Noname manuscript No will be inserted by the editor

Andrew Gacek

Description

Niveau: Supérieur, Doctorat, Bac+8
ar X iv :0 91 1. 29 93 v2 [ cs .L O] 1 Se p 2 01 0 Noname manuscript No. (will be inserted by the editor) A two-level logic approach to reasoning about computations Andrew Gacek · Dale Miller · Gopalan Nadathur the date of receipt and acceptance should be inserted later Abstract Relational descriptions have been used in formalizing diverse computational notions, including, for example, operational semantics, typing, and acceptance by non-deterministic machines. We therefore propose a (restricted) logical theory over relations as a language for specifying such notions. Our specification logic is further characterized by an ability to explicitly treat binding in object languages. Once such a logic is fixed, a natural next question is how we might prove theorems about specifications written in it. We propose to use a second logic, called a reasoning logic, for this purpose. A satisfactory reasoning logic should be able to completely encode the specification logic. Associated with the specification logic are various notions of binding: for quantifiers within formulas, for eigenvariables within sequents, and for abstractions within terms. To provide a natural treatment of these aspects, the reasoning logic must encode binding structures as well as their associated notions of scope, free and bound variables, and capture-avoiding substitution. Further, to support arguments about provability, the reasoning logic should possess strong mechanisms for constructing proofs by induction and co-induction.

• logic


• nominal constant


• capture


• can therefore


• using ?-abstraction


• rule


• specification language


• clause can


• clause


• rules

Sujets

• École polytechnique
• Capture
• Specification language
• Clause

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 27
• Langue: English

Extrait

NonamemanuscriptNo. (willbeinsertedbytheeditor) Atwo-levellogicapproachtoreasoningaboutcomputations AndrewGacek ∙ DaleMiller ∙ GopalanNadathur thedateofreceiptandacceptanceshouldbeinsertedlater Abstract Relationaldescriptionshavebeenusedinformalizingdiversecomputationalnotions,including,for example,operationalsemantics,typing,andacceptancebynon-deterministicmachines.Wethereforeproposea (restricted)logicaltheoryoverrelationsasalanguageforspecifyingsuchnotions.Ourspecificationlogicisfurther characterizedbyanabilitytoexplicitlytreatbindinginobjectlanguages.Oncesuchalogicisfixed,anatural nextquestionishowwemightprovetheoremsaboutspecificationswritteninit.Weproposetouseasecondlogic, calledareasoninglogic,forthispurpose.Asatisfactoryreasoninglogicshouldbeabletocompletelyencodethe specificationlogic.Associatedwiththespecificationlogicarevariousnotionsofbinding:forquantifierswithin formulas,foreigenvariableswithinsequents,andforabstractionswithinterms.Toprovideanaturaltreatment oftheseaspects,thereasoninglogicmustencodebindingstructuresaswellastheirassociatednotionsofscope, freeandboundvariables,andcapture-avoidingsubstitution.Further,tosupportargumentsaboutprovability, thereasoninglogicshouldpossessstrongmechanismsforconstructingproofsbyinductionandco-induction.We providethesecapabilitiesherebyusingalogiccalledGwhichrepresentsrelationsoverλ-termsviadefinitions ofatomicjudgments,containsinferencerulesforinductionandco-induction,andincludesaspecialgeneric quantifier.WeshowhowprovabilityinthespecificationlogiccanbetransparentlyencodedinG.Wealsodescribe aninteractivetheoremprovercalledAbellathatimplementsGandthistwo-levellogicapproachandwepresent severalexamplesthatdemonstratetheefficacyofAbellainreasoningaboutcomputations. 1Introduction Weareinterestedinthispaperinspecifyingcomputationsandthenreasoningaboutthem.Arangeofformalisms havebeenusedasameansforrealizingthefirstoftheseobjectives.Forexample,theexecutionsemanticsofpro- gramminglanguageshavebeendescribeviatheλ-calculus[Reynolds,1972,Plotkin,1976],theπ-calculus[Milner, 1992],andabstractmachines[Landin,1964].Aspecificationformalismthathasbeenparticularlysuccessfuland widelyapplicableisoperationalsemanticsinbothits“small-step”version[Plotkin,1981]andits“big-step”ver- sion[Kahn,1987].Ofthemanymatureandflexiblechoicesthatcanbemade,wepickhererelationalspecifications andtheirdirectencodingastheoriesinrestrictedlogics.Thischoiceallowsustotransparentlyencodeoperational semanticsaswellasarangeofothernotionsincluding,mostnotably,typing.Anotherconsequenceofourchoice isthatourspecificationlanguagewill,infact,beaspecificationlogic.Morespecifically,itwillturnouttobea simple,wellunderstoodlogicthatcanbeinterpretedasalogicprogramminglanguageinthestyleofλProlog [NadathurandMiller,1988]. Afteronehaspickedalanguageforwritingspecifications,thereisstillachoicetobemadeaboutalanguage forreasoningaboutthem.Thechoicesofthesetwolanguagesareoftenrelated.Ifonehasselectedaspecification languagerelyingon,say,processcalculus,thenareasoninglanguagethatexploitsbisimulationsandcongruences wouldbeanaturalchoice:see,forexample,[Sangiorgi,1994].Ifonechoosesabstractmachinesforspecifications A.GacekandD.Miller INRIASaclay-I ˆ le-de-France&LIX/E ´ colePolytechnique,Palaiseau,France E-mail:gacekatlix.polytechnique.fr,dale.milleratinria.fr G.Nadathur DepartmentofComputerScienceandEngineering,UniversityofMinnesota 4-192EE/CSBuilding,200UnionStreetSE,Minneapolis,MN55455USA E-mail:gopalanatcs.umn.edu

