions in terms of the concrete dimensions space and time [cf. Freksa & Habel 1990]; 2 in this way we can exploit our familiarity with those dimensions and convey dependencies in other domains. During the last few years, the central roles of space and time for cognitive systems have been increasingly recognized. In artificial intelligence, a great interest has developed to understand and to model structures and uses of cognitive space and time. The work in this field is carried out in cooperation with other disciplines of cognitive science in an effort to jointly solve the puzzle of space, time, and their representation and use in cognitive systems. Space and Time in Various Disciplines of Cognitive Science Space and time have become of central interest to several branches of cognitive science. Psychologists study the cognition of perceived and imagined visual space, the cognition of large scale space, i.e. space which cannot be perceived from a single view point [cf. Lynch 1960], and the cognition of the duration of events. The relation between subjectively perceived duration and physically measured time in actual experience and in successive recollection hint at complex structures in the cognitive organization of time. Of particular interest in the cognition of space and time are questions of spatial and temporal reference frames, the relation between visual and haptic space, and the role of spatial scale for spatial cognition. Psychological experiments in spatial and temporal cognition are carried out by relating performance of human subjects in spatial and temporal tasks to models of spatial and temporal representation. In the neurosciences, spatial and temporal cognition is investigated mainly by studying the effects of neurological deficits, for example deficits in the ability to correctly order sequences of events, deficits in the cognition of personal space, deficits in the cognition of locations, and deficits in the cognition of objects [cf. Andersen 1987]. Reference systems play a decisive role here too. The research in these areas is not restricted to the study of human spatial cognition, but extends to animals as well. 2 See also: Habel & Eschenbach, Abstract structures in spatial cognition this volume.
3 Linguistics studies adequate and inadequate use of language to describe space and time, for example the use of prepositions to describe and distinguish spatial and temporal arrangements of physical objects or events. Seemingly inconsistent usage of prepositions may be explained by identifying suitable reference systems [Retz- Schmidt 1988]. Philosophers have been asking questions about possible structures of space and time for more than 2500 years. An increasing number of interdisciplinary treatments is published like Landau and Jackendoff s [1993] much-discussed article What and where in spatial language and spatial cognition, to mention one example. Artificial intelligence has traditionally treated space and time mostly implicitly. Realizing that this may cause problems of restricted expressiveness and excessive complexity, AI researchers have been developing explicit representations of knowledge about space and time. Some of these representations may serve as operational models of spatial and temporal cognition and raise interesting questions both to empirical and to analytical approaches to cognitive science. Abstract and Concrete Spaces Two disciplines dealing extensively with space and time are mathematics and physics. One way of axiomatizing space in mathematics is as a structure made up of a set of points. Euclidean geometry builds a system of concepts on the basis of points, lines, and planes: distance, area, volume, and angle. In the context of spatial reasoning, Schlieder [1996] distinguishes between topological information (e.g. information about connectivity), orientation or ordering (e.g. information about convexity), and metric information (e.g. information about distances and angles). Classical physics is concerned with concrete physical space. This space can be described in terms of orthogonal components to solve problems of classical mechanics. Physical space is positively extended, and movement can take place in any spatial direction. Unlike in classical physics, common sense notions of the world generally conceive time as directed (and therefore irreversible). Physical space and its elements are related to other physical quantities in many ways: movement relates time and (spatial) distance, atomic structures relate mass and spatial extension, gravity relates weight and mass, etc. Cognitive systems appear to employ different conceptualizations of space. Central questions are: What are basic entities of the cognitive spatial structures? How are these entities related to one another? Which reference frames are employed? How are the entities and their relations cognitively processed? Which aspects of space are processed separate from others and which aspects are processed in an integrated manner? What is the role of spatial structures in generating and processing spatial metaphors? Conceptions of Space and Time When we speak about space, we refer to notions of location, orientation, shape, size (height, width, length and their combination), connection, distance, neighborhood, etc. When we speak about time, we refer to notions of duration, precedence, concurrency, simultaneity, consequence, etc. Some of the notions have well-defined
4 meanings in disciplines like physics, topology, geometry, and theoretical computer science; but here we are concerned with the question how humans think and talk about them, how they represent such notions to get around in their spatio-temporal environment, how they reason successfully about the environment, and how they solve problems based upon this reasoning. In AI, these questions were first addressed in the framework of naive physics research [cf. Hayes 1978]. There is a multitude of ways in which space and time can be conceptualized, each of which rests on implicit assumptions or explicit knowledge about the physical structure of the world. We will start with a common sense picture, which could be something like: space is a collection of places which stand in unchanging relative position to one another and which may or may not have objects located at them ; time is an ever growing arrow along which changes take place. Implicit in these pictures are the assumptions that the time arrow grows even when no other changes are taking place, that space is there even when there are no objects to fill it, and that spatial relations and changes can be observed and described. As these assumptions cannot be redeemed in practice, it is more reasonable to assume that objects and events constitute space or time, respectively. Another distinction concerns the question whether space or time should be modeled by infinite sets of (extensionless) points or by finite intervals (or regions). If we talk about Rocquencourt being located South-West of Rostock, it is likely that we think of two geometric points (without spatial extension) on a map of Europe. If, in a different situation, we say that you have to follow a certain road through Rocquencourt to reach a particular destination, Rocquencourt will be considered to have a spatial extension. Also, it is not clear from the outset whether a discrete, a dense, or a continuous representation of time and space may be more adequate for human cognition or for solving a given task [Habel 1994]: if we want to reason about arbitrarily small changes, a dense representation seems to be a good choice; if we want to express that two objects touch each other and we do not want anything to get in between them, a discrete representation seems preferable; if on one level of consideration a touching relation and on another level arbitrarily small changes seem appropriate, yet another structure may be required. Nevertheless, a continuous structure (e.g. R 2 ) is often assumed which provides a better correspondence with models from physics. Description in Terms of Quantities and Qualities Space and time can be described in terms of external reference values or by reference to domain-internal entities [e.g. Zimmermann 1995]. For external reference, usually standardized quantities with regular spacing (scales) are used; this is done particularly when precise and objective descriptions are desired; the described situations can be reconstructed accurately (within the tolerance of the granularity of the scale) in a different setting. In contrast, domain-internal entities usually do not provide regularly spaced reference values but only reference values which happen to be prominent in the given domain. The internal reference values define regions which correspond to sets of quantitatively neighboring external values. The system of internally defined regions is domain-specific.
