Andrei Voronkov - Böcker

Handbook of Automated Reasoning - Vol 2

Ershov Informatics Conference, PSI 2019, held in Novosibirsk, Russia, in July 2019. The 18 full papers and 3 short papers presented in this volume were carefully reviewed and selected from 70 submissions.

This book constitutes the refereed proceedings of the 10th International Andrei Ershov Informatics Conference, PSI 2015, held in Kazan and Innopolis, Russia, in August 2015.

This book constitutes the refereed proceedings of the 11th International Andrei P. Ershov Informatics Conference, PSI 2017, held in Moscow, Russia, in June 2017.  The 31 full papers presented in this volume were carefully reviewed and selected from 57 submissions. The papers cover various topics related to the foundations of program and system development and analysis, programming methodology and software engineering and information technologies.

ThisvolumecontainsthepaperspresentedattheNinthInternationalC- ferenceonLogicforProgramming,Arti?cialIntelligence,andReasoning(LPAR 2002),heldonOctober14-18,2002,attheUniversityofTbilisi(Georgia),- getherwiththeThirdInternationalWorkshoponImplementationofLogics. Therewere68submissions,ofwhicheightbelongedtothespecialsubmission categoryofexperimentalpapers,intendedtodescribeimplementationsorc- parisonsofsystems,orexperimentswithsystems. Eachsubmissionwasreviewed byatleastthreeprogramcommitteemembersandanelectronicprogramc- mitteemeetingwasheldviatheInternet. Thenumberofsubmissionsandshort reviewingperiodcausedalargeamountofwork,andweareverygratefultothe other22PCmembersfortheire?ciencyandforthequalityoftheirreviewsand discussions. Finally,thecommitteedecidedtoaccept30papers. Theprogramme alsoincludesthreeinvitedlecturesandapostersession. Apartfromtheprogrammecomittee,wewouldalsoliketothanktheother people who made LPAR 2002 possible: the additional referees, and the local arrangementschairsKhimuriRhukia,KotePhakadze,GelaChankvetadze,and JemalAntidze. TheInternet-basedsubmissionsoftwareandtheprogramcommitteedisc- sionsoftwarewereprovidedbythesecondco-chair.August2002 MatthiasBaaz AndreiVoronkov ConferenceOrganization ProgramChairs MatthiasBaaz(TechnischeUniversit..atWien) AndreiVoronkov(UniversityofManchester) ProgramCommittee ElviraAlbert(UniversidadPolitecnicadeValencia) FranzBaader(TechnischeUniversit..atDresden) MauriceBruynooghe(KatholiekeUniversiteitLeuven) PatrickCousot(ENSParis) MaartendeRijke(UniversityofAmsterdam) HaraldGanzinger(Max-PlanckInstitut,Saarbruc ..ken) JeanGoubault-Larrecq(ENSdeCachan) MikiHermann(EcolePolytechnique) MatejaJamnik(UniversityofCambridge) NeilJones(KobenhavnsUniversitet) DeepakKapur(UniversityofNewMexico) MaurizioLenzerini(Universit'adiRoma"LaSapienza") GiorgioLevi(Universit'adiPisa) LeonidLibkin(UniversityofToronto) RobertNieuwenhuis(TechnicalUniversityofCatalonia) LeszekPacholski(WroclawskiUniwersytet) MichelParigot(Universit'eParis7) PavelPudlak(MathematicalInstitute,Prague) CarstenSchuermann(YaleUniversity) J..orgSiekmann(DFKISaarbruc ..ken) WolfgangThomas(RWTHAachen) MichaelZakharyaschev(King'sCollege) LocalOrganization KhimuriRhukia KotePhakadze GelaChankvetadze JemalAntidze VIII ConferenceOrganization