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City, University of London Institutional Repository

Citation

:

Howe, J. M. (2001). Proof search in Lax Logic. Mathematical Structures in

Computer Science, 11(4), pp. 573-588. doi: 10.1017/S0960129501003334

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(2)

Proof Searh in Lax Logi

JaobM.Howe

y

ComputingLaboratory, UniversityofKent

Reeived13June2000

AGentzensequentalulusfor LaxLogiispresented,theproofsinwhihnaturally

orrespondina1{1waytothenormalnaturaldedutionsforthelogi.Thepropositional

fragmentofthisalulusisusedasthebasisfor anotheralulus,onewhihusesa

historymehanisminordertogiveadeisionproedureforpropositional LaxLogi.

1. Introdution and Bakground

Proof searh an be used with either of twomeanings. It an either be used to mean

the searh for all proofs of a formula (proof enumeration), or to mean the searh for

a yes/no answer to a query (theorem proving). This paperdesribestwo new sequent

aluliforLaxLogi.OnealulusisforproofenumerationforquantiedLaxLogi,the

otheralulusis fortheorem provinginpropositionalLaxLogi.

Lax Logi is an intuitionisti modal logi rst introdued by Curry (Curry, 1952)

to illustrate ut-elimination in the presene of modalities. The logi was redisovered

by Mendler,who developed the logi in the ontext of hardware veriation to enable

abstrat veriation of iruits (Mendler, 1993). The logi has a single modality (Æ,

somehow)axiomatisedby

SÆS; ÆÆSÆS; (ST)(ÆSÆT)

The modality is unusual in having properties of both neessity and possibility. It an

be thought of asexpressing orretness up to a onstraint, abstrating awayfrom the

detail(hene the hoie of name, Lax Logi). A formula ÆP an be read as\forsome

onstraint,P holdsunder ".Theprooftheoryandsemantisof LaxLogi,inluding

Gentzen aluli, naturaldedutionaluliand Kripkesemantis, arefurther developed

in(Fairtlough andMendler,1997;FairtloughandWalton,1997;Bentonetal.,1998).

TheabilityofLaxLogitogiveanabstratexpressionofonstraintshasbeenutilised

bothinhardwareveriationandtogiveaprooftheoretisemantisforonstraintlogi

programminglanguages.Inhardwareveriation,thetimingonstraintsthatneedtobe

satisedin airuitanbeabstrated away asinstanesof themodalityand reasoned

aboutseparatelyfrom thelogialanalysis oftheiruit(Mendler,1993;Fairtlough and

Mendler,1994).Inonstraintlogiprogramming,LaxLogihasbeenusedtoextendthe

viewoflogiprogrammingasbakwardsproofsearhinonstrutivelogis(Milleretal.,

(3)

1991). Inessene, this approah takesnormal naturaldedution astheprooftheoreti

semantisforlogiprogramming.Constraintsanbeabstratedawayasmodalitiesand

the queryan be reasonedaboutlogially. The logi is used to give proofs of queries.

In turn, these proofs givethe onstraints to be satised. The onstraintsan then be

analysedseparately(Fairtloughet al.,1997;Walton,1998).

Natural dedution hasa pragmatidrawbak.In searhing bakwards fora proof of

a formula, it is notalwaysobvious whih rule to apply. For example, in Intuitionisti

Logiitisnotobviousfromtheonlusionthatrule(

"

)shouldbeapplied.Evenwhen

the rule has been xed, it is hard to determine the formulae in the premiss. Cut-free

Gentzen sequent alulus systems(Gentzen, 1969)are muh better from this point of

view.Whenaprinipalformulahasbeenhosen,therulesappliablearerestrited.The

appliationoflogialrulesis diretedbythesyntaxoftheprinipalformula.Strutural

rules an often be built into the sequent system. In suh a system, when a prinipal

formulahasbeenhosen,thenextruleappliation isexatlydetermined bythesyntax

ofthatformula.

Therearewellknowntranslations(Prawitz,1965)betweennormalnaturaldedutions

and sequent proofs. Therefore, one an searh for proofs in sequent alulus systems

and thentranslatethe resultingproofs to normalnaturaldedutions. Thedrawbak is

that manysequentproofstranslatetothesamenormalnaturaldedution.Hene when

oneis tryingto enumerate allproofsofaformula,thesameproof isfoundmanytimes.

Thisgivesonemotivationfor`permutation-free'sequentaluli(introduedin(Herbelin,

1995)forIntuitionistiLogi).Thesearesequentaluli(enablingsyntaxdiretedproof

searh)whoseproofsanbetranslatedina1{1waywiththenormalnaturaldedutions

forthelogi.Permutation-freealulihavetheadvantagesofasequentalulussystem,

whilstreetingthestrutureofnormalnaturaldedutions.Therstalulusdesribed

in this paper, PFLAX, is a proof enumeration alulus for rst-order quantied Lax

Logi.PFLAXisapermutation-freealulusforLaxLogi{thesequentproofsnaturally

orrespondina1{1waytothenormalnaturaldedutions.