5 Which of the two ways of representing knowledge about a physical environment is more useful for a cognitive system? In our modern world of ever-growing standardization we have learned that common reference systems and precise quantities are extremely useful for a global exchange of information. From an external perspective, the signals generated in receptor cells of (natural and artificial) perception systems also provide quantitative information to the successive processing stages. But already in the most primitive decision stages, for example in simple threshold units, rich quantitative information is reduced to comparatively coarse qualitative information, when we consider the threshold as an internal reference value. We can learn from these considerations, that information reduction and abstraction may be worthwhile at any level of processing. As long as we stay within a given context, the transition from quantitative to qualitative descriptions does not imply a loss of precision; it merely means focusing on situation-relevant distinctions. By using relevant entities from within a given environment for reference, we obtain a customized system able to capture the distinctions relevant in the given domain. Customization as information processing strategy was to be considered expensive when information processing power was centralized; but with decentralized computing, as we find in biological and in advanced technical systems, locally customized information processing may simplify computation and decision-making considerably. Local & Global Reference Systems and Conceptual Neighborhood Significant decisions frequently are not only of local relevance; thus it must be possible to communicate them to other environments. How can we do this if we have opted for qualitative local descriptions? To answer this question, we must first decide which are the relevant aspects that have to be communicated. Do we have to communicate precise quantitative values as, for example, in international trade or do qualitative values like trends and comparisons suffice? In cognitive systems, a qualitative description of a local decision frequently will suffice to get the picture of the situation; the specific quantities taken into account may have no particular meaning in another local context. Qualitative descriptions can convey comparisons from one context to another, provided that the general structures of the two contexts agree. If the descriptions refer to the spatio-temporal structures of two different environments, this will be the case [cf. Freksa 1980]. Now consider qualitative spatio-temporal descriptions in a given environment. As they compare one entity to another entity with respect to a certain feature dimension, they form binary (or possibly higher-order) relations like John is taller than the arch or Ed arrived after dinner was ready. In concrete situations in which descriptions serve to solve certain tasks, it only makes sense to compare given entities to specific other entities. For example, comparing the size of a person to the height of an arch is meaningful, as persons do want to pass through arches and comparing the arrival time of a person to the completion time of a meal may be meaningful, as the meal may have been prepared for consumption by that person; on the other hand, it may not make sense to compare the height of a person to the size of a shoe, or the arrival time of a person at home with the manufacturing date of some tooth paste. For this reason,
6 we frequently abbreviate binary spatial or temporal relations by unary relations (leaving the reference of the comparison implicit). Thus, to a person understanding the situation context, the absolute descriptions John is tall and Ed arrived late in effect may provide the same information as the previous statements in terms of explicit comparisons. As long as it is clear from the situation context, that there is only one meaningful reference object and dimension for the implicit comparison, the descriptions can be considered crisp (as the description either is fully true or fully false in the reference world). However, in more realistic settings, situations are not completely specified and it is therefore not completely clear which should be considered the single relevant reference object. In fact, usually even the producer of a description himself or herself will not be able to precisely indicate which is the unique correct reference object. For example, when I assert John is tall, I am not able to uniquely specify with respect to which reference value I consider John to be tall. Why is the description still meaningful and why is it possible to understand the meaning of such a description? In many cases, the reason that qualitative descriptions with indeterminate reference value work is that potential reference candidates provide a neighborhood of similar values, or in terms of the terminology of qualitative reasoning the values form a conceptual neighborhood [Freksa 1991]. Conceptual neighbors in spatial and temporal reasoning have the property that they change the result of a computation very little or not at all, in comparison to the original value. For example, in interpreting the description John is tall as an implicit comparison of the height of John with the height of other objects, it may not be critical whether I use as reference value the height of other people in the room, the average height of persons of the same category, or the median, provided that these values are in the same ballpark of values. On the other hand, if the context of the situation does not sufficiently specify which category of values provides an adequate reference for interpreting a given description, we may be in trouble. For example, if the description John is tall is generated in the context of talking about pre-school children, it may be inappropriate to interprete it with reference to the average height of the male population as a whole. Crisp Qualities and Fuzzy Quantities We have discussed descriptions like John is tall in terms of qualitative descriptions. The theory of fuzzy sets characterizes descriptions of this type in terms of fuzzy possibility distributions [Zadeh 1978]. What is the relation between the fuzzy set interpretation and the interpretation in terms of qualitative attributes? 3 The fuzzy set tall describes objects in terms of a range of (external objective) quantities. Having a range of quantities instead of a single quantity reflects the fact that the value is not precisely given with respect to the precision of the scale used. This is a simple granularity effect implicit also in crisp quantitative descriptions. For example, distance measures in full meters usually imply a range of 100 possible actual values in full centimeters or 1000 possible actual values in full millimeters, etc. The graded 3 See also: Hernández, Qualitative vs. quantitative representations of spatial distance this volume.