ListofReferees GianlucaAmato CarlosAreces M.Arenas OferArieli RobertoBarbuti ChristophBenzmueller MartaBilkova BrunoBlanchet AlexandreBoisseau AhmedBouajjani DmitriChubarov PabloCordero VeroniqueCortier AgostinoDovier RachidEchahed MorenoFalaschi C'esarFerri OlivierGasquet BernhardGramlich Jos'eManuelG'omez PhilippedeGroote RainerHaehnle JiriHanika JuanHeguiabehere JoseHernandez-Orallo ThomasHillenbrand JoeHurd EmilJerabek ValentinJijkoun KonstantinKorovin G. Lakemeyer DominiqueLarchey-Wendling OlivierLaurent StefanLeue LuigiLiquori ChristofLoeding CarstenLutz ChristopherLynch PatrickMaier PaoloMancarella FlorenceMaraninchi MaartenMarx Jesus ' Medina EricaMelis AntoineMin'e RalfM..oller PaulinDeNaurois HansdeNivelle MichaelNorrish ManuelOjeda-Aciego NikolayPelov GuyPerrier ReinhardPichler ChristopherPollett Germ'anPuebla Jean-PierreRessayre ChristianRetor'e AlexandreRiazanov MichalRossler PaulRozi'ere NikitaSakhanenko UlrikeSattler StefanSchlobach ManfredSchmidt-Schauss KonradSlind Anni-YasminTurhan ChristianUrban WimVanhoof YdeVenema So?eVerbaeten UweWaldmann KlausWeich ClausPeterWirth StefanWoehrle ConferenceOrganization IX ConferencesprecedingLPAR2001 RCLP'90,Irkutsk,SovietUnion RCLP'91,Leningrad,SovietUnion,aboardtheship"MichailLomonosov" LPAR'92,St. Petersburg,Russia,aboardtheship"MichailLomonosov" LPAR'93,St. Petersburg,Russia LPAR'94,Kiev,Ukraine,aboardtheship"MarshalKoshevoi" LPAR'99,Tbilisi,RepublicofGeorgia LPAR2000,ReunionIsland,France LPAR2001,Havana,Cuba TableofContents ImprovingOn-DemandStrategyAnnotations ...1 M. Alpuente,S. Escobar,B. Gramlich,S. Lucas First-OrderLogicasaConstraintProgrammingLanguage...19 K. R. Apt,C. F. M. Vermeulen MaintenanceofFormalSoftwareDevelopmentsby Strati?edVeri?cation...36 SergeAutexier,DieterHutter ANoteonUniversalMeasuresforWeakImplicit ComputationalComplexity ...53 ArnoldBeckmann ExtendingCompositionalMessageSequenceGraphs...68 BenediktBollig,MartinLeucker,PhilippLucas SearchingforInvariantsUsingTemporalResolution ...86 JamesBrotherston,AnatoliDegtyarev,MichaelFisher, AlexeiLisitsa ProofPlanningforFeatureInteractions:APreliminaryReport...102 ClaudioCastellini,AlanSmaill AnExtensionofBDI withFunctional CTL DependenciesandComponents...115 MehdiDastani,LeendertvanderTorre BindingLogic:ProofsandModels ...130 GillesDowek,Th'er'eseHardin,ClaudeKirchner DirectedAutomatedTheoremProving...145 StefanEdelkamp,PeterLeven AFrameworkforSplittingBDIAgents...160 XiaocongFan,JohnYen OntheComplexityofDisjunctionandExplicitDe?nability PropertiesinSomeIntermediateLogics...175 MauroFerrari,CamilloFiorentini,GuidoFiorino UsingBDDswithCombinationsofTheories ...190 PascalFontaine,E. PascalGribomont XII TableofContents OnExpressiveDescriptionLogicswithCompositionofRolesin NumberRestrictions...202 FabioGrandi QueryOptimizationofDisjunctiveDatabaseswithConstraints throughBindingPropagation ...216 GianluigiGreco,SergioGreco,IrinaTrubtsyna,EsterZumpano ANon-commutativeExtensionofMELL...231 AlessioGuglielmi,LutzStrassburger ProceduralSemanticsforFuzzyDisjunctivePrograms...247 Du?sanGuller PushdownSpeci?cations...262 OrnaKupferman,NirPiterman,MosheY. Vardi TheoremProvingwithSequenceVariablesandFlexible AritySymbols...