Propositionallogisareusually deidableandthereforeitisdesirabletondeetive

deisionproeduresforsuhlogis.Here,bystudyingthenatureofnon-terminating

bak-wardssearhtoseewhereoneanstopthesearh,adeisionproedureforpropositional

LaxLogiisgiven;thistheoremprovingalulusisalledPFLAX

Hist

.Thealulususes

atehniquefordetetingloopsusingahistorymehanism,buildingonworkof

Heuerd-ingetal (Heuerdingetal.,1996;Heuerding,1998;Howe,1997).Itusesthepropositional

fragment of PFLAX asthe base alulus to whih a historymehanism is added,

giv-ingthedeisionproedure.Thetehniqueisgeneralandmaybeappliedto manyother

propositionallogis.WeknowofnootherdeisionproedureforpropositionalLaxLogi.

2. Natural Dedution

This setion gives the relevant material on natural dedution needed to develop the

permutation-freealulusforLaxLogi(theproofsinwhihorrespondina1{1wayto

(4)

;P`P (ax)

`>

(>I)

`?

`P

(?")

;P`Q

`P Q

( I

)

`PQ `P

`Q

( "

)

`P `Q

`P^Q

(^ I

)

`P^Q

`P

(^ "1

)

`P^Q

`Q

(^ "2

)

`P

`P_Q

(_ I

1 )

`Q

`P_Q

(_ I

2 )

`P_Q ;P `R ;Q`R

`R

(_ "

)

`P

`ÆP

(ÆI)

`ÆP ;P`ÆQ

`ÆQ

(Æ")

`P[u=x℄

`8xP

(8I)y

`8xP

`P[t=x℄

(8 "

)

`P[t=x℄

`9xP

(9I)

`9xP ;P[u=x℄`R

`R

(9")y

yunotfreein

Fig.1.SequentstylepresentationofnaturaldedutionforLaxLogi.

Normalnaturaldedutionsare theobjetsofinterest.The-redutionandommuting

onversionstepsof normalisationaregivenin(Bentonet al.,1998).Theextraasesfor

? "

and9

"

anbeadded,and aretobefoundin(Howe,1998;Howe,1999).

Denition1Anaturaldedutionissaidtobein;-normalformwhenno-redutions

andnoommuting onversionsareappliable.

WepresentarestritedversionofnaturaldedutionforLaxLogi.Inthisalulus,the

onlydedutionspossiblearein;-normalform.Thisalulushastwokindsof`sequent',

dierentiatedbytheironsequenerelations,and.Rulesareappliableonlywhen

thepremisses have theappropriate onsequenerelation. The onlusions havea xed

onsequenerelation.Thusvaliddedutionsareofarestritedform.Thisalulus,whih

weallNLAX,isgiven(withtheprooftermsgiveninthenextsetion)in Figure2.

Proposition 1Thealulus NLAXonlyallowsdedutionstowhihno-redutionsand

noommutingonversionsareappliable.Moreover,itallows all;-normaldedutions.

2.1. Term Assignment

We give a proof term system for NLAX. The term system is needed to prove the

re-sults givenin setion 4.In (Moggi,1989) Moggigavea-alulus, whihhe alled the

omputational -alulus. As is shown in (Benton et al., 1998), this alulusnaturally

mathesLaxLogi.Moreabouttheomputational-alulusandLaxLogi(therealled

omputationallogi)anbefoundin(Bentonetal.,1998).

Prooftermsforunrestrited naturaldedution forLaxLogianbe foundin(Howe,

1998; Moggi, 1989). We are interested in the `real' proofs for Lax Logi { the normal

[image:4.612.97.475.84.255.2]
(5)

;x:Pvar(x):P (ax)

A:P

an(A):P

(M)

:>

(>I)

A:?

efq(A):P

(? "

)

;x:PN:Q

x:N:P Q

(I)

A:PQ N:P

ap(A;N):Q

( "

)

N1:P N2:Q

pr(N

1

;N

2

):P^Q

(^ I

)

A:P^Q

fst(A):P

(^ "

1 )

A:P^Q

snd(A):Q

(^ "

2 )

N:P

i(N):P_Q

(_I 1

)

N:Q

j(N):P_Q

(_I 2

)

A:P_Q ;x1:PN1:R ;x2:QN2:R

wn(A;x

1 :N

1

;x

2 :N

2

):R

(_ "

)

N:P

smhi(N):ÆP

(ÆI)

A:ÆP ;x:PN:ÆQ

smhe(A;x:N):ÆQ

(Æ")

N:P[u=x℄

u:N:8xP

(8I)y

A:8xP

apn(A;t):P[t=x℄

(8 "

)

N:P[t=x℄

prq(t;N):9xP

(9 I

)

A:9xP ;y:P[u=x℄N:R

ee(A;u:y:N):R

(9 "

)y

yunotfreein

Fig.2.NLAXwithproofannotations.

ome in two syntatiategories,A and N. V is theategory of variables (proofs),U

istheategoryofvariables(individualsin formulae),andTtheategoryofterms.The

prooftermsaregivenwithanabstratsyntax,withthenotationhosentobesuggestive

oftheassoiatedproofrules.Hene smhi(N)forthetermassoiatedwiththesomehow

introdutionrule. Theextraonstrutoran(A)mathesthe(M)ruleofNLAX.

