Intersection Logic in sequent calculus style

0.12 MB·English

by  Simona Ronchi Della Rocca

Intersection Logic in sequent calculus style
SimonaRonchi DellaRocca AlexisSaurin
DipartimentodiInformatica LaboratoireCNRSPPS&INRIApi.r2
UniversitadiTorino Paris,France
IT-10149Torino,Italy
[email protected]
[email protected]
YiorgosStavrinos AnastasiaVeneti
GraduatePrograminLogicandAlgorithms(MPLA)
DepartmentofMathematics
UniversityofAthens
GR-15784Zografou,Greece
[email protected] [email protected]
1 Introduction
TheintersectiontypeassignmentsystemIT(showninFigure1)isadeductivesystemthatassignsformu-
lae(builtfromtheintuitionisticimplication→andtheintersection∩)astypestotheuntypedl -calculus.
IthasbeendefinedbyCoppoandDezani[6],inordertoincreasethetypability powerofthesimpletype
assignment system. In fact, IT has a strong typability power, since it gives types to all the strongly
normalizing terms [17, 11]. Intersection types, supported by a universal type and a suitable pre-order
relation, hasbeenusedfordescribing thel -calculus semantics indifferent domains(e.g. Scottdomains
[3,7],coherencespaces[14]),allowingtocharacterizeinterestingsemanticalnotionslikesolvabilityand
normalizability, bothincall-by-name andcall-by-value settings[18].
Differently from other well known type assignments for l -calculus, like for example the simple or
thesecondorderone,IThasnotbeendesignedasdecorationofalogicalsystem,followingthesocalled
Curry-Howardisomorphism,relatinginparticulartypeswithlogicalformulae. Infact,despitetothefact
that it looks like a decoration of the implicative and additive fragment of the intuitionistic logic, this is
not true. Indeed, while the connective → has the same behaviour of the intuitionistic implication, the
intersection ∩doesnotcorrespondtotheintuitionistic conjunction, asHindleypointedoutin[10],since
theruleintroducingithasthemeta-theoreticalconditionthatthetwoproofsofthepremisesbedecorated
bythesameterm,andthisconstraint isnomoreexplicitwhentermsareerased.
It is natural to ask if there is a logic with a natural correspondence with intersection types. Some
attempts have been made, which will be briefly recalled in Section 7. Our work follow the line started
from [19] and [15, 16], where the authors defined Intersection Logic IL and Intersection Synchronous
Logic ISL, respectively. ISL is a deductive system in natural deduction style proving molecules which
arefinitemultisetsofatomswhich,inturn,arepairsofacontextandaformula. Thenotionofmolecule
allows to distinguish between two sets of connectives, global and local, depending on whether they
are introduced and eliminated in parallel on all atoms of a molecule or not. Note that global and local
connectives arecalled asynchronous and synchronous, respectively, in[15,16]. Inparticular, twokinds
of conjunction are present: the global one corresponding to intuitionistic conjunction and the local one
corresponding to intersection. Then, the intersection type assignment system is obtained by decorating
ISLwithl -termsintheCurry-Howardstyle. Roughlyspeaking,amoleculecanbemappedintoasetof
E.Pimentel,B.VenneriandJ.Wells(Eds.):Workshopon
IntersectionTypesandRelatedSystems2010(ITRS2010). (cid:13)c S.RonchiDellaRocca,A.Saurin,etal.
EPTCS45,2011,pp.16–30,doi:10.4204/EPTCS.45.2
S.RonchiDellaRocca,A.Saurin,etal. 17
(A)
G ,x:s ⊢ x:s
IT
G ⊢ M:s G ⊢ M:t G ⊢ M:s ∩s
IT IT (∩I) IT L R (∩E )k=L,R
G ⊢ M:s ∩t G ⊢ M:s k
IT IT k
G ,x:s ⊢ITM:t (→I) G ⊢ITM:s →t G ⊢ITN :t (→E)
G ⊢ l x.M:s →t G ⊢ MN:t
IT IT
Figure1: Theintersection typeassignment systemIT.
intuitionistic proofs which are isomorphic, in the sense that some rules are applied in parallel to all the
elementsintheset. Theisomorphism isobtained bycollapsing thetwoconjunctions.
ISLputintoevidencethefactthattheusualintuitionistic conjunction canbesplittedintotwodiffer-
entconnectives,withadifferentbehaviour,andtheintersectioncorrespondstooneofit. Sowethinkthat
ISLisalogicthatcanbeinteresting initself,andsowewouldliketoexploreitfromaproof-theoretical
point of view. To this aim, we present here a sequent calculus formulation of it, that we call ISC, and
we prove that it enjoys the property of cut elimination. The proof is not trivial, since the play between
globalandlocalconnectives iscomplicated andnewstructural rulesareneeded.
Stavrinos andVeneti[21,23]furtherenhanced ILandISLwithanewconnective: thelogical coun-