ThisvolumecontainsthepaperspresentedattheTenthInternationalConference on Logic for Programming, Arti?cial Intelligence, and Reasoning (LPAR 2003), heldonSeptember22-26,2003,inAlmaty,Kazakhstan,togetherwiththeFourth International Workshop on Implementation of Logics. There were 65 submissions, of which 8 belonged to the special submission category of experimental papers, intended to describe implementations or c- parisons of systems, or experiments with systems. Each submission was reviewed by at least three program committee members and an electronic program c- mitteemeetingwasheldviatheInternet.Weareverygratefultothe32program committee members for their e?orts and for the quality of their reviews and d- cussions. Finally, the committee decided to accept 27 papers. The program also included ?ve invited talks, by Franz Baader, Serikzhan Badaev, Dexter Kozen, Sergei Goncharov, and Thomas Wilke. Apart from the program committee, we would also like to thank the other people who have made LPAR 2003 possible: the external reviewers and the local organizers Serikzhan Badaev and Anna Romina.The Internet-based submission software and the program-committee disc- sion software were provided by the second co-chair.

This volume contains the papers presented at the 11th International Conference on Logic for Programming, Arti?cial Intelligence, and Reasoning (LPAR), held from March 14 to 18, 2005, in Montevideo, Uruguay, together with the 5th - ternational Workshop on the Implementation of Logics (organized by Stephan Schulz and Boris Konev) and the Workshop on Analytic Proof Systems (or- nized by Matthias Baaz). The call for papers attracted 77 paper submissions, each of which was - viewed by at least three expert reviewers. The ?nal decisions on the papers were taken during an electronic Program Committee meeting held on the Internet. The Internet-based submission, reviewing, and discussion software EasyChair, provided by the second PC co-chair, supported each stage of the reviewing p- cess. But the most important work was, of course, done by the 34 PC members and their external reviewers, who provided high-quality reviews. After intense discussions to resolve con?icts among the reviewers, the Program Committee decided to accept 33 papers. The conference program also included 4 invited talks, by Jurgen ..Giesl, Al- ander Leitsch, Helmut Seidl, and Igor Walukiewicz, which are documented by short or extended abstracts in these proceedings. In addition, Mart' ?n Abadi held a tutorial on Reasoning About Security Protocols, and Ian Horrocks on Description Logic Reasoning.

This volume containsthe full papers presentedat the 12th InternationalConf- ence on Logic for Programming, Arti?cial Intelligence, and Reasoning (LPAR), held 2-6 December 2006,in MontegoBay,Jamaica.The callfor papersattracted 108 full paper submissions, each of which were reviewed by at least three - viewers. The Program Committee accepted the 46 papers that appear in these proceedings. The conference program also included 4 invited talks, by Tom Ball of Microsoft Research, Doug Lenat of Cycorp, Roberto Nieuwenhuis of the U- versidad Polit' ecnica de Cataluna " , and Allen Van Gelder of the University of California at Santa Cruz. Papers or abstracts for the invited talks are in these proceedings. In addition to the main program, the conference o?ered a short paper track, which attracted 13 submissions, of which 12 were accepted, and the Workshop on Emperically Successful Higher Order Logic (ESHOL). Thanks go to: the authors (of both accepted and rejected papers); the P- gramCommitteeandtheirreviewers;theinvitedspeakers;ChristophBenzmuller .. , John Harrison, and Carsten Schur .. mann for organizing ESHOL; Celia Alleyne- Ebanks for administering the conference in Jamaica; the Honorable Minister Phillip Paulwell of the Ministry of Commerce, Science and Technology for op- ing the conference (and Daphne Simmonds for introducing us to the minister); the Mona Institute of Applied Sciences at the University of the West Indies for their support; Microsoft Research for sponsorship of student regsitrations; the Kurt Godel .. Society for taking registrations; and EasyChair for hosting the review process.

This volumecontains the papers presentedatthe SeventhInternationalC- ference on Logicfor Programmingand Automated Reasoning (LPAR 2000)held onReunionIsland,France,6-10November2000,followedbythe ReunionWo- shop on Implementation of Logic. Sixty-?ve papers were submitted to LPAR 2000 of which twenty-six papers were accepted. Submissions by the program committee members were not - lowed. There was a special category of experimental papers intended to describe implementations of systems, to report experiments with implemented systems, orto compareimplementedsystems.Eachof thesubmissionswasreviewedbyat least three program committee members and an electronic program committee meeting was held via the Internet. In addition to the refereed papers, this volume contains full papers by two of the four invited speakers, Georg Gottlob and Micha. el Rusinowitch, along with an extended abstract of Bruno Courcelle's invited lecture and an abstract of Erich Gr. adel's invited lecture. WewouldliketothankthemanypeoplewhohavemadeLPAR2000possible.We are grateful to the following groups and individuals: the program and or- nizing committees; the additional referees; the local arrangements chair Teodor Knapik; PascalManoury, who was in chargeof accommodation; Konstantin - rovin,whomaintainedthe programcommittee Webpage;andBillMcCune,who implemented the program committee management software.