7 membership values associated with different actual values in addition account for the fact that in many descriptions not all actual values are possible to the same degree. Nevertheless, the description in terms of quantities with graded membership in a fuzzy set is a quantitative description. The qualitative view directly accounts for the fact that the granulation of a scale in terms of meters, centimeters, or millimeters is somewhat arbitrary and does not directly reflect relevant differences in the domain described. Thus, the description John is tall can be viewed as a statement about the height category which John belongs to and whose distinction from other height categories is relevant in a certain context for example when walking through an arch. In the qualitative view, the fact that the category tall can be instantiated by a whole range of actual height values is not of interest; the view abstracts from quantities. As fuzzy sets relate the qualitative linguistic terms to a quantitative interpretation, they provide an interface between the qualitative and the quantitative levels of description. Levels of Interpretation We have discussed a qualitative and a quantitative perspective on spatial and temporal representations and I have argued that the qualitative view uses meaningful reference values to establish the categories for the qualitative descriptions while the quantitative view relies on pre-established categories from a standard scale, whose categories are not specifically tuned to the application domain. In this comparison, the two approaches reflected by the two views are structurally not very different: both rely on reference values the first from the domain, the second from an external scale. As a result of the adaptation to the specific domain, qualitative categories typically are coarser than their quantitative counterparts. So what else is behind the qualitative / quantitative distinction? Let us consider again the qualitative / quantitative interface manifested in the fuzzy set formalism. A problem with the use of fuzzy sets for modeling natural language or other cognitive systems is that the interpretation of qualitative concepts is carried out through the quantitative level linked via the interface to the qualitative concept while cognitive systems seem to be able to abstract from the specific quantitative level [Freksa 1994]. For example, from X is tall and Y is much taller than X we can infer Y is very tall, independent of a specific quantitative interpretation of the category tall. How is this possible? The answer is simple. The semantic correspondence between spatial and temporal categories should be established on the abstract qualitative level of interpretation rather than on a specific quantitative level. This requires that relations between spatial and temporal categories are established directly on the qualitative level, for example in terms of topological relations or neighborhood relations. In this way, the essence of descriptions can be transmitted and evaluated more easily. An interesting aspect of transferring spatial and temporal operations from the quantitative level to the qualitative level is that fuzzy relations on the quantitative level may map into crisp relations on the qualitative level. For example, to compute the relation between the spatial categories tall and medium-sized on the quantitative
8 level, fuzzy set operations have to be carried out. On the other hand, on the qualitative level, we may provide partial ordering, inclusion, exclusion, overlap information, etc. to characterize the categories and to infer appropriate candidate objects matching the description. From a philosophical point of view, there is an important difference between the quantitative and the qualitative levels of interpretation. While quantitative interpretations for example in terms of fuzzy sets implicitly refer to a dense structure of target candidate values, qualitative interpretations directly refer to singular discrete objects which actually exist in the target domain. For this reason, certain issues relevant to fuzzy reasoning and decision making do not arise in qualitative reasoning. This concerns particularly the problem of setting appropriate threshold values for description matching. Horizontal Competition Vertical Subsumption Creating and interpreting object descriptions with respect to a specific set of objects calls for the use of discriminating features rather than for a precise characterization in terms of a universally applicable reference system. This is particularly true when the description is needed for object identification where the reference context is available (as opposed to object reconstruction in the absence of a reference context). When the reference context is available for generating and interpreting object descriptions, this context can be employed in two interesting ways. First of all, the context can provide the best discriminating features for distinguishing the target object from competing objects; second, the set of competing objects can provide the appropriate granulation for qualitatively describing the target object in relation to the competing objects. For example, when a description is needed to distinguish one person from one other person in a given situation, a single feature, say height, and a coarse granulation into qualitative categories, say tall and short, may suffice to unambiguously identify the target; if, on the other hand, in the interpretation of a description there are many competing objects, the description tall may be interpreted on a much finer granulation level distinguishing the categories very tall, tall, mediumsized, short, very short, which are subsumed by the coarser categories [Freksa & Barkowsky 1996]. Conclusion and Outlook I have presented a view of cognitive representations of spatial and temporal structures and I have sketched a perspective for exploiting these structures in spatial and temporal reasoning. The discussion focused on the description and interpretation of actually existing concrete situations. The general arguments should carry over to non-spatial and non-temporal domains. However, difficulties can be expected in using the ideas for describing hypothetical worlds and impossible situations, as the approach relies heavily on the reference to spatially and/or temporally representable situations. By carrying the principle of context-sensitive interpretation of the semantics of spatial and temporal categories a step further, we may get a rather different
9 explanation of the meaning of semantic categories: whether a given category is applicable or not may not depend so much on their fulfilling certain absolute qualifying criteria as on the availability of potentially competing categories for description. Acknowledgments I thank Bettina Berendt, Reinhard Moratz, Thomas Barkowsky, and Ralf Röhrig for valuable comments and suggestions. References Andersen, R.A Inferior parietal lobule function in spatial perception and visuomotor integration. In Handbook of physiology, The nervous system VI, Higher functions of the brain, part 2, ed. Plum, Bethesda, Md.: American Physiological Society. Freksa, C Communication about visual patterns by means of fuzzy characterizations. Proc. XXIInd Intern. Congress of Psychology, Leipzig. Freksa, C Conceptual neighborhood and its role in temporal and spatial reasoning. In Decision Support Systems and Qualitative Reasoning, ed. Singh and Travé-Massuyès, Amsterdam: North-Holland. Freksa, C Fuzzy systems in AI. In Fuzzy systems in computer science, ed. Kruse, Gebhardt, and Palm. Braunschweig/Wiesbaden: Vieweg. Freksa, C., and Barkowsky, T., On the relation between spatial concepts and geographic objects. In Geographic objects with indeterminate boundaries, ed. Burrough and Frank, London: Taylor and Francis. Freksa, C., and Habel, C Warum interessiert sich die Kognitionsforschung für die Darstellung räumlichen Wissens? In Repräsentation und Verarbeitung räumlichen Wissens, ed. Freksa and Habel, 1-15, Berlin: Springer-Verlag. Habel, C Discreteness, finiteness, and the structure of topological spaces. In Topological foundations of cognitive science, FISI-CS workshop Buffalo, NY, Report 37, ed. Eschenbach, Habel, and Smith, Hamburg: Graduiertenkolleg Kognitionswissenschaft. Hayes, P The naive physics manifesto. In Expert systems in the microelectronic age, ed. Michie. Edinburgh: Edinburgh University Press. Landau, B. and Jackendoff, R What' and where in spatial language and spatial cognition, Behavioral and Brain Sciences 16: Lynch, K The image of the city. Cambridge, Mass.: The MIT Press. Retz-Schmidt, G Various views on spatial prepositions. AI Magazine, 4/88: Schlieder, C Räumliches Schließen. In Wörterbuch der Kognitionswissenschaft, ed. Strube, Becker, Freksa, Hahn, Opwis, and Palm, Stuttgart: Klett-Cotta. Zadeh, L.A Fuzzy sets as a basis for a theory of possibility. Fuzzy sets and systems 1, Zimmermann, K Measuring without measures: the delta-calculus. In Spatial information theory. A theoretical basis for GIS, ed. Frank and Kuhn, Berlin: Springer.
