adjointlogic and its concurrent semanticsfp/talks/edinburgh17-talk.pdfadjointlogic and its...

Adjoint LogicandItsConcurrentSemantics

FrankPfenningABCDMeeting,Edinburgh,December18-19,2017JointworkwithKlaas Pruiksma andWilliamChargin

Outline

• Proofsasprograms• Linearsequentproofsasconcurrentprograms

• Take1:standardsequentcalculus=synchronouscommunication• Take2:unary⊗Rand–oR• Take3:positiveaxiomaticform=asynchronouscommunication

• Adjoint logic• Structuralpropertiesandmodesoftruth• DeclarationofIndependence

• Concurrentoperationalsemantics• Contraction,weakening,multicut,andmulti-identity• Garbagecollection

12/19/17 ABCDMeeting,Edinburgh 2

ProofsasPrograms

• Codesign oflanguageanditsreasoningprinciples• Threelevelsofcorrespondence

• Propositionsastypes• Proofsasprograms• Proofreductionascomputation

• Styleofproofsystemiscriticaltocharacterizecomputation• Axiomaticstyle⟺ combinators andcombinatoryreduction[Curry’35]• Naturaldeduction⟺ λ-calculusandsubstitution[Howard’69]• Sequentcalculus⟺ explicitsubstitutions[Herbelin’94]

• Allintuitionisticandstructural(admittingweakening&contraction)


LinearProofsasSession-TypedPrograms

• Threelevelsofcorrespondence• Linearpropositionsassessiontypes• Sequentproofsasconcurrentprograms• Cutreductionascommunication

• Intuitionisticvariant:provider/clientmodel[Caires &Pf’10]• Noneedtodualize types• Dependenttypes[Toninho etal.’11]• Monadicintegrationwithfunctionallanguage(SILL)[Toninho’15][Griffith’15]• Integrationinimperativelanguage(CC0)[Willsey etal.16]• Polymorphismandlogicalrelations[Perezetal.’13]• Exploitingcategoricalview(thistalk)

• Classicalvariant:symmetriccommunication• [Wadler’12][Caires etal.’16]


Outline






� ` c : A �0, c : A ` e : C

�,�0 ` e : Ccut

CutasParallelComposition

c:A

provideschannelc:AprocessP processQ

usedby

ΔΔ’

e :C

whichprovidese:C

processconfigurationalwaysformsatree


CutasProcessSpawn

c:A

Δ’

e :C

Δ

x :A

ΔΔ’

e :C

� ` c : A �0, c : A ` e : C

�,�0 ` e : Ccut


IdentityasIdentification

d : A ` c : Aid

d:A c:A

d=c:A

transitionofconfigurationmodelscutofanyproofwithidentity


Tensor,Original

x :A

ΔΔ’

c:A⊗ B

� ` x : A �0 ` c : B�,�0 ` c : A⌦B

⌦R

c:B

Δ

Δ’ c:B

d :A


Tensor,Simplified

c:A⊗ B

c:B

d :A

� ` c : B�, d : A ` c : A⌦B

⌦R⇤

cutreductionpredisposestosynchronoussemantics

d :A


Interderivable usingIdentityandCut

A ` Aid

� ` B�, A ` A⌦B

⌦R

� ` A�0 ` B

�0, A ` A⌦B⌦R⇤

�,�0 ` A⌦Bcut


TensorasaMessage

d : A, c0 : B ` c : A⌦B⌦RA

c:A⊗ B

c’:B

d :A

c’:B

d :A

asynchronousmessagesendingd:Aalongc:A⊗ Bwithcontinuationc’:B

useaxiomaticformforpositiveprovidersornegativeclients(”senders”)


Interderivable with2cuts,2ids

A ` Aid

B ` Bid

A,B ` A⌦B⌦R

� ` A

�0 ` B A,B ` A⌦B⌦RA

�0, A ` A⌦Bcut

�,�0 ` A⌦Bcut

because⊗RAhasnocontinuationanasynchronoussemanticsseemsforced!


