Automated Deduction in Geometry: 9th International Workshop, ADG 2012, Edinburgh, UK, September 17-19, 2012. Revised Selected Papers PDF
Preview Automated Deduction in Geometry: 9th International Workshop, ADG 2012, Edinburgh, UK, September 17-19, 2012. Revised Selected Papers
Tetsuo Ida Jacques Fleuriot (Eds.) 3 Automated Deduction 9 9 7 AI in Geometry N L 9th International Workshop, ADG 2012 Edinburgh, UK, September 2012 Revised Selected Papers 123 Lecture Notes in Artificial Intelligence 7993 Subseries of Lecture Notes in Computer Science LNAISeriesEditors RandyGoebel UniversityofAlberta,Edmonton,Canada YuzuruTanaka HokkaidoUniversity,Sapporo,Japan WolfgangWahlster DFKIandSaarlandUniversity,Saarbrücken,Germany LNAIFoundingSeriesEditor JoergSiekmann DFKIandSaarlandUniversity,Saarbrücken,Germany Tetsuo Ida Jacques Fleuriot (Eds.) Automated Deduction in Geometry 9th International Workshop, ADG 2012 Edinburgh, UK, September 17-19, 2012 Revised Selected Papers 1 3 VolumeEditors TetsuoIda UniversityofTsukuba DepartmentofComputerScience Tsukuba305-8573,Japan E-mail:[email protected] JacquesFleuriot TheUniversityofEdinburgh SchoolofInformatics InformaticsForum 10CrichtonStreet EdinburghEH89AB,UK E-mail:[email protected] ISSN0302-9743 e-ISSN1611-3349 ISBN978-3-642-40671-3 e-ISBN978-3-642-40672-0 DOI10.1007/978-3-642-40672-0 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2013948305 CRSubjectClassification(1998):I.2.3,I.3.5,F.4.1,F.3,G.2-3,D.2.4 LNCSSublibrary:SL7–ArtificialIntelligence ©Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection withreviewsorscholarlyanalysisormaterialsuppliedspecificallyforthepurposeofbeingenteredand executedonacomputersystem,forexclusiveusebythepurchaserofthework.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheCopyrightLawofthePublisher’slocation, initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Permissionsforuse maybeobtainedthroughRightsLinkattheCopyrightClearanceCenter.Violationsareliabletoprosecution undertherespectiveCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Whiletheadviceandinformationinthisbookarebelievedtobetrueandaccurateatthedateofpublication, neithertheauthorsnortheeditorsnorthepublishercanacceptanylegalresponsibilityforanyerrorsor omissionsthatmaybemade.Thepublishermakesnowarranty,expressorimplied,withrespecttothe materialcontainedherein. Typesetting:Camera-readybyauthor,dataconversionbyScientificPublishingServices,Chennai,India Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Automated Deduction in Geometry (ADG) is a well-established international series of workshops that bring together researchers interested in the theoretical and practical aspects of machine-based geometric reasoning. Since its inception in 1992 – initially as a workshop on purely algebraic approaches for geometric reasoning – ADG has continually evolved, enabling it to add a diversity of new topics and applications to its scope and establish its status as the foremost biennial event for automated reasoning in geometry. In2012,theADGworkshopwasheldintheSchoolofInformaticsattheUni- versity of Edinburgh, UK, during September 17–19. Each of the 15 submissions acceptedfor presentationwasreviewedby atleasttwomembersofthe Program Committee chaired by T. Ida, resulting in a lively 3-day program that also in- cluded invited talks by M. Beeson from San Jose State University, USA, and D. Wang from Universit´e Pierre et Marie Curie - CNRS, France. As is custom- ary, the workshop was followed by a new, open call-for-papers, and the present proceedings, with its 10 accepted articles detailing work at the forefront of re- searchinautomateddeductioningeometry,istheresultofafreshrefereeingand paper-revisionprocess.Webrieflyreviewthevariouscontributionsnext,withthe understanding that our broad categorization is not meant to be exhaustive or absolute. The first group of papers is mainly concerned with advances in and applica- tions of algebraic techniques: J. Yang, D. Wang, and H. Hoong continue their investigation of reparameterization of plane curves and present a method that uses C1 piecewise Mo¨bius transformation in an effort to improve angular speed uniformity. S. Moritsugu presents an extension of Descartes Circle Theorem to Steiner’s porism involving n circles and, with the help of Gr¨obner bases and resultant methods, produces several novel results along the way. P. Mathis and P. Schreck discuss an application of Cayley-Menger determinants to yield sys- tems of equations that can be used for geometric constraints involving points and hyperplanes. C. Borcea and I. Streinu address issues in a generalized form of rigidity theory by considering volume frameworks on hypergraphs. J. Bowers andI.StreinuinvestigatecomputationalorigamiusingLang’sUniversalMolecule algorithm and present several new results about the 3D foldability of origami. F. Ghourabi, A. Kasem, and C. Kaliszyk provide a rigorous reformulation of Huzita’s axioms for fold operations in computational origami and give a single generalized fold operation that replaces all the reformulated operations. The second group of papers broadly looks at geometric reasoning from a logical and theorem proving standpoint: C. Brun, J.