Spatial Thinking with Geographic Maps: An Empirical Study *
Spatial Thinking with Geographic Maps: An Empirical Study * Bettina Berendt 1, Reinhold Rauh 2, Thomas Barkowsky 1 1 Universität Hamburg, Fachbereich Informatik, Arbeitsbereich WSV, Vogt-Kölln-Str. 30,
More informationDIMENSIONS OF QUALITATIVE SPATIAL REASONING
In Qualitative Reasoning in Decision Technologies, Proc. QUARDET 93, N. Piera Carreté & M.G. Singh, eds., CIMNE Barcelona 1993, pp. 483-492. DIMENSIONS OF QUALITATIVE SPATIAL REASONING Christian Freksa
More informationPreferred Mental Models in Qualitative Spatial Reasoning: A Cognitive Assessment of Allen s Calculus
Knauff, M., Rauh, R., & Schlieder, C. (1995). Preferred mental models in qualitative spatial reasoning: A cognitive assessment of Allen's calculus. In Proceedings of the Seventeenth Annual Conference of
More informationMappings For Cognitive Semantic Interoperability
Mappings For Cognitive Semantic Interoperability Martin Raubal Institute for Geoinformatics University of Münster, Germany raubal@uni-muenster.de SUMMARY Semantic interoperability for geographic information
More informationMetrics and topologies for geographic space
Published in Proc. 7 th Intl. Symp. Spatial Data Handling, Delft, Netherlands. Metrics and topologies for geographic space Michael F Worboys Department of Computer Science, Keele University, Staffs ST5
More informationConvex Hull-Based Metric Refinements for Topological Spatial Relations
ABSTRACT Convex Hull-Based Metric Refinements for Topological Spatial Relations Fuyu (Frank) Xu School of Computing and Information Science University of Maine Orono, ME 04469-5711, USA fuyu.xu@maine.edu
More informationAdding ternary complex roles to ALCRP(D)
Adding ternary complex roles to ALCRP(D) A.Kaplunova, V. Haarslev, R.Möller University of Hamburg, Computer Science Department Vogt-Kölln-Str. 30, 22527 Hamburg, Germany Abstract The goal of this paper
More informationFuzzy Systems. Introduction
Fuzzy Systems Introduction Prof. Dr. Rudolf Kruse Christoph Doell {kruse,doell}@iws.cs.uni-magdeburg.de Otto-von-Guericke University of Magdeburg Faculty of Computer Science Department of Knowledge Processing
More informationFuzzy Systems. Introduction
Fuzzy Systems Introduction Prof. Dr. Rudolf Kruse Christian Moewes {kruse,cmoewes}@iws.cs.uni-magdeburg.de Otto-von-Guericke University of Magdeburg Faculty of Computer Science Department of Knowledge
More informationAlexander Klippel and Chris Weaver. GeoVISTA Center, Department of Geography The Pennsylvania State University, PA, USA
Analyzing Behavioral Similarity Measures in Linguistic and Non-linguistic Conceptualization of Spatial Information and the Question of Individual Differences Alexander Klippel and Chris Weaver GeoVISTA
More informationMeasurement Independence, Parameter Independence and Non-locality
Measurement Independence, Parameter Independence and Non-locality Iñaki San Pedro Department of Logic and Philosophy of Science University of the Basque Country, UPV/EHU inaki.sanpedro@ehu.es Abstract
More informationSpatial Computing. or how to design a right-brain hemisphere. Christian Freksa University of Bremen
Spatial Computing or how to design a right-brain hemisphere Christian Freksa University of Bremen 1 Acknowledgments 2 Some Examples of Spatial Problems (How) can I get the piano into my living room? How
More informationPairing Transitive Closure and Reduction to Efficiently Reason about Partially Ordered Events
Pairing Transitive Closure and Reduction to Efficiently Reason about Partially Ordered Events Massimo Franceschet Angelo Montanari Dipartimento di Matematica e Informatica, Università di Udine Via delle
More informationDynamic Fuzzy Sets. Introduction
Dynamic Fuzzy Sets Andrzej Buller Faculty of Electronics, Telecommunications and Informatics Technical University of Gdask ul. Gabriela Narutowicza 11/12, 80-952 Gdask, Poland e-mail: buller@pg.gda.pl
More informationConceptual Modeling in the Environmental Domain
Appeared in: 15 th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, Berlin, 1997 Conceptual Modeling in the Environmental Domain Ulrich Heller, Peter Struss Dept. of Computer
More informationIntelligent GIS: Automatic generation of qualitative spatial information
Intelligent GIS: Automatic generation of qualitative spatial information Jimmy A. Lee 1 and Jane Brennan 1 1 University of Technology, Sydney, FIT, P.O. Box 123, Broadway NSW 2007, Australia janeb@it.uts.edu.au
More informationEuler s Galilean Philosophy of Science
Euler s Galilean Philosophy of Science Brian Hepburn Wichita State University Nov 5, 2017 Presentation time: 20 mins Abstract Here is a phrase never uttered before: Euler s philosophy of science. Known
More informationAn Introduction to Description Logics
An Introduction to Description Logics Marco Cerami Palacký University in Olomouc Department of Computer Science Olomouc, Czech Republic Olomouc, 21.11.2013 Marco Cerami (UPOL) Description Logics 21.11.2013
More informationOn Terminological Default Reasoning about Spatial Information: Extended Abstract
On Terminological Default Reasoning about Spatial Information: Extended Abstract V. Haarslev, R. Möller, A.-Y. Turhan, and M. Wessel University of Hamburg, Computer Science Department Vogt-Kölln-Str. 30,
More informationUncertainty and Rules
Uncertainty and Rules We have already seen that expert systems can operate within the realm of uncertainty. There are several sources of uncertainty in rules: Uncertainty related to individual rules Uncertainty
More informationThe Problem. Sustainability is an abstract concept that cannot be directly measured.