A::= var(V)jap(A;N)j fst(A)jsnd(A)japn(A;T)

N::= jefq(A)jan(A)jV:N jpr(N;N)ji(N)jj(N)jwn(A;V:N;V:N)

smhi(N)jsmhe(A;V:N)jU:N jprq(T;N)jee(A;U:V:N)

NLAXtogetherwithproofannotationsfornormaltermsanbeseeninFigure2.

3. SequentCalulus

Inthis setionwepresentanewGentzen sequent alulusforLaxLogi, PFLAX.The

proofsallowedbyPFLAXnaturallyorrespond ina1{1wayto normalnatural

dedu-tionsforLaxLogi {i.e.theproofsofNLAX. InFigure 3weremindthereaderof the

sequentalulus,extendingthosein(FairtloughandMendler,1997;Bentonet al.,1998)

toquantiers.

Wegiveanewsequentalulus,PFLAX(`permutation-free' LaxLogi).PFLAX

ex-tendsthepermutation-freealulusMJforIntuitionistiLogi(Herbelin,1995;Dykho

and Pinto, 1998; Dykho and Pinto, 1999) to a alulusfor Lax Logi. Like MJ this

alulushastwoformsof judgment, )R and

Q

[image:5.612.95.455.78.327.2]
(6)

;P )P (ax)

;P;P )R

;P)R

(C)

)>

(> R

)

;?)P

(? L

)

;P )Q

)PQ

( R

)

)P ;Q)R

;P Q)R

( L

)

)P )Q

)P^Q

(^R)

;P )R

;P^Q)R

(^L 1

)

;Q)R

;P^Q)R

(^L 2

)

)P

)P_Q

(_R 1

)

)Q

)P_Q

(_R 2

)

;P )R ;Q)R

;P_Q)R

(_L)

)P

)ÆP

(ÆR)

;P )ÆR

;ÆP)ÆR

(ÆL)

)P[u=x℄

)8xP

(8 R

)y

;P[t=x℄)R

;8xP )R

(8 L

)

)P[t=x℄

)9xP

(9 R

)

;P[u=x℄)R

;9xP )R

(9 L

)y

yunotfreein

Fig.3.SequentalulusforLaxLogi.

sequent in bakwards proof searh. The seond kindof sequent has aformula (on the

left) in aprivileged position alled thestoup,following(Girard, 1991). The formulain

thestoupisalwaysprinipalin theonlusionof aninferenerule. Thestoupisaform

offousing;proof searhis restritedsothat, wheneverpossible,theativeformulaeof

aninfereneare prinipal inthe premiss. Inbakwardsproof searh, leftrulesare only

appliableto stoupsequents.PFLAX(togetherwith theproof termsgiven in thenext

setion)isdisplayedinFigure 4.WegiveasimpleexampleofaderivationinPFLAX:

B;B;ÆB^(BA)

B

!B

(ax)

B;B;ÆB^(BA))B

(C)

B;B;ÆB^(BA)

A

!A

(ax)

B;B;ÆB^(BA)

BA

! A

( L

)

B;B;ÆB^(BA)

ÆB^(BA) ! A (^ L 2 )

B;B;ÆB^(BA))A

(C)

B;B;ÆB^(BA))ÆA

(Æ R

)

B;ÆB^(BA)

ÆB

!ÆA

(Æ L

)

B;ÆB^(BA)

ÆB^(BA)

! ÆA

(^ L1

)

B;ÆB^(BA))ÆA

(C)

3.1. Term Assignmentfor Sequent Calulus

WegiveatermassignmentsystemforPFLAX.Thetermsystemisasimpleextensionof

thatgivenin(Herbelin,1995;DykhoandPinto,1996;DykhoandPinto,1998).The

term alulushas two syntatiategories, M and Ms. V is the ategory of variables

[image:6.612.101.446.85.260.2]
(7)

P

![℄:P

(ax)

;x:P

P

!Ms:R

;x:P )(x;Ms):R

(C)

):>

(>R)

?

!ae:?

(?L)

;x:P)M:Q

)x:M :P Q

(R)

)M :P

Q

!Ms:R

PQ

! (M::Ms):R

( L

)

)M

1

:P )M

2

:Q

)pair(M1;M2):P^Q

(^ R

)

P

!Ms:R

P^Q

!p(Ms):R

(^ L

1 )

Q

!Ms:R

P^Q

!q(Ms):R

(^ L

2 )

)M :P

)inl (M):P_Q

(_R 1

)

)M :Q

)inr(M):P_Q

(_R 2

)

;x

1

:P )M

1

:R ;x

2

:Q)M

2

:R

P_Q

!when(x

1 :M

1

;x

2 :M

2

):R

(_L)

)M :P

)smhr(M):ÆP

(Æ R

)

;x:P)M :ÆR

ÆP

!smhl (x:M):ÆR

(Æ L

)

)M:P[u=x℄

)u:M :8xP

(8 R

)y

P[t=x℄

! Ms:R

8xP

!apq(t;Ms):R

(8 L

)

)M :P[t=x℄

)pairq(t;M):9xP

(9R)

;y:P[u=x℄)M:R

9xP

!spl (u:y:M):R

(9 L

)y

yunotfreein

Fig.4.ThesequentalulusPFLAX,withproofannotations.

Inpartiular,[℄isusedfortheaxiomtermand(M ::Ms)forthetermassoiatedwith

impliationontheleft,givingalistofMtermssuggestiveoftheorderingofrulesinthe

alulus.