theninductivedefinitionsareanaturalchoiceforareasoninglanguage.Inthispaper,ourreasoninglanguage willbealogicthatcontainsstandardbutpowerfulmechanismsforinductionandco-inductionaswellasthe∇- quantifier(bothinformulasandindefinitions)[MillerandTiu,2005,Gaceketal.,2009].Ourchoiceofreasoning logichasanumberofappealingaspects,chiefamongthembeingthatitispowerfulenoughtocapturewithinit manyotherreasoningtechniquessuchasbisimulationandinductivedefinitions. Theapproachweshalldescribeinthispaperisthuscharacterizedbytheuseoftwologics,oneforspecifying computationsandtheotherforreasoningabouttheselogicspecifications.Wepickbothoflogicstobeintuition- isticherebutotherchoicesarealsosensible:forexample,McDowellandMiller[2002]usedalinearlogicasa specificationlanguageinordertoprovidedeclarativespecificationsforaprogramminglanguagewithstate.The logicalsymbolsofthesetwologicswillbeseparatedinourtreatment:infact,provabilityinthespecificationlogic willbeaninductivelydefinedpredicateofthereasoninglogic.Althoughwedistinguishthelogicsinthisway,the termstructuresthattheyusewillbeidentical:inparticular,theconstructionoftermsinbothlogicswillusethe sameapplicationandabstractionoperations.Asaresult,termequalityinthereasoninglogicwillimmediately reflecttermequalityinthespecificationlogic. Inmanycommonlyusedapproaches,itisproblematictotreatspecification-languageabstractionthrough reasoning-logicabstraction.Thereasoninglogicofteninvolvesfunctiontypesthatcontainrecursivefunctions; thisisthecase,forexample,inCoqandIsabelle/HOL.Iffunctionabstractionatthetwolevelsareidentified, functiontypesinthespecificationlanguagewouldalsohavetocontainrecursivelydefinedfunctions.Sincethe specificationlanguageisintendedtotreatsyntacticexpressionsandnotgeneralfunctions,thisraisesissuesabout theadequaterepresentationofsyntax:see,forexample,thediscussionabout“exoticterms”in[Despeyrouxetal., 1995].Inthesettingthatweshallsoonunfold,wegetaroundsuchproblemsbymakingfunctiontypesinboth thespecificationandthereasoninglogicweak:whiletermequalitywillstillbegovernedbytherulesforαβη- conversion,thiswillhappenwithinthesimplytypedλ-calculusthatdoesnotincludestrongerprinciplessuchas recursion.Torecovertheloststrength,wewilluseinductivelyandco-inductivelydefinedpredicatesforreasoning aboutcomputations.However,atthepredicatelevel,thespecificationlogicandthereasoninglogicwillbestrictly separated.Insummary,functiontypesinbothlogicswillbeweakandwillbeusedexclusivelytorepresentsyntax thatmaycontainbindings.FollowingMiller[2000],weshallcallthisstyleofencodingdatawithbindingsthe λ-treeapproachtoabstractsyntax. Whenwedevelopthetwo-levellogicapproachindetail,theformulasofthespecificationlogicwillbecome termsofthereasoninglogic.Thisapproachtoencodinganobjectlogicwithinasecondlogicshouldbecontrasted totheapproachofprovabilitylogic(see,forexample,[Smorynski,2004])wherenaturalnumbersareusedto denotesyntacticobjectsofanobject-logicandprimitiverecursivefunctionsareusedtoparseandmanipulate thoseobjects-cum-natural-numbers.Ourencodingismoredirect:bothtermsandformulasofthespecificationlogic arerepresentedbytermsinthereasoninglogic,withsimpletypesbeingusedtoseparatetermsfromformulas. Moreover,thepresenceofbindinginthetermsofthereasoninglogicmakesitpossibletorepresentquantified formulasinthespecificationlogicinanimmediateandnaturalmanner. Thereareanumberofadvantagestothetwo-levellogicapproachtoreasoningandtheparticularrealization ofitthatwediscussinthispaper.First,becauseofthetermstructuresusedinthereasoninglogic,onlymild encodingtechniquesareneededtoembedthespecificationlogicinit:forexample,specification-leveltermequality isdirectlycaptured,quantificationinthespecificationlogicistreatedbyusingλ-abstractiontobindthequantified variable,andtheinstantiationofquantifiersisrealizedthroughβ-conversion.Second,sincespecificationsare writteninalogicandsincesuchalogictypicallyhasmeta-theoreticproperties(suchascut-admissibility)that canbeformalizedinthereasoninglogic,powerfultechniquesbecomeavailableforreasoningaboutdescriptions presentedinthespecificationlogic.Third,asaseriesofexamplesillustrates,thistwo-levellogicapproachcan resultinnatural,readable,andcompletelyformalproofsofwell-knowntheoremsaboutcomputationalsystems. Finally,whenonemovestoimplementingtheoremproversbasedonthisarchitecture,onlyonenotionofbinding, variable,termequality,substitution,andunificationneedstobetreatedforbothlogics. InthenextsectionwedescribetheaspectsofthereasoninglogicGthatweshalluseinthispaper.Section3 presentsthespecificationlanguagehH 2 andshowshowcut-freesequentcalculusprovabilityforitcanbegiven anadequateencodinginG.Section4describesbrieflythestructureofatheorem-provercalledAbellathatcan beusedtointeractivelyconstructsequentcalculusproofsinG.ThisdescriptionisthenexploitedinSection5 topresentexamplesoftheuseofthetwo-levellogicapproach.Section6describesrelatedworkandSection7 concludeswithanindicationofsomefuturedirections. 2