Measurement, Interpretation, and Assessment Applied Ecosystem Services, Inc. (Copyright c 2005 Applied Ecosystem Services, Inc.) The Problem is an abstract concept that cannot be directly measured. There
More informationOntologies and Domain Theories
Ontologies and Domain Theories Michael Grüninger Department of Mechanical and Industrial Engineering University of Toronto gruninger@mie.utoronto.ca Abstract Although there is consensus that a formal ontology
More informationGEO-INFORMATION (LAKE DATA) SERVICE BASED ON ONTOLOGY
GEO-INFORMATION (LAKE DATA) SERVICE BASED ON ONTOLOGY Long-hua He* and Junjie Li Nanjing Institute of Geography & Limnology, Chinese Academy of Science, Nanjing 210008, China * Email: lhhe@niglas.ac.cn
More informationSpatio-Temporal Relationships in a Primitive Space: an attempt to simplify spatio-temporal analysis
Spatio-Temporal Relationships in a Primitive Space: an attempt to simplify spatio-temporal analysis Pierre Hallot 1 1 Geomatics Unit / University of Liège (Belgium) P.Hallot@ulg.ac.be INTRODUCTION Nowadays,
More informationMathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur. Lecture 1 Real Numbers
Mathematics-I Prof. S.K. Ray Department of Mathematics and Statistics Indian Institute of Technology, Kanpur Lecture 1 Real Numbers In these lectures, we are going to study a branch of mathematics called
More informationA General Framework for Conflation
A General Framework for Conflation Benjamin Adams, Linna Li, Martin Raubal, Michael F. Goodchild University of California, Santa Barbara, CA, USA Email: badams@cs.ucsb.edu, linna@geog.ucsb.edu, raubal@geog.ucsb.edu,
More informationComputational Tasks and Models
1 Computational Tasks and Models Overview: We assume that the reader is familiar with computing devices but may associate the notion of computation with specific incarnations of it. Our first goal is to
More informationWhy Is Cartographic Generalization So Hard?
1 Why Is Cartographic Generalization So Hard? Andrew U. Frank Department for Geoinformation and Cartography Gusshausstrasse 27-29/E-127-1 A-1040 Vienna, Austria frank@geoinfo.tuwien.ac.at 1 Introduction
More informationcis32-ai lecture # 18 mon-3-apr-2006
cis32-ai lecture # 18 mon-3-apr-2006 today s topics: propositional logic cis32-spring2006-sklar-lec18 1 Introduction Weak (search-based) problem-solving does not scale to real problems. To succeed, problem
More informationPairing Transitive Closure and Reduction to Efficiently Reason about Partially Ordered Events
Pairing Transitive Closure and Reduction to Efficiently Reason about Partially Ordered Events Massimo Franceschet Angelo Montanari Dipartimento di Matematica e Informatica, Università di Udine Via delle
More informationCoins and Counterfactuals
Chapter 19 Coins and Counterfactuals 19.1 Quantum Paradoxes The next few chapters are devoted to resolving a number of quantum paradoxes in the sense of giving a reasonable explanation of a seemingly paradoxical
More informationDesigning and Evaluating Generic Ontologies
Designing and Evaluating Generic Ontologies Michael Grüninger Department of Industrial Engineering University of Toronto gruninger@ie.utoronto.ca August 28, 2007 1 Introduction One of the many uses of
More informationModeling Temporal Uncertainty of Events: a. Descriptive Perspective
Modeling Temporal Uncertainty of Events: a escriptive Perspective Yihong Yuan, Yu Liu 2 epartment of Geography, University of California, Santa Barbara, CA, USA yuan@geog.ucsb.edu 2 Institute of Remote
More informationChapter 2 Background. 2.1 A Basic Description Logic
Chapter 2 Background Abstract Description Logics is a family of knowledge representation formalisms used to represent knowledge of a domain, usually called world. For that, it first defines the relevant
More informationRepresentation of Geographic Data
GIS 5210 Week 2 The Nature of Spatial Variation Three principles of the nature of spatial variation: proximity effects are key to understanding spatial variation issues of geographic scale and level of
More informationImplementing Visual Analytics Methods for Massive Collections of Movement Data
Implementing Visual Analytics Methods for Massive Collections of Movement Data G. Andrienko, N. Andrienko Fraunhofer Institute Intelligent Analysis and Information Systems Schloss Birlinghoven, D-53754
More informationME 534. Mechanical Engineering University of Gaziantep. Dr. A. Tolga Bozdana Assistant Professor
ME 534 Intelligent Manufacturing Systems Chp 4 Fuzzy Logic Mechanical Engineering University of Gaziantep Dr. A. Tolga Bozdana Assistant Professor Motivation and Definition Fuzzy Logic was initiated by
More information35 Chapter CHAPTER 4: Mathematical Proof
35 Chapter 4 35 CHAPTER 4: Mathematical Proof Faith is different from proof; the one is human, the other is a gift of God. Justus ex fide vivit. It is this faith that God Himself puts into the heart. 21
More informationDomain Modelling: An Example (LOGICAL) DOMAIN MODELLING. Modelling Steps. Example Domain: Electronic Circuits (Russell/Norvig)
(LOGICAL) DOMAIN MODELLING Domain Modelling: An Example Provision of a formal, in particular logical language for knowledge representation. Application of these means to represent the formal structure
More informationA new Approach to Drawing Conclusions from Data A Rough Set Perspective