M::=j(V;Ms)jV:Mjpair(M;M)jinl(M)jinr(M)jsmhr(M)jU:Mjpairq(T;M)

Ms::=[℄jaej(M ::Ms)jp(Ms)jq(Ms)jwhen(V:M;V:M)

smhl(V:M)japq(T;Ms)jspl(U:V:M)

ThesetermsaneasilybetypedbyPFLAX,asseeninFigure 4.

4. Equivaleneof the Caluli

WeprovetheequivaleneofthetermaluliandthesoundnessandadequayofPFLAX.

These results provethedesired orrespondene.The proofs are extensions of those for

theMJalulusforIntuitionistiLogi(DykhoandPinto,1998),henemostdetailis

omitted.Westartbygivingpairsoffuntionsthatdenetranslationsbetweentheterm

assignmentsystems fornaturaldedution (NLAX) andsequentalulus(PFLAX),

[image:7.612.94.473.74.362.2]
(8)

SequentCalulus ! NaturalDedution:

:M!N

0

:AMs!N

(x;Ms)=

0

(var(x);Ms)

0

(A;[ ℄)=an(A)

(x:M)=x:(M)

0

(A;(M ::Ms))=

0

(ap(A;(M));Ms)

(smhr(M))=smhi((M))

0

(A;smhl(x:Ms))=smhe(A;x:(M))

()=

0

(A;ae)=efq(A)

(pair(M 1 ;M 2 ))=pr((M 1 );(M 2 )) 0

(A;p(Ms))=

0

(fst(A);Ms)

(inl(M))=i((M))

0

(A;q(Ms))=

0

(snd(A);Ms)

(inr(M))=j((M))

0

(A;apq(t;Ms))=

0

(apn(A;t);Ms)

(u:M)=u:(M)

0

(A;spl(u:y:M))=ee(A;u:y:(M))

(pairq(t;M))=prq(t;(M))

Natural Dedution ! Sequent Calulus:

:N!M

0

:AMs!M

(an(A))= 0

(A;[℄)

0

(var(x);Ms)=(x;Ms)

(x:N)=x: (N)

0

(ap(A;N);Ms)=

0

(A;( (N)::Ms))

(smhe(A;x:N))=

0

(A;smhl(x: (N)))

0

(fst(A);Ms)=

0

(A;p(Ms))

(smhi(N))=smhr( (N))

0

(snd(A);Ms)=

0

(A;q(Ms))

()=

0

(apn(A;t);Ms)=

0

(A;apq(t;Ms))

(efq(A))= 0 (A;ae) (pr(N 1 ;N 2 ))=pair( (N 1

); (N

2 )) (i(N))=inl( (N)) (j(N))=inr( (N)) (wn(A;x 1 :N 1 ;x 2 :N 2 ))= 0 (A;when(x 1 : (N 1 );x 2 : (N 2 )))

(u:N)=u: (N)

(prq(t;N))=pairq(t; (N))

(ee(A;u:y:N))=

0

(A;spl(u:y: (N)))

We give two lemmas demonstrating the equivalene of the term aluli, that is, the

translationsfrom onesystemtotheotherare1{1.

Lemma1 i) ((M)) = M; ii) (

0

(A;Ms)) =

0

(A;Ms).

Proof. Bysimultaneous indution onthestrutureofM andMs.Forfull details see

(Howe,1999).

Lemma2 i)( (N)) = N; ii)(

0

(A;Ms)) =

0

(A;Ms).

Proof. Bysimultaneous indution on the struture of N and A. Forfull details see

(Howe,1999).

Thefollowingtwotheoremsstatesoundnessandadequay.Theyshowthatthe

(9)

Theorem1(Soundness) Thefollowing rulesare admissible:

)M :R

(M):R

i)

A:P

P

!Ms:R

0

(A;Ms):R

ii)

Proof. Bysimultaneous indution onthestrutureofM andMs.Forfull details see

(Howe,1999).

Theorem2(Adequay)Thefollowing rulesare admissible:

N:R

) (N):R

i)

A:P

P

!Ms:R

) 0

(A;Ms):R

ii)

Proof. Bysimultaneous indution on the struture of A and N. Forfull details see

(Howe,1999).

Sine the term systems are in 1{1orrespondene (lemma 1 and lemma 2) and the

translations preserve provability (theorem 1 and theorem 2), the 1{1 orrespondene

betweenPFLAXandNLAXhasbeenestablished.Thisisstatedinthefollowingtheorem.

Theorem3The normal natural dedutions of Lax Logi (the proofs of NLAX) are in

1{1orrespondene tothe proofsof PFLAX.

Animmediateorollaryoftheorem3isthatPFLAXissoundandompletewithrespet

to naturaldedutionfor LaxLogi. QuantiedLax Logiis demonstrated tobesound

and omplete with respet to ertain lassesof Kripkemodel-strutures in (Fairtlough

andWalton,1997).

