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Integration of Symbolic Computation and Mechanized Reasoning, September 10-12 2003, Roma, Italy
Thérèse Hardin, Renaud Rioboo
To cite this version:
Thérèse Hardin, Renaud Rioboo. CALCULEMUS-2003 - 11th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, September 10-12 2003, Roma, Italy. Thérèse Hardin; Renaud Rioboo. CALCULEMUS-2003 - 11th Symposium on the Integration of Symbolic Computation and Mechanized Reasoning, Sep 2003, Rome, Italy. LIP6; Aracne Editrici, 2003. �hal- 02549766�
CALCULEMUS-2003
11th Symposium on the Integration of Symbolic
Computation and Mechanized Reasoning
September 10-12 2003
Roma, Italy
Therese Hardin and Renaud Rioboo (Eds)
TheuseofComputerAlgebrasystemsisnowwide-spreadnotonlyineducation
orscienticcontextsbutalsoin industry,wheremathematicalsoftwaresystems
helpengineerstodesignsystems.Inthesameway,thegrowingneedsforamore
formalapproachinsoftwareindustryrequirepowerfuldeductionsystems,help-
ingengineerstoprovethatthedevelopmentsagreewiththeirrequirements.The
combination of automated mathematical computation and automated mathe-
matical deduction is the majortopic of theCALCULEMUS symposium. This
includes development ofmorereliableand accuratecomputeralgebrasystems,
morepowerfuland exiblededuction systems.Butessentially, theCALCULE-
MUS symposium is intended to researchersand developersinterestedin coop-
eration and unication between the two families of mathematical based soft-
wareandof theircommunities. Forthese reasons,CALCULEMUS symposium
co-locateinalternateyearswitheitheraComputerAlgebraconferenceorade-
ductionconferences.Thisisthecasein2003:CALCULEMUSisco-locatedwith
TABLEAUX2003andTPHOL2003.WethankMartaCielda,thelocalorganiser
ofthisjoined conferences.
Wewouldliketo thankthemembersoftheprogram committeeandallthe
refereesfortheirimportantworkin selectingthe submittedpapers.Wehad29
submissionsoutofwhichweselected6longpapersand9shortpapers.Thebest
paperswillbepublishedinaspecialissueoftheLondonMathematicalSociety's
JournalofComputation andMathematics.Submissionswill berequired
ThereseHardinandRenaudRioboo
Co-chairs
Program Commitee
Chairs ThereseHardin
RenaudRioboo
Members AndreaAsperti
HenkBarendregt
ChrisBenzmuller
OlgaCaprotti
JamesDavenport
WilliamFarmer
HoonHong
FairouzKamareddine
MichaelKohlhase
SteveLinton
LoicPottier
RobertoSebastiani
VolkerSorge
ThomasSturm
StephenWatt
WolfgangWindsteiger
Additional referees
PhilippeAubry
GillesAudemard
QuocBaoVo
MarcoBozzano
JacquesCarette
Veronique Donzeau-Gouge
CatherineDubois
HermanGeuvers
DimitarP Guelev
ManfredKerber
TemurKutsia
RoyMcCasland
ValerieMenissier-Morain
MiladNiqui
MartinPollet
BasSpitters
JeremieWajs
Freek Wiedijk
Claus-PeterWirth
Sponsoring Institutions
http://www.colognet.org/
TheEuropeanNetworkof Excellencyin ComputationalLogic
Local Organization
MartaCialdeaMayer
TheCalculemusResearchTrainingNetwork|AshortOverview ::::: 1
Christoph Benzm uller
QueryingDistributedDigitalLibrariesofMathematics::::::::::::::::: 17
Ferruccio Guidi,ClaudioSacerdotiCoen
?
