Automated Theory Interpretation by Immanuel Normann Submitted as PhD Thesis in Computer Science 19 November 2008 Jacobs University Bremen School of Engineering and Science Date of Defense: 8 December 2008 Dissertation Committee Prof. Dr. Michael Kohlhase, Jacobs University Bremen (Supervisor) Prof. Dr. Herbert Jaeger, Jacobs University Bremen PD Dr. Till Mossakowski, German Research Center for Artificial Intelligence, Bremen Abstract Automated reasoning is a computer aided technology to support human reasoning based on formalized content. The main benefits expected from automated reasoning are relia- bility regarding its correctness and reasoning speed. Speed matters if reasoning is applied on large formalized knowledge bases and reliability matters if critical decisions depend on reasoning or if reliability of reasoning is per se desirable as in mathematics. This thesis presents novel methods from automated reasoning to find reusable knowl- edge and knowledge overlaps in large formalized knowledge bases. The focus is on for- malized mathematics, but the methods are applicable on any kind of formalized content as long as the application of classical logic inference rules on this content is admissible. ormalized as well as traditional mathematics is advantageously organized in theories as its biggest knowledge units. As history of mathematics has shown, appearently unre- lated theories happen to share a common core !i.e. knowledge overlap" once the con- cepts of each theories are translated appropriately. If all valid sentences of a theoryA also hold in another theoryB with an appropriate t ranslationT , thenA is said to be included inB $ orA can be interpreted inB. Since all sentences of a theory are deriv- able from a few of it, namely from its axioms, we know thatA is included inB as soon as we have shown that the axioms ofA hold inB. #n this case all deriva ble sentences in A automatically hold inB too, i.e. the knowledge fromA can be reused inB. #n this thesis an algorithm for automated theory interpretation search is presented. It is based on semantic formula matching by means of normalization taken from term rewriting and a novel standardization techni%ue for associative and commutative terms. &oreover, a practical notion of theory intersection is introduced and an algorithm for the construction of such intersections is presented. 'oth algorithms are implemented in a prototype system and experiments are con- ducted on the largest library of formalized mathematics $ the &izar &athematical (ibrary $ which demonstrates the scalability of the algorithms and also reveal thousands of theory interpretations. )arts of this thesis are based on material previously published* +,-, ,./. Acknowledgement The first time # learned about automated theorem proving was about -0 years ago $ when # finished my studies in physics. 1uring this period # developed a particular interest in formalized mathematical content. # read many articles about this field $ in particular about &ichaels chief work* 2&1oc $ the markup language for 2pen &athe- matical 1ocuments. At the same time however, # thought it was too late in my career to make a new start in research. irst, because # already had a 3ob in industry and second, # was neither a mathematician nor a computer scientist. # am most grateful to &ichael, first of all, because he gave me this second chance to follow my passion for research in formal methods. 4is ideas always inspired me and his patient support during critical phases of my research made it possible to turn my former vague research dreams into a concrete scientific work. All the programming aspects of my research were supervised by Till, whom # thank for his always responsive support, not 3ust in programming issues but also concerning theo- retical %uestions. 1ue to discussions with 4erbert 5aeger # felt encouraged not to forget a philosophical view on mathematics. As one of the earliest members of the working group 678A9: !7nowledge Adapta- tion and 9easoning for :ontent"; # was pleased to see it growing. # en3oyed particularly enthusiastic discussions with <ormen &=ller about various research ideas in endless ritz sessions and (akatos nights. 'ut # also want to thank all other members of the 78A9: group* :hristoph (ange for his support in programming issues and the review of my thesis, lorian 9abe for theoretical insights related to category theory, :hristine &=ller for constructive criti%ues. As member of 5ohn 'ateman>s working group, # want to thank 5ohn and Alexander ?arcia who released me from extra work load during the final phase of my thesis, and Oliver Kutz for many late night discussions. (ast but not least # am deeply indebted to my family* &aren, 7asimir and (orenz. @ou endured so much, but $ 6change has come; $ as someone recently said $ and that>s for sure in our case. Table of contents 1 Introduction.............. ............... ............... A 2 Logical Reasoning.............. ............... .......... -B ..