PositiveAxiomaticFormulation

• Forces(?)asynchronoussemantics• Restoresynchronouscommunicationviamode-neutralshifts[Pf&Griffith’15]• Canwriteall“sending”rulesasaxioms

• Rightrulesforpositiveconnectives• Leftrulesfornegativeconnectives

• Canhidethisfromsurfacesyntax,ifdesired


Outline






StructuralRules

• Weakening:donotneedtouseantecedents(affinelogic)• Contraction:mayreuseantecedents(strictlogic)• Weakening+contraction=structurallogic

• Compromisestheusualcutreductionandcutelimination!

� ` e : C�, c : A ` e : C

W�, c1 : A, c2 : A ` e : C

�, c : A ` e : CC


Adjoint Logic[Reed’09]

• Wouldliketohaveourcakeandeatit,too!• Allowweakening,contractionwheredesirableornecessary

• Challenges• Howdowemakesystemcoherent:linearremainslinear,etc.• Combinationshouldbeconservativeoveritsparts• Logically:cutelimination,identityelimination• Operationally:sessionfidelity,globalprogress(evenwithrecursion)• Uniformsyntaxandsemantics


ModesofTruth

• Modesoftruthk,m,n• EverypropositionAm hasanintrinsicmodeoftruthm• Everymodempossessesstructuralpropertiesσ(m)⊆{W,C}

• Itispossibletoaddexchangeasanoption,startingfromorderedlogic

• Modesarerelatedbypreorder≤,wherem≤kimpliesσ(m)⊆σ(k)


ModesofTruth,continued

• Syntaxofpropositions(andproofs)isuniformacrossallmodes

• Shift↑Ak referencespropositionAk inmodem,form≥k• Shift↓An referencespropositionAn inmodem,forn≥m

mknm

Am, Bm ::= Pm | Am ( Bm | N{` : A`m}`2L | "mk Ak

| Am ⌦Bm | 1 | �{` : A`m}`2L | #nmAn


Example:IntuitionisticLinearLogic

• UnrestrictedmodeUwithσ(U)={W,C}• LinearmodeLwithσ(L)={}• L<U• ModeUpopulatedonlybyshifts

AU ::= "ULAL

AL ::= AL ( BL | · · · | #ULAU

!AL , #UL "ULAL


Example:LNL[Benton’94]

• UnrestrictedmodeUwithσ(U)={W,C},withallconnectives• LinearmodeLwithσ(L)={}• L<U

m ::= L | U with L < U

Am, Bm ::= Pm | Am ( Bm | N{` : A`}`2L | "ULAL

| Am ⌦Bm | 1 | �{` : A`}`2L | #ULAU

AU ! BU , AU ( BU


OtherExamples

• IntuitionisticS4~stagedcomputation• U<V,σ(U)=σ(V)={W,C}

• Laxlogic~monadicencapsulation• X<U,σ(X)=σ(U)={W,C}

• Subexponential logic• Likeadjoint logic,distinguishedmodeL• Allothermodesmcontainonly↑AL

• !mAL ≜ ↓↑AL

mLm

LmL


TheDeclarationofIndependence

• Keytoobtainingcoherence,conservativity,cutelimination,uniformity• Ψ ::=ε |Am |Ψ1,Ψ2 where‘,’isassociative,commutative,w/unitε• Ψ ≥mifk≥mforeveryAk inΨ

• TruthofAm mustbeindependentfromallAk fork≥m

Ψ ⊢ Am iswell-formedonlyifΨ ≥m


RulesforAdjoint Logic

• Logicalrulesunchanged,stayatsame,butarbitrarymodem• Structuralrulesdependonmodeproperties