FoCDoc:TheDocumentationSystemofFoC ::::::::::::::::::::::::: 31
ManuelMaarek, Virgile Prevosto
BrokersandWeb-ServicesforAutomaticDeduction:aCaseStudy :::::: 43
ClaudioSacerdoti Coen,Stefano Zacchiroli
TrustableCommunicationBetweenMathematicsSystems :::::::::::::: 58
JacquesCarette,William M.Farmer,Jeremie Wajs
SystemDescription:Analytica2 :::::::::::::::::::::::::::::::::::: 69
Edmund Clarke, Michael Kohlhase, JoelOuaknine,Klaus Sutner
ANewInterfacetoPVS ::::::::::::::::::::::::::::::::::::::::::: 74
A.A.Adams
IntegratingComputationalPropertiesattheTermLevel::::::::::::::: 78
Martin Pollet, Volker Sorge
Towardsahigherreasoninglevelin formalizedHomological Algebra::::: 84
Jes usAransay, ClemensBallarin,Julio Rubio
Makingproofsinahierarchyofmathematicalstructures::::::::::::::: 89
Virgile Prevosto, MathieuJaume
Formalproofsandcomputationsin niteprecisionarithmetic::::::::::: 101
SylvieBoldo, MarcDaumas, LaurentThery
Inductivedenitionsversusclassicaldependent choice in theMinlog
system::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 112
Ulrich Berger and Monika Seisenberger
fu.berger,[email protected]
Building Convex Hullsby Combining SAT Solving andAlgebraic
Computing::::::::::::::::::::::::::::::::::::::::::::::::::::::: 118
SilvioRanise
RingsandModulesinIsabelle/HOL :::::::::::::::::::::::::::::::: 124
ExploringanAlgorithmforPolynomialInterpolationin theTheorema
System :::::::::::::::::::::::::::::::::::::::::::::::::::::::::: 130
WofgangWindsteiger
SomeGrandMathematicalChallengesin MechanizedMathematics:::::: 137
JacquesCalmet
| A short Overview
?
ChristophBenzmuller
FRInformatik,UniversitatdesSaarlandes,66041Saarbrucken,Germany
1 Introduction
This papersketchesthestructureand scienticcontributionsof theCalcule-
musResearchTrainingNetwork(CalculemusRTN)sinceitsstartinSeptem-
ber 2000. It has been reproduced from the networks midterm report [22] and
credit is dueto all researchersof theCalculemusRTN. Morethan28 young
visiting researchers (with asum of approx. 150nanced person-months)have
been supported by the network so far and approx. 47 senior researchers are
involved in the training measures at the dierent partner sites. Figure 1 pro-
videsthelistof theCalculemusRTN partnersites.Thenetwork'shomepage
ishttp://www.eurice.de/calculemus/.
2 Motivation
The long-term motivation of the Calculemus research initiative (see www.
calculemus.net)is tofoster thedevelopmentof anewgenerationof assistant
systemsfor mathematics and formal methods. Somekeycharacteristicsof the
systems Calculemusis aiming at arecompiled in the following (incomplete)
list:
{ Combinedsupportforsymbolic reasoningandsymboliccomputation.
{ Interoperabilitywithemergingdecentralisedandsharedmathematicalknowl-
edgebases.
{ Support mechanisms for the exploration, validation, and maintenance (in
particularmanagementofchange)ofdomain specic knowledge.
{ Supportforexibleintegrationofheterogeneousspecialistreasonersassub-
systems(includingclassicalautomated theoremprovers,modelgenerators,
decisionprocedures,etc.).
{ Provisionofrichandexpressiverepresentationlanguagesandcommunication
meanstotheusersside(probablyincludingratherinformalorevennatural
languagebasedrepresentations)incombinationwithhuman-oriented,multi-
modaluserinterfaces.
?
This work is supported by the EU Research Training Network CALCULEMUS
USAAR
SaarlandUniversity,Saarbrucken,Germany(JorgSiekmann,Christoph
Benzmuller)
UED TheUniversityofEdinburgh,Scotland(AlanBundy)
UKA KarlsruheUniversity,Germany(JacquesCalmet)
RISC
Research Institute for Symbolic Computation, Linz, Austria (Bruno
Buchberger)
TUE
EindhovenUniversityofTechnology,Netherlands(ArjehCohen)
UniversityofNijmegen,Netherlands(HenkBarendregt)
ITC-IRST
InstitutoperlaRicercaScienticaeTecnologica,Trento,Italy(Fausto
Giunchiglia)
UWB UniversityofBialystok, Poland(AndrzejTrybulec)
UGE UniversitadegliStudidiGenova(AlessandroArmando)
UBIR TheUniversityofBirmingham,England(ManfredKerber)
Fig.1.TheCalculemusRTN
{ Support for transformationsbetweenthe expressiveand user-orientedrep-
resentations employed in the assistantsystem and the usually highly spe-
cialised machine-oriented representations employed by the integrated spe-
cialistreasoners.
{ Developmentandutilisationofopensystemarchitecturesfosteringinterop-
erabilityandtoolexchangebetweendierentassistantsystems(forexample,
intheemergingmathematicalsemanticweb).