-F. Dufourd, and N. Ma- gaud use the Coq proof assistant to mechanize an incremental procedure for determining the convex hull of a set of points and discuss the derivation by hand of its associated imperative C++ program. G. Braun and J. Narboux VI Preface carry out a study in the mechanization of Hilbert’s and Tarski’s axiomatics in Coq and show how Hilbert’s axioms can be derived from Tarski’s. U. Siddique, V. Avrantinos, and S. Tahar explore the formalization of geometrical optics in theproofassistantHOLLightandusetheirnewmechanicalframeworktoverify the stability of two widely-used optical resonators. S. Stojanovic proposes two heuristics for improving the efficiency of forward chaining in the coherent logic theorem prover ArgoCLP and shows that these improve automated proofs in a Hilbert-like geometry. Ontheinvitedpapersfront,weinclude theabstractofD.Wang’sADG2012 invited talk, which covered historical aspects of the mechanization of geometry and discussed the engineering foundations needed to develop the next genera- tion of geometric reasoning software. In a happy marriage between algebra and logic, the article by M. Beeson discusses the relationships between proof and computation in geometry from both theoretical and practical standpoints. He introduces a new vector geometry theory and reports on experiments that solve several automated reasoning challenges in Tarski’s geometry set by Quaife in 1990. We thank the School of Informatics for hosting ADG 2012and for providing generousfinancial support. Our gratitude also extends to Suzanne Perryfor her valuablehelp inorganizingthe workshopandits associatedevents. Last,but by no means least, we thank the Program Committee and secondary referees for their reviews and discussions – both for the workshop papers and the current proceedings. Their dedication is what makes ADG a continuing success. May 2013 Tetsuo Ida Jacques Fleuriot Organization Program Committee Hirokazu Anai Fujitsu Laboratories Ltd., Japan Francisco Botana Universidad de Vigo, Spain Xiaoyu Chen Beihang University, China Giorgio Dalzotto Universita` di Pisa, Italy Jacques Fleuriot University of Edinburgh, UK Laureano Gonzalez-Vega Universidad de Cantabria, Spain Hoon Hong North Carolina State University, USA Tetsuo Ida University of Tsukuba, Japan Andres Iglesias Prieto Universidad de Cantabria, Spain Predrag Janicic University of Belgrade, Serbia Deepak Kapur University of New Mexico, USA Ulrich Kortenkamp Martin-Luther-Universit¨at Halle-Wittenberg, Germany Shuichi Moritsugu University of Tsukuba, Japan Julien Narboux Universit´e de Strasbourg, France Pavel Pech University of South Bohemia, Czech Republic Tomas Recio Universidad de Cantabria, Spain Georg Regensburger JohannRadonInstituteforComputationaland Applied Mathematics, Austria Ju¨rgen Richter-Gebert Technische Universita¨t Mu¨nchen, Germany Pascal Schreck Universit´e de Strasbourg, France Meera Sitharam University of Florida, USA Thomas Sturm Max-Planck-Institut fu¨r Informatik, Saarbru¨cken, Germany Additional Reviewers Montes Antonio Robert Joan Arinyo Filip Maric Uwe Waldmann Christoph Zengler Table of Contents Proof and Computation in Geometry ............................... 1 Michael Beeson Automation of Geometry — Theorem Proving, Diagram Generation, and Knowledge Management ........................... 31 Dongming Wang Improving Angular Speed Uniformity by C1 Piecewise Reparameterization .............................................. 33 Jing Yang, Dongming Wang, and Hoon Hong Extending the Descartes Circle Theorem for Steiner n-Cycles.......... 48 Shuichi Moritsugu Equation Systems with Free-Coordinates Determinants ............... 59 Pascal Mathis and Pascal Schreck Formal Proof in Coq and Derivation of an Imperative Program to Compute Convex Hulls......................................... 71 Christophe Brun, Jean-Franc¸ois Dufourd, and Nicolas Magaud From Tarski to Hilbert ........................................... 89 Gabriel Braun and Julien Narboux Realizations of Volume Frameworks ................................ 