4.1. CutElimination

Wenowbriey disuss ut forPFLAX. Inthe usual sequent alulus,ut may be

for-mulatedasfollows:

)P ;P )Q

)Q

(ut)

InPFLAX,thetwojudgmentformsleadtothefollowingfourutrules(asfor

Intuition-istiLogi in(Herbelin, 1995;Dykho andPinto,1998)):

Q

!P

P

!R

Q

!R

(ut 1

)

)P ;P

Q

!R

Q

!R

(ut 2

)

)P

P

!R

)R

(ut 3

)

)P ;P )R

)R

(ut 4

)

WeallPFLAXextendedwiththefourutrulesPFLAX

ut

.Weangiveredutionrules

for PFLAX

ut

and provethe weak ut elimination theorem for the logi. We an also

provestrongnormalisation forthe termsystemassoiatedwith thelogi, henestrong

(10)

Theorem4The rules(ut 1

);(ut

2

);(ut

3

);(ut

4

)areadmissible in PFLAX.

Theorem5The utredutionsystemfor PFLAXstronglynormalises.

5. DeidingLax Logi

It is usefuland interesting to haveadeisionproedure foranylogi. This setion

de-sribesadeisionproedurefor propositionalLaxLogi. Tothe best ofourknowledge,

nodeisionproedure forpropositionalLaxLogihasbeenpresentedbefore.

Thenewalulususesahistorymehanismtoensureterminationofbakwardsproof

searh.Historymehanismswereintroduedin(Heuerdingetal.,1996;Heuerding,1998).

Therenedhistorymehanismusedhereanbefoundin (Howe,1997;Howe,1998).

Another approah to deiding propositional logis is by the use of `ontration-free'

sequentaluli,suhastheoneforpropositionalIntuitionistiLogigivenin(Dykho,

1992; Hudelmaier, 1993). If suh a deision proedure for Lax Logi ould be found,

we would expet it to be faster than one involving a history mehanism. An

investi-gation of ontration-free alulifor Lax Logi an be found in (Avellone and Ferrari,

1996). Unfortunately, this investigation did not sueed in nding a ontration-free

alulus. We believe that a ontration-free alulus for Lax Logi annot be found,

as (for arbitrary n) examples an be onstruted whih require an entire formula in

a sequent to be ontrated n times in a proof. As an example, onsider the sequent

B(ÆAC)ÆA;ÆB;ÆAC)C,whereÆACneedstodupliatedinitsentirety

inorderto provethesequent.

5.1. DeidingPropositional LogisUsing HistoryMehanisms

Oneapproah to ndinga deisionproedure for apropositionallogi is to plae

on-ditionson thesequentalulusto ensureterminationof searh. Itis elegant tobeable

tobuild theontentof theseonditionsintothesequentalulusitself. Thisis howthe

alulusfortheorem provinginthissetionisdeveloped.

In order to ensuretermination of bakward proof searh, weneed to hek that the

same sequent (modulo number of ourrenes of idential formulae) does not appear

againonabranh,that is,proof searh doesnotloop.Avoidingloopsanalsoprevent

theunneessaryomputationarisingfromanitenumberofpassesthroughaloopina

suessfulderivation.Weneedamehanialwaytodetetsuhloops.Onewaytodothis

istoaddahistory toasequent.Thehistoryisthesetofallsequentstohaveourredso

faronabranhofaprooftree.Aftereahbakwardsinferenethenewsequent(without

itshistory)is hekedtosee whetheritis amemberofthis set.Ifit iswehavelooping

and baktrak. If not, the new history is the extension of the old history by the old

sequent(without thehistoryomponent), andwetryto provethenewsequent,andso

on.Unfortunately,this methodisspaeineÆientasitrequireslonglistsofsequentsto

be stored by the omputer,and all of this list hasto be heked at eah stage.When

(11)

Thebasisofthereduedhistoryistherealisation,asin(Heuerdingetal.,1996),that

oneneedonlystoregoalformulae(agoalformulaisthesuedentofasequent)inorder

toloop-hek.Inthealulidealt withinthis paper,oneaformulaisin theontextit

willbeintheontextofallsequentsaboveitintheprooftree.Wesaythat thealulus

has inreasing ontext. For two sequents to be the same they need to have the same

ontext (up to multiple ourrenesof formulae).Therefore wemayempty thehistory

everytimetheontextis (properly)extended.All weneedstoreinthehistoryaregoal

formulae.If we ometo a sequentwhose goalis already in thehistory, then it hasthe

samegoalandthesameontextasanothersequent{thereisaloop.

Therearetwoslightlydierentapproahesapturingthis.Thereisthestraightforward

extensionofthealulusdesribedin (Heuerdinget al.,1996),whih weallthe`Swiss

history';moreonthisloop-hekingmethodanbefoundin(Heuerding,1998).Thereis

alsorelatedworkonhistoriesforIntuitionistiLogiin(Gabbay,1991).Anotherapproah

involvesstoringslightlymoreformulaeinthehistory,butwhihforsomealulidetets

loops morequikly. This we desribe asthe `Sottish history'(Howe, 1997); it an,in

manyases, bemoreeÆientthan the Swiss method. In this paper wegive aalulus

forLaxLogi using theSottish historyaswebelievethis to be thebettermethod for

intuitionistilogis(Howe,1997).

Thegeneralityofthis approahisattrative.Thehistorymehanismanbeattahed

tomanyalulitogivedeisionproedures;appliationsanbefoundin(Howe,1998).