{ Directsupportforthepreparationandvalidationofmathematicaltextsand
publications.
{ Applicationsin mathematics,mathematicseducation,andformalmethods.
These research goals are ambitious and call for the combination of resources
and the mutual exchange of scientic expertise between theinvolved scientic
communities. Totacklethem, Calculemus isbasically pursuinga bottom-up
approachstartingfromsingleresearchaspectsasmentionedaboveandfromthe
existingand emergingtoolsoftheinvolvedresearchgroups.
The current scientic focus is on the integration of symbolic computation
andsymbolicreasoningwhichhasbeenidentiedasamajorissue.Thesociolog-
ical goalof theCalculemusRTNis tocombinethescienticexpertise ofthe
involvedresearchersinordertooptimallytrainanddevelopanewgenerationof
youngresearchersin considerationoftheimpliedscienticchallenges.
3 Calculemus RTN: Research Objectives and Results
A predominant research objective of the Calculemus RTN is to foster the
integration of deduction systems (DS) and computer algebra systems (CAS),
bothataconceptualandatapracticallevel.Thepointoforiginforthiskindof
researchis alandscapeofheterogeneousapproachesand systemsonbothsides
ofthespectrum,wherethediversityontheDSssideisgreaterthanontheside
Sinceitsstartin September2000theCalculemusRTNhascontributedto
theconvergenceof DSsandCASs throughits researchonunifyingframeworks
forencodingandcombiningcomputationanddeduction,theidenticationofthe
architecturalrequirementsforanewgenerationofreasoningsystemswithcom-
binedreasoningandcomputationalpower,andtheprototypicalimplementation
and application oftheimprovedsystems.However,asinglepredominant theo-
reticalframeworkiscurrentlynotpossible.Suchanapproachwouldparticularly
involvepredominantsolutionstothestillratherdivergingsystemsatbothsides
of thespectrum betweenDSsand CASs.Therefore astrong line ofresearchin
the CalculemusRTN focuses on themodelling and integration of CASs and
DSsatthesystemslayer.Inthisresearchdirection,signicantprogresshasbeen
madeandseveralsystemsofprojectpartnersandotherresearchinstituteshave
beenconnectedin order toform networksof cooperatingmathematicalservice
systems. Thebenetsand impacts of such integrations havebeeninvestigated
in prototypicalcasestudies.
The researchers of the Calculemus RTN and the Calculemus interest
groupalsofosteredtheMathematicalKnowledgeManagement(MKM,EUMKM-
NET)researchinitiative;see[40,8].Thisrelativelyyounglineofresearchadopts
abroaderperspectiveonthefuture of mathematics(e.g. researchand publica-
tion practice, education, and knowledge maintenance) in the 21st century. A
signicantamountofCalculemusresearchisMKM relevantand iscurrently
being takenup in this community in order to adopt and integrate it into the
MKMperspective.
TheextensiveresearchactivitiesoftheCalculemusNetworkandtheCal-
culemusInterestGrouparefurthermoreshowninteraliabythreespecialissues
of the Journalof SymbolicComputation [101,4,78] and the following interna-
tionalevents:CalculemusSymposium2000inSt.Andrews,Scotland[69,101],
Calculemus Symposium 2001 in Siena, Italy [78],Calculemus Symposium
2002in Marseilles, France [45,49], CalculemusAutumn School2002in Pisa,
Italy[23{25,128].TheCalculemusSymposium 2003 1
will beheldin Septem-
berin Rome,Italy,anditwilljoin IJCARconferencein2004.
In the following paragraphs we sketch the highlights of the Calculemus
RTNsinceitsstartinSeptember2000;formoredetailedreportstoalltaskswe
referto[22].
Task1.1:MathematicalFrameworks TUEandNijmegenUniversityinves-
tigatedtypetheoryforthepurposeofformalisingmathematics:Barendregtand
Geuvers[21] givean overview of typetheory, how it is used to representlogic
and mathematicsand what issuesand choices comeup. Typetheory(encoded
in OpenMath) as a way for communicating mathematics is proposed in [20]
and in [48] it is shown how aproofpresentation can begenerated from afor-
malised proof in typetheory. This paper argues that `formal contexts' in Coq
canbeusedasabasisforinteractivemathematicaldocuments.Thistopicisalso
1
treatedin[99].Anin-depthdiscussionofthevariouswaystotreatcomputations
intheorem proversisgivenin [19]andfurtherrelatedworkispresentedin [36].