W 2 �(m) � ` e : Cr

�, c : Am ` e : CrW

C 2 �(m) �, c1 : Am, c2 : Am ` e : C

�, c : Am ` e : CrC


RulesforShift

• Derivedentirelyfromdeclarationofindependence


` Ak

` "mk Ak"R k � r , Ak ` Cr

, "mk Ak ` Cr"L

� n ` An

` #nmAn#R , An ` Cr

, #nmAn ` Cr#L

JudgmentalRules

• Cutandidentityarethemostinterestingrules• Standardargumentforcuteliminationdoesnotworkinthepresenceofexplicitcontraction• ReturntoGentzen (1935):Multicut


Multicut

• Somerestrictionsforcedbydeclarationofindependence

• IfWϵσ(m),thenn=0isallowed• IfCϵσ(m),thenn>1isallowed• Provideris“awareof”allclientchannelsc


c = (c1, . . . , cn) � m � r ` c : Am 0, c1:Am, . . . , cn:Am ` e : Cr

, 0 ` e : Crmcut⇤

Multi-identity

• Providermightbecommunicatingwithmultipleclients

• IfWϵσ(m),then|c|=0isallowed• IfCϵσ(m),then|c|>1isallowed

d : Am ` c : Amid⇤


Outline






ProcessInterpretation

• Transitionsdonotcareaboutmodesatruntime• However,somerulescareaboutthenumberofclientsaprocesshas• Inthisuniformsemantics,computationatallstructuralpropertiesisimplementedbymessagepassing• Transitionsofprocessconfigurationsmimiccutreductions(notcutelimination)• Shiftssendandreceive’shift’messageswhichsynchronize[Pf&Griffith’15]


Multicut,ProcessInterpretation

c1 :A

cn :A

c:A ci :A

c=c1,…,cn

providershouldknowaboutallclientchannels

oneproviderwithmultipleclients


withoutmodes,toemphasizedynamicmodeindependence

� ` c : A �0, c1:A, . . . , cn:A ` e : C

�,�0 ` e : Cmcut

KeyIdea:Contraction=Identity+Multicut

c : A ` c : Aid

�, c1 : A, c2 : A ` e : C

�, c : A ` e : Cmcut

c:A

c:Ac1 :A

c2 :A

c:A


KeyIdea:Weakening=Identity+Multicut

c:A

c:A ϵ :A


KeyIdea:IdentityPropagation

• Helpstoimplementbothdrop(W)andcopy(C)• Distributedgarbagecollection“forfree”

b :A

dn :A

b,d:A d1 :A

c:A

dn :A

c,d:A d1 :A

c1 :Ack :A

c1 :A

ck :A


KeyIdeas:MulticastandCopy-on-Receive

• Positivetypes(e.g.,A⊗ B)domulticast,iftherearemultipleclients• Negativetypes(e.g.,A–oB)copy-on-receive,iftherearemultipleclients• Previousworkon!Aonlyperformedcopy-on-receive


OngoingandFutureWork

• Exploitindependenceprinciplefornon-uniformsemantics• S=shared,U=unrestricted,L=linearwithS≤U,U≤S,L<U,L<Sandσ(S)=σ(U)={C,W},σ(L)={}[Balzer &Pf,2017]

• H=locallyheap-allocated,L=linear,L<Hwithσ(H)=?,σ(L)={}• Proofirrelevance,intensional equality[Pf’01]• Ghostmessagesforcontractsandreasoningaboutdistributedcomputation

• Isthereageneraltheoremaboutnon-uniformcompatibility?• Surfacesyntax?(Uniformispossible,uptoapoint)• Generalimplementation


Summary

• Adjoint logiccombineslogicscoherentlyandconservatively• Shiftscomposetoamonad(oneorder)andcomonad (otherorder)• DeclarationofIndependenceenableskeyresults

• Cutelimination,identityelimination• Conservativeextensionovercombinedfragments

• Uniformmessage-passingsemanticsviamulticut andidentity• Contraction=identity+multicut forn>1• Weakening=identity+multicut forn=0• Distributedgarbagecollection=identitypropagation


mk


adjointlogic and its concurrent semanticsfp/talks/edinburgh17-talk.pdfadjointlogic and its...

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