5.2. PFLAX

Hist

This setion desribes a history alulus for propositional Lax Logi. The alulus is

an extension of that for Intuitionisti Logi given in (Howe, 1997). The modality is

handled similarlyto disjuntion (disjuntionisnotoveredin (Heuerding et al.,1996),

andrequiresspeialtreatment).ItusesthealulusPFLAXasabaseonwhihtobuild

the alulusas this alulus has already reduedthe searh spae to a ertain extent.

PFLAXhastheinreasingontextrequiredfortheappliationofthehistorymehanism.

However,amoreusualformulationouldhavebeenusedinstead.PFLAX

Hist

anbeseen

in Figure 5.Observe thetworules for(

R

). These orrespond to thetwoases where

thenew formulais oris notin theontext.As notedabove, thisis veryimportantfor

historymehanisms.Notiethat thenumberofformulaein thehistoryisat mostequal

tothelengthof theformulawehekforprovability.

A sequent is mathed againsttheonlusions of rightrules until the goalformula is

eitherapropositionalvariable,falsum,disjuntionoraÆformulaThishasbeenensured

bytherestritionongoalformulaegiveninthealulus(notethattherulesfordisjuntion

ontherightandsomehowontherightareonlypossibleonbaktraking,orwithanempty

ontext). A formula from the ontext is then seleted using therule (C) and mathed

againsttheleft rules of thealulus.The Sottish aluluskeeps(as aset)a omplete

reord of goal formulae between ontext extensions. At eah of the plaes where the

historymight be extended, the new goal is heked against the history. If it is in the

(12)

P

!P;H

(ax)

;P

P

!D;H

;P )D;H

(C)

)>;H

(>R)

?

!D;H

(?L)

;P)Q;fQg

)P Q;H

( R1

) ifP2=

)Q;(Q;H )

)PQ;H

( R2

) ifP2 andQ2=H

;P )?;f?g

):P;H

(: R1

) ifP 2=

)?;(?;H )

):P;H

(: R2

) ifP 2 and?2=H

)P;(P;H)

Q

!D;H

PQ

! D:H

(L) ifP 2=H

)P;(P;H )

:P

!D;H

(: L

) ifP2=H

)P;(P;H) )Q;(Q;H)

)P^Q;H

(^ R

) ifP;Q2=H

P

!D;H

P^Q

!D;H

(^L 1

) Q

!D;H

P^Q

!D;H

(^L 2

)

)P;(P;H)

)P_Q;H

(_R 1

) ifP2=H

)Q;(Q;H )

)P_Q;H

(_R 2

) ifQ2=H

;P)D;fDg ;Q)D;fDg

P_Q

!D;H

(_ L

) ifP 2= andQ2=

)P;(P;H )

)ÆP;H

(ÆR) ifP2=H

;P )ÆR ;fÆR g

ÆP

!ÆR ;H

(ÆL) ifP2=

Diseitheranatom,?,disjuntionoraÆformula.Wherethehistoryhasbeenextended

wehaveparenthesised(P;H )for emphasis.

Fig.5.ThealulusPFLAX

Hist

(Sottish).

essary, theleft rules haveside onditions to ensure that the ontext is inreasing. For

the(

R

)rule(whihattemptstoextendtheontext)therearetwoasesorresponding

to whentheontextis andwhenit isnotextended. Somethingsimilar ishappeningin

the left rules. Take (_

L

) as anexample. In both premisses of the rule a formula may

be added to ontext. If both ontexts really are extended, then we ontinue building

theprooftree. If oneorbothontexts are notextended then thesequent,S, with the

non-extendedontext, will be thesameassomesequentat alesserheightin the proof

tree{there isaloop(whih wedesribeasatrivialloop).Thisiseasytosee:sinethe

ontextandthegoalofS arethesameasthat oftheonlusion,theremust bealower

sequent(theonlusionofaninstaneof(C))thesameasthepremissS.Asanexample

wegiveaderivationin PFLAX

Hist

ofthesequentintheexampleinsetion3:

B;ÆB^(BA)

B

!B

(ax)

B;ÆB^(BA))B

(C)

B;ÆB^(B A)

A

!A

(ax)

B;ÆB^(BA)

BA

! A

( L

)

B;ÆB^(BA)

ÆB^(BA)

! A

(^ L2

)

B;ÆB^(BA))A

(C)

(Æ R

[image:12.612.97.466.89.340.2]
(13)

Note that thederivation of this sequent given in setion 3would be preventedby the

historymehanism,asitontainsaloop.

It is now demonstrated that PFLAX

Hist

is equivalent to PFLAX, in terms of

prov-ability. The equivaleneis provedvia anintermediate alulusPFLAX

D

. Thealulus

PFLAX D

is thealulusPFLAXwhere therule (C) isrestritedso that itis only

ap-pliablewhenthegoalformulaisanatom,adisjuntion,falsumorasomehowformula.

Proposition 2The alulus PFLAX is equivalent to the alulus PFLAX

D

. That is,

sequent )Gisprovable in PFLAXi )Gisprovable in PFLAX

D .

Thefollowinglemmaisneededintheproofoftheorem6.

Lemma3(Contration)The following rulesareadmissible in PFLAX

Hist :

;P;P )R ;H

;P )R ;H

(C 0

)

;P;P

Q

!R ;H

;P

Q

!R ;H

(C 00

)

Proof. Bysimultaneousindution ontheheightsofderivationsofpremisses.