TheCalculemusRTNhasalsostudiedotherapproachestotheoremproving
and their capacities to integrate computations (see also [122]). This includes
proofplanning,asdevelopedandemployedbythenodesUSAARandUED.In
the mega system [104], at USAAR, symbolic calculationscanbe integrated
into proof planning in two ways: (i) to guide the proof planner and to prune
thesearchspace bycomputinghintswithcontrol rulesand (ii)to shorten and
simplify the proofs by calling a CAS within the application of a method to
solveequations.Asaside-eectbothcasescanrestrictpossibleinstantiationsof
meta-variables.Theseapproachesarediscussedin[52,107,84,105].
Aninvestigationintotheuseofdeductionfortheimplementationofcorrect
computations within computer algebra system was considered at UGE and is
presentedin [1].
TheTheorema system,developedat RISC,aimsat providingone mathe-
maticalframeworkencompassingallaspectsofalgorithmicmathematics,notably
theaspectsofproving, computing,andsolving;see[39,37,38].
In[70,71]itiscriticallyarguedbyUBIRthataspectsofmathematicalcon-
cepts,includingproceduralknowledge,arehardtoreconstructfromtheformal-
isationindeductionsystems.Thisworkpointstolimitationsoftheexibilityof
mathematicalrepresentationswhichapplyto allourcurrentapproaches.
Task 1.2: Denition of Mathematical Service The primary goal of this
Task is the enhancement of existing computer algebrasystems and deductive
systemsby turning them into open systems capableof using and/orproviding
mathematical services. After a preliminary analysis of the state-of-the-art of
reasoningsystems, it wasdecided to tacklethe problem, in parallel,by atop-
downandabottom-upapproach.
Inthetop-downapproach,newinfrastructures(bothattheconceptual,spec-
ication, and architectural level) for the seamless integration of mathematical
serviceshavebeeninvestigated.Thiswasintendednotonlyforcurrentsystems,
but also and in particular for future implementations. To this extent particu-
laremphasiswasonthedenitionofframeworks(languages,protocols,semantic
specications,architecturalschemata)suitableformakingmathematicalservices
accessible overthe web.The relevant top-downapproachesare: OMRS (Open
Mechanised Reasoning Systems) developed by UGE and ITC-IRST [2], LBA
(LogicBrokerArchitecture)developedbyUGE[6,7],MathWeb-SB(MathWeb
SoftwareBus)developedbyUSAAR[129],MathBrokerdevelopedbyRISC[81].
These networks canthemselvesbe coupled again as, for instance, exemplarily
investigatedin[127].
Inthebottom-upapproach,wehaveinvestigatedhowcomplexmathematical
servicescanbebuiltoutofsimplerones.Aparticularemphasishasbeendevoted
to decision procedures,and in particular to theintegrationof procedures spe-
bottom upapproachesare CCR(ConstraintContextual Rewriting)developed
byUGEandMathSat[61,11,10,9,12],developedbyITC-IRST.
InTask 1.2 the Calculemusnetwork alsoclosely cooperateswith theEU
projectMONET.InMONETspecialontologiescomprisingmathematicalprob-
lems,queriesandserviceshavebeendened andinvestigated.
Task2.1:IntegrationofCASsandDSsviaProtocols Cooperationamong
severalsoftwaresystemscanbeachievedwithindirect,unidirectionalandbidi-
rectional communication. The goalof this task is to investigatehowprotocols
canbedenedtoprovideasemanticsaswellassoundnessresultsforsystemsex-
changingmathematicalinformation.Thisdenitionhintsat severalothertasks
in the Calculemus RTN dealingwith very similar problems. This is for ex-
ample truewhen dening acontextforacomputation andis partlycoveredin
Task 1.Unidirectionalandbidirectionalcommunicationprotocols aredesigned
when coupling directly dierent modules. Although there are no direct links
betweentheserviceswithindirect communication,interactionispossiblewhen
systemscancommunicate withacommonuserinterface,centralunit,mediator
or evaluator. This approach, which is partlybased on ajoint work with ITC-
IRSTonOMSCS(OpenMechanisedSymbolicComputationSystems),hasbeen
investigatedwithintheKometsystematUKAsee[44,76,55,46].
Asemanticscanbeprovidedbyatleastthreeapproaches:(a)deneamathe-
maticalsoftwarebus,(b)deneacontextfromwhichasemanticcanbederived,
(c)formulate theproblem asaknowledgerepresentationparadigm.