- 1erivability . ............... ............... .......... -, ... Theories . ............... ............... .......... -C ..B <otions of Theory in (ogic and )ractice ............... .......... -D .., The 9ole of Theory Interpretation in &athematics .......... ......... -A 3 State of the Art in Theory Management.............. -E B.- (ittle Theories in ormal &ethods . ............... .......... -E B.. Imps.............. ............... ............... .0 The mathematical database ........... ............... .- 9ecent theoretical development based on Imps.............. .. B.B Maya and the 1evelopment ?raph ........... ............... .. The development graph ............... ............... .. The development graph calculus ......... ............... .B 'asic operations on the development graph. ............... .B The diFerence analysis ............... ............... .B B., Hets.............. ............... ............... ., The architecture of the Hetssystem ......... ............ ., B.C #sabelle . ............... ............... .......... .C B.D :onclusion . ............... ............... .......... .D B.A :riti%ue . ............... ............... .......... .A B.G 9emark ...... ............... ............... .......... .E 4 Theory Interpretation.............. ............... B- ,.- The Hxplored )art ofaTheory ...... ............... .......... BC ,.. Automated Search for Theory Interpretations ............. ......... BD ,...- Algorithm 2utline ............. ............... BD ,.... An #llustrative Hxample . ............... BA ,...B ormal 1evelopment of the Algorithm ............... BE 5 Theory Intersection.............. ............... ,B C.- Algorithm for Theory #ntersection Search ............... .......... ,A C.-.- Algorithm 2utline ............. ............... ,A C.-.. &aximum #ntersection as &aximum :li%ue )roblem . ,A C.-.B #llustrative Hxample ............ ............... ,E 6 ormal Language.............. ............... .......... CB B , Table of contents ! Matching.............. ............... ............... CA A.- #ntroduction to the Simple 9enaming )roblem ............ ......... CG A.. #ntroduction to the H%uational 9enaming )roblem .......... ......... CE A.B #ntroduction to the A:-9enaming )roblem ............... D0 A., ormula Abstraction ............. ............... .......... D. A.,.- Skeleton and )arametrisation of a ormula ............. DB A.C Standardisation Algorithms . ............... .......... DC A.C.- Term 2rdering ............... ............... DC A.C.. Algorithm for Simple Standardisation . ............... DD A.C.B Algorithm for A:-Standardisation . ............... DA " #ormali$ation.............. ............... ............. AB G.- Simple (ogical (anguage .......... ............... .......... AB G.. )reliminaries from Term 9ewriting Theory ............... ......... A, G.B 2verview of <ormalization Steps . ............... .......... AG G., )renex <ormal orm............. ............... .......... AE G.,.- rom &inimal Scope 'ack to )renex <ormal orm . G. G.C 'oolean 9ing <ormalization . ............... ......... GB G.D :ombining the 9ewrite Systems . ............... ......... GC G.A inalizing <ormalization with A:-Standardization . .......... GA % Theory &ompletion.............. ............... ......... GE 1' System (escription.............. ............... EB -0.- The 1idactic System ............ ............... .......... EB -0.-.- 4askell in a <utshell ........... ............... E, -0.-.. (imits of the 1idactic System . ............... EA -0.-.B 2verview . ............... ............... EA -0.-., <ormalization ............... ............... EG -0.-.C Indexing . ............... ............... EE -0.-.D Search . ............... ............... EE -0.-.A Theory Intersection ........... ............... -0. -0.. eatures of the Hnvisioned System . ............... ......... -0, -0...- Input and 2utput of a 9eal System . ............... -0, -0.... Supporting Theories in 1iFerent (ogics and ormats . -0D -0.B unctions Implemented in the )rototype System . ......... -0G -0.B.- Support of 1iFerent ormats in the )rototype System . -0G -0.B.. Input and 2utput in the )rototype System ........... -0G -0.B.B )articular 9estrictions in the )rototype System . -0E 11 )*periments.............. ............... ............ --- --.- Hxperiments on Automated Theory #nterpretation . .. ......... --- --.-.- The &izar System and its (ibrary . ............... --- --.-.. &&( as Axiomatic (ibrary . ............... --. --.-.B &&( in an Intyped irst-Order (ogic ............... --B --.-., Translating