Theequivaleneproof below, althoughlong,hasasimplestruture. An algorithmto

turnaPFLAXprooftreeinto aPFLAX

Hist

prooftree isdesribed indetail. Asimple

indutionargumentshowsthatthealgorithmterminates, provingtheresult.

Theorem6Thealuli PFLAXand PFLAX

Hist

areequivalent.Thatis,sequent )G

isprovable in PFLAXisequent )G;fGgisprovable in PFLAX

Hist .

Proof. FromProposition2weknowthatitisenoughtoshowthatPFLAX

D

is

equiv-alent to PFLAX

Hist

. It is trivialthat any sequent provablein PFLAX

Hist

is provable

in PFLAX

D

. (Useontration(C

0

)aboveinstanesof (

R2

)andthen simplydropthe

historypartofthesequent).Weprovetheonverse.

Take any proof tree forsequent )Gin PFLAX

D

. By denition this proof tree is

nite,with n>0nodes.Usingthis prooftree,weonstrut(from therootup) aproof

treeforthesequent )G;fGginPFLAX

Hist

.ThemajordierenebetweenPFLAX

D

andPFLAX

Hist

isthattheformerpermitsloops,whereasthelatterdoesnot.Essentially

wetakethePFLAX

D

prooftreeandgiveareipefor`snippingout'theloops:removing

thesequentsthat formtheloop.

Theonstrutionuses`hybridtrees'.AhybridtreeisafragmentofaPFLAX

Hist proof

tree with allbranhesthat do not have(ax), (>) or(?) leavesending with PFLAX

D

prooftrees.ThesePFLAX

D

prooftreeshaverootswhihanbeobtainedbybakwards

appliation of a PFLAX

D

rule to the top history sequent (ignoring its history). We

analyse eah ase of atopmost history sequent with non-historypremiss(es) resulting

from appliation of rule (R ) in the sequent tree. We write

0

when the set of

formulaein multisets and

0

arethe same(althoughthe numberofourrenesmay

bedierent).Wedenoteaseriesofzeroormoreinstanesofrule(R ) by(R )

(14)

{ The root of the PFLAX D

tree. We hange (non-history) sequent )G to history

sequent )G;fGg.

{ (R ) isoneof(ax), (C), i.e.arule whih inPFLAX

Hist

hasno sideonditions.The

premissishangedbyaddingtheappropriatehistory.Itbeomesthehistorysequent

obtainedbyapplying(bakwards)thePFLAX

Hist

ruletotheoriginalonlusion.For

example,ifthesituation weareanalysingis

;P

P

!D

;P )D;H

(C) thenitbeomes

;P

P

!D;H

;P)D;H

(C)

Wenowhaveanewhybridtree.

{ (R ) is (

R

). Ifthe ontext is extended, then add theappropriate history, allowing

the replaement of the instane of (

R

) in PFLAX

D

by an instane of (

R1

) in

PFLAX Hist

. If the ontext is not extended and the new goal is notin thehistory,

then again addthe appropriatehistory, allowingthe replaement ofthe instane of

( R

) in PFLAX

D

by aninstane of (

R 2

) with a(C

0

) in PFLAX

Hist

. If the new

goal isin the history, there is aloop,whih thehistorymehanismprevents.If the

historyonditionisnotmet,thenbelowtheonlusionthehybridtreehastheform:

;P )G

)P G;H

( R ) . . . . 0

)G;H

0

where G 2 H

0

, H

0

H and

0

. The historyis notreset at any point in this

fragment. This an easily be seen to ontain the loop whih is the reason for the

history onditionnot being met. It is transformed by removingall sequentsabove,

but notinluding,

0

)G;H

0

(alongwith anysubtrees aboveexised sequents) up

to ;P)G .Addingtheappropriatehistorytothissequentandusing(oneormore

instanesof)(C

0

)givesthenewhybridtree:

;P )G;H

0

0

)G;H

0 (C

0 )

{ (R ) is (

L

). If the historyondition is satised, then add the appropriatehistory,

allowingthereplaementofinstaneof (

L

)in PFLAX

D

byaninstane of(

L

)in

PFLAX Hist

. Ifthe historyondition isnot satised,then belowthe onlusionthe

hybridtreehastheform:

)P

Q

!R

PQ

! R ;H

( L ) . . . . 0

)P;H

0

where P 2 H

0

, H

0

H and

0

. The historyis not reset at any point in this

fragment. It is transformed by removingall the sequents above, but not inluding,

0

)P;H

0

(alongwith any subtrees aboveexisedsequents) upto )P. The

(15)

history to )P and using (zero ormore instanes of) (C 0

) gives the newhybrid

tree:

)P;H

0

0

)P;H

0 (C

0 )

{ (R ) is (Æ

R

). If the history ondition is satised, then add the appropriate history,

allowingthereplaementoftheinstane of(Æ

R

)in PFLAX

D

byaninstaneof(Æ

R )

inPFLAX

Hist

.Ifthehistoryonditionisnotsatised,thenbelowtheonlusionthe

hybridtreehasform:

)P

)ÆP;H

(Æ R

)