This book constitutes the refereed proceedings of the 8th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2001, held in Havana, Cuba, in December 2001. The 40 revised full papers presented together with an invited paper were carefully reviewed and selected from 112 submissions. The book offers topical sections on verification, guarded logic, agents, automated theorem proving, non-classical logics, types, experimental aspects, foundations of logic, CSP and SAT, nonmonotonic reasoning, semantics, termination, knowledge-based systems, analysis of logic programs, databases and knowledge bases, and program analysis and proof planning.

This book constitutes the refereed proceedings of the 18th International Conference on Automated Deduction, CADE - 18, held in Copenhagen, Denmark, in July 2002. The 27 revised full papers and 10 system descriptions presented together with three invited contributions were carefully reviewed and selected from 70 submissions. The book offers topical sections on description logics and the semantic Web, proofcarrying code and compiler verifications, non-classical logics, system descriptions, SAT, model generation, CASC, combination and decision procedures, logical frameworks, model checking, equational reasoning, and proof theory.

This book constitutes the refereed proceedings of the 13th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2006, held in Phnom Penh, Cambodia in November 2006. The 38 revised full papers presented together with one invited talk were carefully reviewed and selected from 96 submissions.

This volume contains the proceedings of two Russian conferences on logic programming, held in 1990 in Irkutsk and in 1991 in St. Petersburg. The aim of the conferences was to bring together researchers from the Russian and the international logic programming communities. Topics covered in the volume include: logic programming, automated theorem proving, non-monotonic reasoning, applications of mathematical logic to computer science, deductive databases, implementation of declarative concepts, and programming in non-classical logics. This is the first volume to represent activity in the field of logic programming in the countries of the former Soviet Union.

This volume contains the proceedings of LPAR '92, theinternational conference on logic programming and automatedreasoning held in St. Petersburg in July 1992. The aim ofthe conference was to bring together researchers from theRussian and the international logic programming and theoremproving communities. The topics of interest covered by papers inthe volumeinclude automated theorem proving, non-monotonic reasoning,applications of mathematical logic to computer science,deductive databases, implementation of declarative concepts,and programming in non-classical logics. LPAR '92 is the successor of the First and Second RussianConferences on Logic Programming held in 1990 and 1991,respectively, the proceedings of which were publishedinLNAI Vol. 592.

LPAR is an international conference series aimed at bringingtogether researchers interested in logic programming andautomated reasoning. The research in logic programming grewout of the research in automated reasoning in the early1970s. Later, the implementation techniques known from logicprogramming were used in implementing theorem provingsystems. Results from both fields applied to deductivedatabases. This volume contains the proceedings of LPAR '93, which wasorganized by the Russian Association for Logic Programming. The volume contains 35 contributed papers selected from 84submissions, together with an invited paper by Peter Wegnerentitled "Reasoning versus modeling in computer science".

This book constitutes the thoroughly refereed and revised post-workshop proceedings of two international meetings devoted to deductive databases and logic programming.The technical papers presenting original research results were carefully reviewed and selected for inclusion in the present book, together with several survey articles summarizing the state of the art in the area. Thus, the book is the ultimate reference for anybody interested in the theory and application of deductive databases and logic programming.

This book constitutes the refereed proceedings of the 6th International Conference on Logic Programming and Automated Reasoning, LPAR'99, held in Tbilisi, Georgia in September 1999.The 23 revised full papers presented were carefully reviewed and selected from 44 submissions. Among the topics addressed are logic programming, constraint logic programming, complexity aspects, logical inference search, model checking, formal specification, rewriting, practical reasoning, Horn clauses, linear logic, partial evalutation, etc.