Motto: Let the data speak for themselves R.A. Fisher A new Approach to Drawing Conclusions from Data A Rough et Perspective Zdzisław Pawlak Institute for Theoretical and Applied Informatics Polish Academy
More informationChapter One. The Real Number System
Chapter One. The Real Number System We shall give a quick introduction to the real number system. It is imperative that we know how the set of real numbers behaves in the way that its completeness and
More informationExploring Spatial Relationships for Knowledge Discovery in Spatial Data
2009 International Conference on Computer Engineering and Applications IPCSIT vol.2 (2011) (2011) IACSIT Press, Singapore Exploring Spatial Relationships for Knowledge Discovery in Spatial Norazwin Buang
More informationRule-Based Fuzzy Model
In rule-based fuzzy systems, the relationships between variables are represented by means of fuzzy if then rules of the following general form: Ifantecedent proposition then consequent proposition The
More informationDIAGRAMMATIC SYNTAX AND ITS CONSTRAINTS. 1. Introduction: Applications and Foundations
DIAGRAMMATIC SYNTAX AND ITS CONSTRAINTS ANDREI RODIN 1. Introduction: Applications and Foundations In their recent papers Zynovy Duskin and his co-authors stress the following problem concerning diagrammatic
More informationMeasurement Theory for Software Engineers
Measurement Theory for Software Engineers Although mathematics might be considered the ultimate abstract science, it has always been motivated by concerns in the real, physical world. Thus it is not surprising
More informationA Boolean Lattice Based Fuzzy Description Logic in Web Computing
A Boolean Lattice Based Fuzzy Description Logic in Web omputing hangli Zhang, Jian Wu, Zhengguo Hu Department of omputer Science and Engineering, Northwestern Polytechnical University, Xi an 7007, hina
More informationTransformation of Geographical Information into Linguistic Sentences: Two Case Studies
Transformation of Geographical Information into Linguistic Sentences: Two Case Studies Jan T. Bjørke, Ph.D. Norwegian Defence Research Establishment P O Box 115, NO-3191 Horten NORWAY jtb@ffi.no Information
More informationCOMP3702/7702 Artificial Intelligence Week1: Introduction Russell & Norvig ch.1-2.3, Hanna Kurniawati
COMP3702/7702 Artificial Intelligence Week1: Introduction Russell & Norvig ch.1-2.3, 3.1-3.3 Hanna Kurniawati Today } What is Artificial Intelligence? } Better know what it is first before committing the
More informationIncompatibility Paradoxes
Chapter 22 Incompatibility Paradoxes 22.1 Simultaneous Values There is never any difficulty in supposing that a classical mechanical system possesses, at a particular instant of time, precise values of
More information1 Propositional Logic
CS 2800, Logic and Computation Propositional Logic Lectures Pete Manolios Version: 384 Spring 2011 1 Propositional Logic The study of logic was initiated by the ancient Greeks, who were concerned with
More informationFrom Bi-facial Truth to Bi-facial Proofs
S. Wintein R. A. Muskens From Bi-facial Truth to Bi-facial Proofs Abstract. In their recent paper Bi-facial truth: a case for generalized truth values Zaitsev and Shramko [7] distinguish between an ontological
More informationLecture notes (Lec 1 3)
Lecture notes (Lec 1 3) What is a model? We come across various types of models in life they all represent something else in a form we can comprehend, e.g. a toy model of car or a map of a city or a road
More informationAbstract. 1 Introduction. 2 Archimedean axiom in general
The Archimedean Assumption in Fuzzy Set Theory Taner Bilgiç Department of Industrial Engineering University of Toronto Toronto, Ontario, M5S 1A4 Canada taner@ie.utoronto.ca Abstract The Archimedean axiom
More informationHardy s Paradox. Chapter Introduction
Chapter 25 Hardy s Paradox 25.1 Introduction Hardy s paradox resembles the Bohm version of the Einstein-Podolsky-Rosen paradox, discussed in Chs. 23 and 24, in that it involves two correlated particles,
More informationMODAL LOGICS OF SPACE
MODAL LOGICS OF SPACE Johan van Benthem, Amsterdam & Stanford, http://staff.science.uva.nl/~johan/ Handbook Launch Spatial Logics, Groningen, 14 November 2007 Pictures: dynamically in real-time on the
More informationA Computable Language of Architecture
A Computable Language of Architecture Description of Descriptor Language in Supporting Compound Definitions Introduction Sora Key Carnegie Mellon University, Computational Design Laboratory, USA http://www.code.arc.cmu.edu
More informationThe Importance of Spatial Literacy
The Importance of Spatial Literacy Dr. Michael Phoenix GIS Education Consultant Taiwan, 2009 What is Spatial Literacy? Spatial Literacy is the ability to be able to include the spatial dimension in our
More informationE QUI VA LENCE RE LA TIONS AND THE CATEGORIZATION OF MATHEMATICAL OBJECTS
E QUI VA LENCE RE LA TIONS AND THE CATEGORIZATION OF MATHEMATICAL OBJECTS ANTON DOCHTERMANN Abstract. In [2] Lakoff and Nuñez develop a basis for the cognitive science of embodied mathematics. For them,
More informationPROOF-THEORETIC REDUCTION AS A PHILOSOPHER S TOOL
THOMAS HOFWEBER PROOF-THEORETIC REDUCTION AS A PHILOSOPHER S TOOL 1. PROOF-THEORETIC REDUCTION AND HILBERT S PROGRAM Hilbert s program in the philosophy of mathematics comes in two parts. One part is a
