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 do I get from A to B? Which is closer: from A to B or from A to C? Which is (the area of) my land? Is the tree (walkway, driveway) on my property or on your property? 3
Many / most spatial problems come without numbers Do we have to formulate spatial problems in terms of numbers in order to solve them ( left-brain computing )? Or can we find ways to process spatial configurations directly ( right-brain computing )? 4
Plan for my talk Qualitative temporal and spatial reasoning Conceptual neighborhood SparQ toolbox From relations to configurations Spatial computing (vs. propositional computing) Interaction most welcome! 5
Starting Point: Allen Relations (1983) (Previously published by C. Hamblin, 1972) 6
13 Qualitative Interval Relations Relation Symbol Pictorial Example before after equal meets met by overlaps overlapped by during contains starts started by finishes finished by < > = m mi o oi d di s si f fi 7
Allen s Composition Table for Temporal Relations
... applied to 1-D Perception Space, arranged by conceptual neighborhood spatially inhomogeneous categories: intervals points compare: human perception human memory human concepts human language 9
Interval relations characterized by relations between beginnings and endings Interval relations characterized by beginnings and endings 10
Spatial and Conceptual Neighborhood spatial neighborhood between locations conceptual neighborhood between relations static structure process structure 11
Features of Conceptual Neighborhood Coarse relations = CNs of fine relations CNs define conceptual hierarchies for representing incomplete knowledge Efficient non-disjunctive reasoning Incremental refinement as knowledge is gained Natural correspondence to everyday concepts Spatio-temporal inferences form conceptual neighborhoods Reduce computational complexity from exponential to polynomial Can be defined at arbitrary granularity 12
Incomplete knowledge as coarse knowledge Example: Disjunction of the relations before or meets or overlaps (<, m, o) can be considered incomplete knowledge as it cannot be reduced to a single interval relation. It can be considered coarse knowledge as the three relations form a conceptual neighborhood that defines the coarse relation 13
Coarse relations as semi-interval relations I 14
Coarse relations as semi-interval relations II 15
Neighborhood-based coarse reasoning 16
Composition Table for Coarse Reasoning 17
Inference based on coarse relations 18
Fine reasoning based on coarse relations 19
Closed composition table for fine and coarse relations 20
A Multitude of Specialized Calculi Topology 4-intersection, 9-intersection (Egenhofer et al.) RCC-5, RCC-8 (Randell, Cohn et al.) Orientation point-based (double cross, FlipFlop, QTC, dipole) extended objects Position Ternary Point Configuration Calculus (TPCC) Measurement Delta-Calculus 21
Generic Toolbox SparQ for Spatial Qualitative Reasoning D Wolter, F Dylla, L Frommberger, JO Wallgrün Calculus specification base relations / operations in list notation or: algebraic specification (metric space) Functional list notation Interfacing: command line or TCP/IP Available under GNU GPL license www.sfbtr8.spatial-cognition.de/project/r3/sparq/ manual included 22
Modular SparQ Architecture syntax: sparq <module> <calculus> <operation> <input> 23
Boat Race [Ligozat 2005] Example: qualify sparq qualify point-calculus all ((A 0) (B 10.5) (C 7) (D 7) (E 17)) ((A < B) (A < C) (A < D) (A < E) (B > C) (B > D) (B < E) (C = D) (C < E) (D < E)) 24
Boat Race Ex: compute-relation sparq compute-relation point-calculus composition < < (<) sparq compute-relation point-calculus converse (< =) (> =) 25
Boat Race Ex: constraint-reasoning sparq constraint-reasoning pc scenarioconsistency first ((E > B) (A < B) (A < C) (D = C)) ((C (=) D) (A (<) D) (A (<) C) (B (>) D) (B (>) C) (B (>) A) (E (>) D) (E (>) C) (E (>) A) (E (>) B)) 26
Boat Race Ex: constraint-reasoning sparq constraint-reasoning pc scenarioconsistency first ((E > B)(A < B)(A < C)(D = C) (X < C) (B < X)) NOT CONSISTENT 27
Boat Race Ex: constraint-reasoning sparq constraint-reasoning pc scenarioconsistency all < < <five scenarios found> 28
Spatial Configurations Example: quantify experimental sparq quantify flipflop ((A B l C) (B C r D)) ((A 0 0) (B 7.89 15.36) (C -4.98 1.14) (D -36.75 21.25)) 29
SparQ - Summary generic qualitative reasoning toolbox binary and ternary calculi algebraic calculus specification determines operations automatically calculus verification qualitative reasoning more effective / efficient than general theorem proving challenges are welcome! available under GNU GPL license www.sfbtr8.spatial-cognition.de/project/r3/sparq/ manual included 30
Challenge Knowing which tool to select for a given problem Meta-knowledge about spatial reasoning 31
Spatial Configurations 32
Computation by Abstraction Example: Trigonometry γ Given: a=5; b=3; c=6 Compute:,,, A,... α β A 33
Computation by Diagrammatic Construction 34
Computation by Diagrammatic Construction: A Form of Analogical Reasoning Universal properties of spatial structures: Trigonometric relations hold on all flat surfaces Flat diagrammatic media provide suitable spatial properties to directly compute trigonometric relations Static spatial structures can replace computational processes of geometric algorithms Computational operations are built into spatial structures Constraints in spatial structures act instantaneously; i.e., no constraint solving procedures are required 35
Computing Space 36
Diagrammatic vs. Formal Reasoning concrete vs. abstract time formal task stage formal specification formal reasoning solution stage formal result language / formal level formalization instantiation formalization instantiation spatial spatial configuration image level no time (instantaneous) 37
Elementary Entities of Cognitive Processing geometry cognition Composition Aggregation configurations objects areas lines points basic level configurations objects areas lines points Composition Decomposition Aggregation Refinement 38
Spatio-Visual Problems 39
Reasoning by Imagination How many degrees is the smallest turn that aligns the cube with its original orientation (corners coincide with corners, edges coincide with edges)?
Diagrammatic Approach the cube viewed from above
Limitations of Spatial Computing? 42
Approach: Implementation of a Visuo-Spatial Sketch-Pad Courtesy: Mary Hegarty 43
Thank you very much for your attention! www.spatial-cognition.de 44
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Application-Perspectives 21.12.2007 06:53 Uhr Schiffsunglück bei Krefeld Sojaschiff rammt Kerosin-Tanker Auf dem Rhein in Krefeld sind Donnerstagnacht drei Schiffe kollidiert. Die Bergungsarbeiten dauern an, die Höhe des Schadens ist noch unklar. Drei Motorschiffe sind am Donnerstagabend auf dem Rhein in Höhe des Krefelder Stadtteils Uerdingen kollidiert. Eines der beteiligten Schiffe drohte zu sinken, doch konnte dies von den Rettungskräften verhindert werden. 46
SailAway International navigation rules regulate right of way for pairs of vessels What happens when more than two vessels are involved? 47
SailAway: Vessels A and B 48
SailAway: Vessels B and C 49
SailAway: Vessels A and C 50
SailAway: Conflicting Rules 51
The Space of Qualitative Values e.g. double cross calculus [Freksa 1992] left front straight ahead right front spatially inhomogeneous categories: areas left abeam right abeam lines points left right compare: human perception left back straight back right back human memory human concepts human language 52