These approachesare shared by several of the partners. Indeed, they lead
to introduce multi-agentsystems, contexts, and ontologies to just quote afew
features(seeforinstancetheLBAandtheMathWeb-SB).
Task2.2: Enhancingthe Reasoning Power ofComputer Algebra Sys-
tems EnhancementofCASwithreasoningpowercanbeattemptedatdierent
levels:(a)enhancementof CASonthe SystemLevel, (b)enhancementof CAS
ontheTheoryLevel,and(c)enhancementofCAS ontheUserLevel.
Direction (a) can be achieved by adding additional reasoning capabilities,
i.e.,logicalinferencesystems,toalgorithmsbuiltintotheCAS.TheConstraint
ContextualRewriting(CCR)frameworkdevelopedbyUGEcanbeusedinorder
to integratetheevaluationmechanismoftheCAS Maplewithanappropriate
decisionprocedureforcheckingside-conditions,see[1]and[5].
Direction(b)canbeachievedbyaddingprovenknowledgeaboutCASfunc-
tionstotheCASknowledgebase.TheHRsystem,developedatUED,hasbeen
used toconjecturepropertiesoffunctions available in theMaplealgorithm li-
braryfromempiricalpatterns detectedincomputationaldata producedbythe
CAS[53].
Direction(c)canbeachievedbygivingtheCASuserthepossibilitytoprove
mathematical statementsusing proof techniques from logic within theCAS in
Calculemus RTN, the work of RISC representsthis aspect of CAS enhance-
ment:TheTheoremasystem,see[41],isanadd-onpackageforthewidespread
and popularCAS Mathematica where theuser formulates mathematical theo-
remsandprovesthementirelywithin theMathematica environment.
Task 2.3: Enhancing the Computation Power of Deductions Systems
UEDinvestigatedthecombinationoftheproof-plannerClam[102]withother
systemsforcomputationallycostlytasks.Thisincludes(a)animplementationof
thegsexibledecisionprocedure systemframeworkin (Teyjus)LambdaProlog
andwithintheClamproofplanningsystem[42]and(b)theintegrationofthe
Clamproof-plannerintotheMathWeb-SBsystem[54].
UED also investigated the combination of systems to discover attacks to
security protocols [108,109]. This work makes use of computational power in
that itgeneratesalargenumberofclausesinitsprocessing.
FurtherrelevantworkhasbeendoneintheClamproof-plannertoconstruct
verylargeand modular proof-plansfor complicatedreal analysistheorems [65,
79,80].
ThemegaproofplanneratUSAARhasbeencoupledwithdierentCASs
viaMathWeb-SB,see[107,84,105].TheantsapproachtointegrateCASsinto
mathematicalassistantsystemsissketchedin[29,28,34,35].Thisworkproposes
anagent-basedmodellingofinferencerulesandexternalsystemsataverybasic
levelwithin theoremprovers.
Finally, work doneat UBIR andUGE which render techniques from auto-
mated reasoning highly eÆcient by using enhanced computational power are
presentedin [66{68]and[9,12,3].Furtherrelevantworkisgivenin [100].
Task 3.1: AutomatedSupport to WritingMathematical Publications
Typically,amathematicalpublication containsthefollowingingredients:natu-
rallanguagetext,mathematical formulae,formaltext (i.e.denitions andthe-
orems), proofs, examples (typically with computations), and graphics (tables,
drawings,sketches,etc.).Intheoptimalcase,asoftwaresystemfor supporting
mathematicalpublicationswould supportallthesefacetsofmathematicalpub-
lications. Severalsystemsandlanguageshavebeenusedforcasestudiesin this
area:
(a)TheMIZARapproach(atUWB)isbasedontwokindsofsoftwarewhich
automatetheprocess ofwritingformal mathematicalpapers:(i) softwareused
to prepare an article as a formal text whose correctness is computer veried
and(ii)thesoftwareforautomatic(orsemi-automatic)translationinto natural
language (particularly English); this includes also the software for translation
into XML-based formats. The cooperation with other Calculemus sites in-
cludes developmentof theMIZAR Mathematical Library (MML) andalso the
abovementionedtranslation into XML formats.Relevantpublications are [88,
60,16{18,94].Recentlypublished MIZAR articlesintheJournal ofFormalized
Mathematicsare[113,74,95,63,117,73,103,15,14,64,89,97,90,111,112,98,93,