More informationSpatial Cognition: From Rat-Research to Multifunctional Spatial Assistance Systems
Spatial Cognition: From Rat-Research to Multifunctional Spatial Assistance Systems Markus Knauff, Christoph Schlieder, Christian Freksa Spatial cognition research is making rapid progress in understanding
More information3 The Semantics of the Propositional Calculus
3 The Semantics of the Propositional Calculus 1. Interpretations Formulas of the propositional calculus express statement forms. In chapter two, we gave informal descriptions of the meanings of the logical
More informationA spatial literacy initiative for undergraduate education at UCSB
A spatial literacy initiative for undergraduate education at UCSB Mike Goodchild & Don Janelle Department of Geography / spatial@ucsb University of California, Santa Barbara ThinkSpatial Brown bag forum
More informationMATH 116, LECTURES 10 & 11: Limits
MATH 6, LECTURES 0 & : Limits Limits In application, we often deal with quantities which are close to other quantities but which cannot be defined eactly. Consider the problem of how a car s speedometer
More informationNon-Markovian Control in the Situation Calculus
Non-Markovian Control in the Situation Calculus An Elaboration Niklas Hoppe Seminar Foundations Of Artificial Intelligence Knowledge-Based Systems Group RWTH Aachen May 3, 2009 1 Contents 1 Introduction
More informationTemporal Extension. 1. Time as Quantity
1 Boris Hennig IFOMIS Saarbrücken Draft version as of July 17, 2005 Temporal Extension Abstract. Space and time are continuous and extensive quantities. But according to Aristotle, time differs from space
More information127: Lecture notes HT17. Week 8. (1) If Oswald didn t shoot Kennedy, someone else did. (2) If Oswald hadn t shot Kennedy, someone else would have.
I. Counterfactuals I.I. Indicative vs Counterfactual (LfP 8.1) The difference between indicative and counterfactual conditionals comes out in pairs like the following: (1) If Oswald didn t shoot Kennedy,
More informationRE-THINKING COVARIATION FROM A QUANTITATIVE PERSPECTIVE: SIMULTANEOUS CONTINUOUS VARIATION. Vanderbilt University
Saldanha, L., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berenson & W. N. Coulombe (Eds.), Proceedings of the Annual
More informationOn the Evolution of the Concept of Time
On the Evolution of the Concept of Time Berislav Žarnić Faculty of Humanities and Social Sciences Research Centre for Logic, Epistemology, and Philosophy of Science University of Split Physics & Philosophy
More informationMathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras. Lecture - 15 Propositional Calculus (PC)
Mathematical Logic Prof. Arindama Singh Department of Mathematics Indian Institute of Technology, Madras Lecture - 15 Propositional Calculus (PC) So, now if you look back, you can see that there are three
More informationReasoning about Cardinal Directions Using Grids as Qualitative Geographic Coordinates *
Reasoning about Cardinal Directions Using Grids as Qualitative Geographic Coordinates * Lars Kulik & Alexander Klippel University of Hamburg, Department for Informatics Vogt-Kölln-Str. 30, D-22527 Hamburg
More informationKey Words: geospatial ontologies, formal concept analysis, semantic integration, multi-scale, multi-context.
Marinos Kavouras & Margarita Kokla Department of Rural and Surveying Engineering National Technical University of Athens 9, H. Polytechniou Str., 157 80 Zografos Campus, Athens - Greece Tel: 30+1+772-2731/2637,
More informationProving Completeness for Nested Sequent Calculi 1
Proving Completeness for Nested Sequent Calculi 1 Melvin Fitting abstract. Proving the completeness of classical propositional logic by using maximal consistent sets is perhaps the most common method there
More informationA Description Logic with Concrete Domains and a Role-forming Predicate Operator
A Description Logic with Concrete Domains and a Role-forming Predicate Operator Volker Haarslev University of Hamburg, Computer Science Department Vogt-Kölln-Str. 30, 22527 Hamburg, Germany http://kogs-www.informatik.uni-hamburg.de/~haarslev/
More informationTwo-Valued Logic Programs
Two-Valued Logic Programs Vladimir Lifschitz University of Texas at Austin, USA Abstract We define a nonmonotonic formalism that shares some features with three other systems of nonmonotonic reasoning
More informationWhy is There a Need for Uncertainty Theory?
Journal of Uncertain Systems Vol6, No1, pp3-10, 2012 Online at: wwwjusorguk Why is There a Need for Uncertainty Theory? Baoding Liu Uncertainty Theory Laboratory Department of Mathematical Sciences Tsinghua
More informationFIXATION AND COMMITMENT WHILE DESIGNING AND ITS MEASUREMENT
FIXATION AND COMMITMENT WHILE DESIGNING AND ITS MEASUREMENT JOHN S GERO Krasnow Institute for Advanced Study, USA john@johngero.com Abstract. This paper introduces the notion that fixation and commitment
More informationA Theoretical Framework for Configuration
Belli (Ed.): Proceedings of the IEA/AIE 92, Paderborn 5th Int. Conf. on Industrial & Engineering Applications of AI Lecture Notes in AI, pp. 441-450, 1992, Springer. ISBN 3-540-55601-X 441 A Theoretical
More informationIt rains now. (true) The followings are not propositions.