. . . . 0

)P;H

0

whereP 2H

0

,H

0

Hand

0

.Thehistoryisnotresetatanypointinthis

frag-ment.Itistransformedbyremovingallsequentsfrom,butnotinluding,

0

)P;H

0

(alongwithanysubtreesaboveexisedsequents)upto )P.Addingthe

appropri-atehistoryandusing(zeroormoreinstanesof)(C

0

)givesthenewhybridtree:

)P;H

0

0

)P;H

0 (C

0 )

{ (R ) is (Æ

L

). If the side ondition is satised, then add the appropriate history,

al-lowingthe replaementof theinstane of (Æ

L

) in PFLAX

D

by aninstane of (Æ

L )

in PFLAX

Hist

. If the side onditionis not satised, then below the onlusion the

hybridtreehasform:

;P )ÆR

ÆP

!ÆR ;H

(Æ L

)

. . . .

)ÆR ;H

where P 2 .Thisis transformedbyremovingallsequentsfrom,but notinluding,

)ÆR ;H(alongwithanysubtreesaboveexisedsequents)upto ;P)ÆR.Adding

theappropriatehistoryandusingasingleinstaneof(C

0

)givesthenewhybridtree:

;P )ÆR ;H

)ÆR ;H

(C 0

)

Sinethenumberofsequentswithoutahistoryinahybridtreeisniteandaseverystep

stritlydereasesthenumberofsequentswithoutahistory,this proessisterminating.

Theinstanesof(C

0

)maybeeliminatedfromtheonstrutedderivation.

Note that the PFLAXderivation given in setion 3is transformedby the algorithm

desribedintheprooftothederivationofthesamesequentinPFLAX

Hist

givenabove.

Wehaveshownthat PFLAX

Hist

issoundandomplete.Toprovethatitisadeision

proedure,weprovethatitisalsoterminating{bakwardsproofsearhinthealulus

endsinsuessorfailureafteranitenumberofsteps.

(16)

Proof. Weassoiatewitheverysequentaquintupleofnaturalnumbers.Withasequent

without astoup, )R ;H, we assoiate: W = (k n;k m;1;0;r). With a sequent

withastoup,

P

!R ;H,weassoiate:W =(k n;k m;0;s;r).Here,kisthenumber

ofelementsin theset ofsubformulaeof( ;R );nisthenumberofelementsintheset of

elementsof ;misthenumberofelementsinH ;risthesizeofgoalformulaRandsisthe

sizeofthestoupformulaP.(Notiethatalthough isamultiset,weountitselements

asaset). Thesequintuplesarelexiographiallyorderedfrom theleft.Byinspetionwe

seethat foreveryinfereneruleW forthepremissesislowerin thelexiographi order

thanW fortheonlusion.Henebakwardproofsearhisterminating.

Whenimplementingatheorem prover,knowledgeof theinvertibilityof theinferene

rulesanbeuseful. Thisinformationisgiveninthefollowingproposition.

Proposition 3Thefollowinginferenerulesof PFLAX

Hist

areinvertible:(

R1

),(

R2 ),

(: R

1

),(:

R2

),(

L

),(:

L

),(^

R

),(_

L

),(Æ

L

).Thefollowinginferenerulesof PFLAX

Hist

arenot invertible: (C), (^

L 1

),(^

L 2

), (_

R 1

), (_

R 2

), (Æ

R ).

6. Conlusion

This paper has presented two proof searh aluli for Lax Logi. The rst, PFLAX,

is a sequent alulus for rst-order quantied Lax Logi. The proofs allowed by this

alulusnaturallyorrespondina1{1waytothenormalnaturaldedutionsforrst-order

quantied LaxLogi. Thealulusis wellsuited forenumerating,withoutredundany,

allproofs in thelogi.This makesthe alulususefulin ontexts whereproof searh is

fornormalnaturaldedutions,suhasin(onstraint)logiprogramming.

The seond alulus, PFLAX

Hist

, builds on the propositional fragment of the rst

alulus to give a deision proedure for propositional Lax Logi. Propositional Lax

Logi has been used in hardware veriation and PFLAX

Hist

ould be of use in this

area.The alulus works by adding a historymehanism to the propositional alulus

to prevent looping. This tehnique is general and may be applied to a wide range of

sequentaluliforpropositionallogisto yielddeisionproedures.Webelievethat,to

date,PFLAX

Hist

istheonlyeetivedeisionproedureforpropositionalLaxLogi.

Aknowledgements IwouldliketothankRoyDykhoforhishelpfuladvieduringmany

usefulandinterestingdisussions.

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Figure

Fig.1.andDe�nition?NormalSequentstylepresentationofnaturaldedu
tionforLaxLogi
.naturaldedu
tionsaretheobje
tsofinterest.The�-redu
tion
onversionstepsofnormalisationaregivenin(Bentonetal.,1998)."and9"
anbeadded,andaretobefoundin(Howe,1998;Ho1Anaturaldedu
tionissaidtobein�;
-normalformno
ommuting
onversionsareappli
able.Wepresentarestri
tedversionofnaturaldedu
tionforLaxLogi
.
Fig.uinofpro
omethethetro
Fig.�anofleft)thesequenufoappli
able
Fig.uInimpli
ation
al
ulus.M::=Ms::=
+2

References

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