This book constitutes the refereed proceedings of the 19th International Conference on Rewriting Techniques and Applications, RTA 2008, held in Hagenberg, Austria, July 15-17, in June 2008 as part of the RISC Summer 2008. The 30 revised full papers presented were carefully reviewed and selected from 57 initial submissions. The papers cover current research on all aspects of rewriting including typical areas of interest such as applications, foundational issues, frameworks, implementations, and semantics.

This volume contains the ?nal proceedings of the Sixth International Andrei Ershov Memorial Conference on Perspectives of System Informatics (PSI 2006), held in Akademgorodok (Novosibirsk, Russia), June 27-30, 2006. The conference was held to honour the 75th anniversary of a member of the Russian Academy of Sciences Andrei Ershov (1931–1988) and his outsta- ing contributions towards advancing informatics. The role of Andrei Ershov in the establishment and development of the theory of programming and systems programming in our country cannot be overestimated. Andrei was one of the founders of the Siberian Computer Science School. He guided and took active part in the development of the programming system ALPHA and the mul- language system BETA, and authored some of the most remarkable results in the theoryofprogramming. Andreiisjustly consideredoneofthefoundersofthe theory of mixed computation. In 1974 he was nominated as Distinguished F- low of the British Computer Society. In 1981 he received the Silver Core Award for services rendered to IFIP. Andrei Ershov’s brilliant speeches were always in the focus of public attention. Especially notablewerehis lectures “Aesthetic and HumanFactorinProgramming”and“Programming—TheSecondLiteracy. ”He was not only an extremely gifted scientist, teacher and ?ghter for his ideas, but also a bright and many-sided personality. He wrote poetry, translated the works of R. Kipling and other English poets, and enjoyed playing guitar and singing. Everyone who had the pleasure of knowing Andrei Ershov and working with him will always remember his great vision, eminent achievements and generous friendship.

This book features the refereed proceedings of the 2nd International Symposium on Computer Science in Russia held in September 2007. The 35 papers cover theory track deals with algorithms, protocols, and data structures; complexity and cryptography; formal languages, automata and their applications to computer science; computational models and concepts; proof theory; and applications of logic to computer science. Many applications are presented.

This book constitutes the refereed proceedings of the 14th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2007, held in Yerevan, Armenia. It contains 36 revised full papers, 15 short papers and three invited talks that were carefully selected from 78 submissions. The papers address all current issues in logic programming, logic-based program manipulation, formal method, automated reasoning, and various kinds of AI logics.

This book constitutes the refereed proceedings of the 15th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2008, which took place in Doha, Qatar, during November 22-27, 2008. The 45 revised full papers presented together with 3 invited talks were carefully revised and selected from 153 submissions. The papers address all current issues in automated reasoning, computational logic, programming languages and their applications and are organized in topical sections on automata, linear arithmetic, verification knowledge representation, proof theory, quantified constraints, as well as modal and temporal logics.

This volume contains the ?nal proceedings of the 7th International Andrei Ershov Memorial Conference on Perspectives of System Informatics Akad- gorodok (Novosibirsk, Russia), June 15–19, 2009. PSI is a forum for academic and industrial researchers, developers and users working on topics relating to computer, software and information sciences. The conference serves to bridge the gaps between di?erent communities whose - searchareasarecoveredbybutnotlimitedtofoundationsofprogramandsystem development and analysis, programming methodology and softwareengineering, and information technologies. PSI 2009 was dedicated to the memory of a prominent scientist, academician Andrei Ershov (1931–1988),and to a signi?cant date in the history of computer science in the country, namely, the 50th anniversary of the Programming - partment founded by Andrei Ershov. Initially, the department was a part of the Institute of Mathematics and later, in 1964, it joined the newly established Computing Center of the Siberian Branch of the USSR Academy of Sciences. Andrei Ershov, who was responsible for forming the department, gathered a team of young graduates from leading Soviet universities. The ?rst signi?cant project of the department was aimed at the development of ALPHA system, an optimizing compiler for an extension of Algol 60 implemented on a Soviet c- puterM-20. Later,theresearchersofthedepartmentcreatedtheAlgibr,Epsilon, Sigma, and Alpha-6 programming systems for the BESM-6 computers. The list of their achievements also includes the ?rst Soviet time-sharing system AIST-0, the multi-language system BETA, research projects in arti?cial intelligence and parallel programming, integrated tools for text processing and publishing, and many others.