110 Ciprian S. Borcea and Ileana Streinu Rigidity of Origami Universal Molecules ............................ 120 John C. Bowers and Ileana Streinu Algebraic Analysis of Huzita’s Origami Operations and Their Extensions ...................................................... 143 Fadoua Ghourabi, Asem Kasem, and Cezary Kaliszyk On the Formal Analysis of Geometrical Optics in HOL ............... 161 Umair Siddique, Vincent Aravantinos, and Sofi`ene Tahar Preprocessing of the Axiomatic System for More Efficient Automated Proving and Shorter Proofs ....................................... 181 Sana Stojanovi´c Author Index.................................................. 193 Proof and Computation in Geometry Michael Beeson San Jos´e StateUniversity,San Jos´e, CA, USA Abstract. We consider the relationships between algebra, geometry, computation,andproof.Computershavebeenusedtoverifygeometrical factsbyreducingthemtoalgebraiccomputations.Butthisdoesnotpro- ducecomputer-checkablefirst-orderproofsingeometry.Wemighttryto produce such proofs directly, or we might try to develop a “back-translation” from algebra to geometry, following Descartes but with computer in hand. This paper discusses the relations between the two approaches, the attempts that have been made, and the obstacles remaining. On the theoretical side we give a new first-order theory of “vector geometry”, suitable for formalizing geometry and algebra and the relations between them. On the practical side we report on some experiments in automated deduction in these areas. 1 Introduction The following diagram should commute: Geometric Theorem Algebraic Translation Geometric Proof Algebraic Proof That diagram corresponds to the title of this paper, in the sense that proof is on the left side, computation on the right. The computations are related to geometry by the two interpretations at the top and bottom of the diagram. In thepast,muchworkhasbeenexpendedoneachofthefoursidesofthediagram, bothintheeraofcomputerprogramsandintheprecedingcenturies.Yet,westill do not have machine-found or even machine-checkable geometric proofs of the theoremsinEuclidBookI,fromasuitablesetoffirst-orderaxioms–letalonethe more complicated theorems that have been verified by computerized algebraic computations.1 In other words, we are doing better on the right side of the diagram than we are on the left. 1 A very good piece of work towards formalizing Euclid is [1], but because it mixes computations (decision procedures) with first-order proofs, it does not furnish a counterexample tothe statement in thetext. T.IdaandJ.Fleuriot(Eds.):ADG2012,LNAI7993,pp.1–30,2013. (cid:2)c Springer-VerlagBerlinHeidelberg2013 2 M. Beeson First-order geometrical proofs are beautiful in their own right, and they give moreinformationthanalgebraiccomputations,whichonlytellusthataresultis true,butnotwhyitistrue(i.e.whataxiomsareneededandhowitfollowsfrom those axioms). Moreover there are some geometrical theorems that cannot be treated algebraically at all (because their algebraic form involves inequalities). We will discuss the possible approaches to getting first-order geometrical proofs, the obstacles to those approaches, and some recent efforts. In partic- ular we discuss efforts to use a theorem-prover or proof-checker to facilitate a “backtranslation”fromalgebratogeometry(alongthebottomofthediagram). This possibility has existed since Descartes defined multiplication and square root geometrically, but has yet to be exploited in the computer age. According toChouet. al.([8],pp.59–60),“nosingletheoremhasbeenprovedinthisway.” To accomplish that ultimate goal, we must first bootstrap down the left side ofthediagramasfarasthedefinitionsofmultiplicationandsquareroot,asthat is needed to interpret the algebraic operations geometrically. We will discuss the progress of an attempt to do that, using the axiom system of Tarski and resolution theorem-proving. 1.1 That Commutative Diagram, in Practice In theory, there is no difference between theory and practice. In practice, there is. – Yogi Berra Here is a version of the diagram, with the names of some pioneers2, and on the rightthe names ofthe computationaltechniquesusedinthe algebraiccomputa- tionsarisingfromgeometry.Thedragon,asinmapsofold,representsuncharted and possibly dangerous territory. Geometric Theorem Algebraic Translation Chou, Wu, Descartes Euclid Tarski Wu-Ritt method Szmielew Narboux Chou’s area method CAD (Collins) Here be dragons Gro¨bner bases Descartes, Hilbert Geometric Proof Algebraic “Proof” 2 Many others have contributed to this subject, including Gelernter, Gupta, Kapur, Ko, Kutzlerand Stifter, and Schwabha¨user.
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