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Logical Representations for Automated Reasoning About Logical Repre s entations for Automate d Re asoning ab out Spatial Relationship s Brandon Bennett Submitte d in accordance with the requirements for the degree of Do ctor of Philo sophy The Univers ity of Lee ds Scho ol of Computer Studie s Octob er The candidate conrms that the work submitte d i s hi s own and that appropr iate cre dit has b een given where reference has b een made to the work of others Ab stract Thi s the s i s inve stigate s logical repre s entations for de scr ibing and reasoning ab out spatial s ituations Previously prop os e d theor ie s of spatial regions are inve stigate d in some detail e sp ecially the storder theory of Randell Cui and Cohn The diculty of achieving eective automate d reasoning with the s e systems i s obs erve d A new approach i s pre s ente d bas e d on enco ding spatial relations in formulae of order pro p os itional logics It i s prove d that entailment which i s valid according to the standard s emantics for the s e logics i s also valid with re sp ect to the spatial interpretation Cons equently wellknown mechani sms for prop os itional reasoning can b e applie d to spatial reasoning Sp ecic enco dings of top ological relations into b oth the mo dal logic S and the intuitioni stic prop os itional calculus are given The complexity of reasoning us ing the intuitioni stic repre s entation i s examine d and a pro ce dure i s pre s ente d which i s shown to b e of O n complexity in the numb er of relations involve d In order to make thi s kind of repre s entation suciently expre ss ive the concepts of model con straint and entailment constraint are intro duce d By means of thi s di stinction a order formula may b e us e d e ither to ass ert or to deny that a certain spatial constraint holds of some s ituation It i s shown how the pro of theory of a order logical language can b e extende d by a s imple metalevel generali sation to accommo date a repre s entation involving the s e two typ e s of formula A numb er of other topics are dealt with a deci s ion pro ce dure bas e d on quantier elimination i s given for a large class of formulae within a storder top ological language reasoning mechani sms bas e d on the composition of spatial relations are studie d the nontop ological prop erty of convexity i s examine d b oth f rom the p oint of view of its storder character i sation and its incorp oration into a order spatial logic It i s sugge ste d that order repre s entations could b e employe d in a s imilar manner to enco de other spatial concepts There is no branch of mathematics however abstract that wil l not eventual ly be applied to the phenomena of the real world Lobachevsky quote d in the Amer ican Mathematical Monthly Feb Acknowle dgements Work on thi s the s i s was partially supp orte d by the CEC under the Bas ic Re s earch Action MEDLAR Pro ject the SERC under grants GRH and GRG and the EPSRC under grant GRK I would like to give sp ecial thanks to Profe ssor Tony Cohn for constant encouragement dur ing my work on thi s the s i s and for providing me with many opp ortunitie s to meet other re s earchers in the eld Thanks also to my mother Gaymer ing Bennett for pro of reading the manuscr ipt Finally I would like to thank my wife Deb orah for b ear ing with me dur ing my long career as a student Contents Intro duction The Domain of Spatial Reasoning Spatial Concepts and Information Geometry of Points and Line s and its Pr imitive Concepts Top ology Shap e Convexity and Containment Pos ition and Or ientation A Br ief Hi story of Spatial Reasoning Or igins Development New Foundations Conceptual and Formal Frameworks Bas ic Elements in a Spatial Theory Mo de s of Formali sm Logical Theor ie s of Space and Spatial Logics Spatial Reasoning in Computer Science Commons ens e Knowle dge Reasoning ab out Phys ical Systems Spatial Reasoning in Rob otics Spatial Reasoning and Computer Vi s ion Temp oral Reasoning Automating Spatial Reasoning Complexity of Mathematical Theor ie s Tractability and Decidability The Content of thi s The s i s Assume d Background and Notations Employe d Axiomatic Theor ie s of Spatial Regions PointSet Top ology The Or igins of RegionBas e d Theor ie s CONTENTS Lesniewskis Mereology Other Mereological Systems Tarskis Geometry of Solids Clarkes Theory Fus ions and Quas iBo olean Op erators Top ological Functions Points The Region Connection Calulus RCC Functional Extens ion of the Bas ic Theory The Sorte d Logic LLAMA Sorts in the RCC Theory Two Additional Axioms Further Development of RCC Other Relevant Work on RegionBas e d Theor ie s de Lagunas Theory Grzegorczyks Undecidability Re sults Some Recent Re s earch in the Field Analys i s of the RCC Theory RCC in Relation to thi s The s i s Identity and Extens ionality The Quas iBo olean Functions The Status of the Function Denitions RCC without Functions or Sorts The Complement Function Relation to Ortho dox Bo olean Algebras A Single Generator for Bo olean Functions Intro duction of a Null Region Atoms and the NTPP Axiom Mo dels of the RCC Theory Graph Mo dels of the C relation Mo dels in PointSet Top ology Interpreting RCC in PointSet Top ology The Bo olean Algebra of Regular Op en PointSets A Dual Top ological Interpretation Completene ss and Categor icity A Revi s e d Vers ion of the RCC Theory CONTENTS A Order Repre s entation Spatial Interpretation of Order Calculi Set Semantics for the Class ical Calculus An Entailment Corre sp ondence Reasoning with NonUniversal Equations Repre s enting Top ological Relationships in C Mo del and Entailment Constraints Cons i stency of C Situation De scr iptions Repre s enting RCC Relations NonNull Constraints Repre s entations of the RCC Relations Reasoning with C Determining Entailments Complexity of the Reasoning Algor ithm A Mo dal Repre s entation The Spatial Interpretation of Mo dal Logics Overview of the Approach Taken Semantics for Order Mo dal Logics Mo dal Logics The Logic S Kr ipke Semantics Mo dal Algebras Algebraic Mo dels PowerSet Algebras.
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