This volume contains the research papers presented at the 17th International Conference on Logic for Programming, Arti?cial Intelligence, and Reasoning (LPAR-17), held in Yogyakarta, Indonesia, October 10-15, 2010, accompanied by the 8th International Workshop on the Implementation of Logic (IWIL-8, organized by Eugenia Ternovska, Stephan Schulz, and Geo? Sutcli?e) and the 5th International Workshop on Analytic Proof Systems (APS-5, organized by Matthias Baaz and Christian Fermuller .. ). The call for papers attracted 133 abstract submissions of which 105 ma- rialized into full submissions, each of which was assigned for reviewing to at least three Program Committee members; 41 papers were accepted after - tense discussions. Once more the EasyChair system provided an ideal platform for submission, reviewing, discussions, and collecting ?nal versions of accepted papers. The program included three invited talks by Krishnendu Chatterjee, Joseph Halpern, and Michael Maher, as well as an invited tutorial by Norbert Preining. They are documented by the corresponding papers and abstract, respectively, in these proceedings, which this year appear for the ?rst time in the ARCoSS subline of the Lecture Notes in Computer Science.

This book constitutes the thoroughly refereed post-conference proceedings of the 16th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR 2010, whichtook place in Dakar, Senegal, in April/May 2010.The 27 revised full papers and 9 revised short papers presented together with 1 invited talk were carefully revised and selected from 47 submissions. The papers address all current issues in automated reasoning, computational logic, programming languages and deal with logic programming, logic-based program manipulation, formal methods, and various kinds of AI logics. Subjects covered range from theoretical aspects to various applications such as automata, linear arithmetic, verification, knowledge representation, proof theory, quantified constraints, as well as modal and temporal logics.

This book constitutes the proceedings of the 18th International Conference on Logic for Programming, Artificial Intelligence, and Reasoning, LPAR-18, held in Merida, Venezuela, in March 2012. The 25 regular papers and 6 tool descriptions and experimental papers presented were carefully reviewed and selected from 74 submissions.The series of International Conferences on Logic for Programming, Artificial Intelligence and Reasoning (LPAR) is a forum where, year after year, some of the most renowned researchers in the areas of logic, automated reasoning, computational logic, programming languages and their applications come to present cutting-edge results, to discuss advances in these fields, and to exchange ideas in a scientifically emerging part of the world.

This book contains thoroughly refereed and revised papers from the 8th International Andrei Ershov Memorial Conference on Perspectives of System Informatics, PSI 2011, held in Akademgorodok, Novosibirsk, Russia, in June/July 2011.The 18 revised full papers and 10 revised short papers presented were carefully reviewed and selected from 60 submissions. The volume also contains 5 invited papers covering a range of hot topics in computer science and informatics. The papers are organized in topical sections on foundations of program and system development and analysis, partial evaluation, mixed computation, abstract interpretation, compiler construction, computer models and algorithms for bioinformatics, programming methodology and software engineering, information technologies, knowledge-based systems, and knowledge engineering.

This Festschrift volume, published in memory of Harald Ganzinger, contains 17 papers from colleagues all over the world and covers all the fields to which Harald Ganzinger dedicated his work during his academic career.The volume begins with a complete account of Harald Ganzinger's work and then turns its focus to the research of his former colleagues, students, and friends who pay tribute to him through their writing. Their individual papers span a broad range of topics, including programming language semantics, analysis and verification, first-order and higher-order theorem proving, unification theory, non-classical logics, reasoning modulo theories, and applications of automated reasoning in biology.

This book constitutes the proceedings of the 19th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, LPAR-19, held in December 2013 in Stellenbosch, South Africa. The 44 regular papers and 8 tool descriptions and experimental papers included in this volume were carefully reviewed and selected from 152 submissions. The series of International Conferences on Logic for Programming, Artificial Intelligence and Reasoning (LPAR) is a forum where year after year, some of the most renowned researchers in the areas of logic, automated reasoning, computational logic, programming languages and their applications come to present cutting-edge results, to discuss advances in these fields and to exchange ideas in a scientifically emerging part of the world.