Chapter 8 Fuzzy Logic Formal language is a language in which the syntax is precisely given and thus is different from informal language like English and French. The study of the formal languages is the
More informationLogic. Quantifiers. (real numbers understood). x [x is rotten in Denmark]. x<x+x 2 +1
Logic One reason for studying logic is that we need a better notation than ordinary English for expressing relationships among various assertions or hypothetical states of affairs. A solid grounding in
More informationSyllogistic Logic and its Extensions
1/31 Syllogistic Logic and its Extensions Larry Moss, Indiana University NASSLLI 2014 2/31 Logic and Language: Traditional Syllogisms All men are mortal. Socrates is a man. Socrates is mortal. Some men
More informationPhilosophy of Science: Models in Science
Philosophy of Science: Models in Science Kristina Rolin 2012 Questions What is a scientific theory and how does it relate to the world? What is a model? How do models differ from theories and how do they
More information5. And. 5.1 The conjunction
5. And 5.1 The conjunction To make our logical language more easy and intuitive to use, we can now add to it elements that make it able to express the equivalents of other sentences from a natural language
More informationOn Likelihoodism and Intelligent Design
On Likelihoodism and Intelligent Design Sebastian Lutz Draft: 2011 02 14 Abstract Two common and plausible claims in the philosophy of science are that (i) a theory that makes no predictions is not testable
More informationUnderstanding Aha! Moments
Understanding Aha! Moments Hermish Mehta Department of Electrical Engineering & Computer Sciences University of California, Berkeley Berkeley, CA 94720 hermish@berkeley.edu Abstract In this paper, we explore
More informationIs Consciousness a Nonspatial Phenomenon?
KRITIKE VOLUME FIVE NUMBER ONE (JUNE 2011) 91-98 Article Is Consciousness a Nonspatial Phenomenon? Ståle Gundersen Abstract: Colin McGinn has argued that consciousness is a nonspatial phenomenon. McGinn
More informationAugust 27, Review of Algebra & Logic. Charles Delman. The Language and Logic of Mathematics. The Real Number System. Relations and Functions
and of August 27, 2015 and of 1 and of 2 3 4 You Must Make al Connections and of Understanding higher mathematics requires making logical connections between ideas. Please take heed now! You cannot learn
More informationDynamical Embodiments of Computation in Cognitive Processes James P. Crutcheld Physics Department, University of California, Berkeley, CA a
Dynamical Embodiments of Computation in Cognitive Processes James P. Crutcheld Physics Department, University of California, Berkeley, CA 94720-7300 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe,
More information1 Modelling and Simulation
1 Modelling and Simulation 1.1 Introduction This course teaches various aspects of computer-aided modelling for the performance evaluation of computer systems and communication networks. The performance
More informationLecture 12: Arguments for the absolutist and relationist views of space
12.1 432018 PHILOSOPHY OF PHYSICS (Spring 2002) Lecture 12: Arguments for the absolutist and relationist views of space Preliminary reading: Sklar, pp. 19-25. Now that we have seen Newton s and Leibniz
More informationLecture Notes on Inductive Definitions
Lecture Notes on Inductive Definitions 15-312: Foundations of Programming Languages Frank Pfenning Lecture 2 September 2, 2004 These supplementary notes review the notion of an inductive definition and
More informationOn the Arbitrary Choice Regarding Which Inertial Reference Frame is "Stationary" and Which is "Moving" in the Special Theory of Relativity
Regarding Which Inertial Reference Frame is "Stationary" and Which is "Moving" in the Special Theory of Relativity Douglas M. Snyder Los Angeles, CA The relativity of simultaneity is central to the special
More informationLecture Notes on Heyting Arithmetic
Lecture Notes on Heyting Arithmetic 15-317: Constructive Logic Frank Pfenning Lecture 8 September 21, 2017 1 Introduction In this lecture we discuss the data type of natural numbers. They serve as a prototype
More informationPropositional Logic. Fall () Propositional Logic Fall / 30
Propositional Logic Fall 2013 () Propositional Logic Fall 2013 1 / 30 1 Introduction Learning Outcomes for this Presentation 2 Definitions Statements Logical connectives Interpretations, contexts,... Logically
More informationCommentary on Guarini
University of Windsor Scholarship at UWindsor OSSA Conference Archive OSSA 5 May 14th, 9:00 AM - May 17th, 5:00 PM Commentary on Guarini Andrew Bailey Follow this and additional works at: http://scholar.uwindsor.ca/ossaarchive
More information185.A09 Advanced Mathematical Logic
185.A09 Advanced Mathematical Logic www.volny.cz/behounek/logic/teaching/mathlog13 Libor Běhounek, behounek@cs.cas.cz Lecture #1, October 15, 2013 Organizational matters Study materials will be posted
More information1. Characterization and uniqueness of products
(March 16, 2014) Example of characterization by mapping properties: the product topology Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/mfms/notes
More informationMcTaggart s Argument for the Unreality of Time:
McTaggart s Argument for the Unreality of Time: A Temporal Logical Analysis Hunan Rostomyan In this paper I want to examine McTaggart s [1908] argument for the unreality of time. McTaggart starts with
More informationMECHANISM AND METHODS OF FUZZY GEOGRAPHICAL OBJECT MODELING
MECHANISM AND METHODS OF FUZZY GEOGRAPHICAL OBJECT MODELING Zhang Xiaoxiang a,, Yao Jing a, Li Manchun a a Dept. of Urban & Resources Sciences, Nanjing University, Nanjing, 210093, China (xiaoxiang, yaojing,
More informationQuantum Computation via Sparse Distributed Representation
1 Quantum Computation via Sparse Distributed Representation Gerard J. Rinkus* ABSTRACT Quantum superposition states that any physical system simultaneously exists in all of its possible states, the number
More informationCS 331: Artificial Intelligence Propositional Logic I. Knowledge-based Agents
CS 331: Artificial Intelligence Propositional Logic I 1 Knowledge-based Agents Can represent knowledge And reason with this knowledge How is this different from the knowledge used by problem-specific agents?
More information