terpart of the union operator on types [2]. It turns out that union is a different kind of disjunction. In
the extensions ofIL and ISLpresented in[21], though, thestatus ofunion isnot so clear; inparticular,
union is not an explicitly local version of intuitionistic disjunction, since its elimination rule involves
a global side-effect. So, the pair union-disjunction does not enjoy the nice characterization of the pair
intersection-conjunction, which are the local and global versions, respectively, of the same connective.
Moving toasequent calculus exposition [23]allowsforaverynice and symmetric presentation ofboth
intersection and union, which are now the local versions of intuitionistic conjunction and disjunction,
respectively. Ournextstepwillbetodoaprooftheoretical investigation ofthisnewcalculus.
The paper is organized as follows. In Section 2 the system ISL is recalled, and its sequent calculus
version ISCis presented. InSection 3the relation between ISCand Intuitionistic Sequent Calculus LJ
is shown. Section 4contains abrief survey of the cut-elimination procedure in LJ. In Section 5 the cut
elimination steps ofISC aredefined, and inSection 6thecut-elimination algorithm isgiven. Moreover
Section 7 contains a brief survey of the other approaches to the problem of a logical foundation of
intersection types.
2 The system ISC
InthissectionthesystemISLpresentedin[15,16]isrecalled. Moreover,anequivalentsequentcalculus
versionisgiven.
Definition2.1(ISL) i) Formulasaregenerated bythegrammar
s ::=a|s →s |s &s |s ∩s
whereabelongs toacountable setofpropositional variables. Formulasarerangedoverbysmall
greekletters.
18 IntersectionLogicinsequentcalculusstyle
M∪N
(Ax) (P)
[(s i;s i)|i∈I] M
[(G i;ti)|i∈I] [(G i1,ti,s i,G i2;r i)|i∈I]
(W) (X)
[(G i,s i;ti)|i∈I] [(G i1,s i,ti,G i2;r i)|i∈I]
[(G i,s i;ti)|i∈I] [(G i;s i→ti)|i∈I] [(G i;s i)|i∈I]
(→I) (→E)
[(G i;s i→ti)|i∈I] [(G i;ti)|i∈I]
[(G i;s i[)(G|ii;∈s iI&]ti)[(|G ii;∈tiI)]|i∈I] (&I) [(G [i(;Gsii;Ls&iks)iR|)i∈|iI∈] I] (&Ek), k=L,R
M∪[(G ;s ),(G ;t )] M∪[(G ;s L∩s R)]
M∪[(G ;s ∩t )] (∩I) M∪[(G ;s k)] (∩Ek), k=L,R
Figure2: ThesystemISL
ii) Contexts are finite sequences of formulas, ranged over by G ,D ,E,Z. The cardinality |G | of a
contextG isthenumberofformulasinG .
iii) Atoms(G ;s )arepairsofacontextandaformula, rangedoverby A,B.
iv) Molecules [(a i,...a i ;a )|1 6 i 6 n] or [(a i,...a i ;a )|i ∈ I] or [(a i,...a i ;a )] are finite
1 m i 1 m i 1 m i i
multisets of atoms such that all atoms have the same context cardinality. Molecules are ranged
overbyM,N .
v) ISL is a logical system proving molecules in natural deduction style. Its rules are displayed in
Figure2. Wewrite⊢ M whenthere isadeduction proving themolecule M, andp :M when
ISL
wewanttodenoteaparticular derivation p proving M.
Somecomments are inorder. Rule(P)isderivable and itisuseful forthe normalization procedure.
Theconnectives→,&areglobal,inthesensethattheyarebothintroducedandeliminatedinalltheatoms
of a molecule by a unique rule, while the connective ∩ is local, since it is introduced and eliminated in
justoneatomofamolecule.
Thefollowingtheorem recallthecorrespondence betweenISLandIT,provedin[16]. Inthetextof
the theorem, we use the notation (G )s for denoting a decoration of the context G through a sequence s
of different variables. More precisely, if G =a ,...,a and s=x ,...,x , (G )s is the set of assignments
1 n 1 n
{x :a ,...,x :a }.
1 1 n n
Theorem2.2(ISLandIT) i) Let ⊢ [(G ;a ),...,(G ;a )], where a and all formulae in G do
ISL 1 1 m m i i
notcontain theconnective &. Thenforeverys,thereisM suchthat(G )s⊢ M:a .
i IT i
ii) If(G i)x1,...,xn ⊢ITM:a i (i∈I),then⊢ISL[(G i;a i)|i∈I].
Nowwewilldefineanequivalent formulation insequent calculusstyle.
Definition2.3 ISC is a logical system proving molecules in sequent calculus style. Its rules are dis-
played inFigure3. Wewrite⊢ M whenthere isaderivation proving molecule M andp :M when
ISC
wewanttodenoteaparticular derivation p proving M.
Notice that the connectives →,& are dealt withrules having amultiplicative behaviour ofcontexts,
but the structural rules ensure an equivalent additive presentation. The connective ∩, though, need to
S.RonchiDellaRocca,A.Saurin,etal. 19
IdentityRules:bothglobal
[(G i;a i)|i∈I] [(D i,a i;b i)|i∈I]
(Ax) (cut)
[(a i;a i)|i∈I] [(G i,D i;b i)|i∈I]
StructuralRules:(W),(X),(C)areglobaland(Fus),(P)arelocal
[(G i;b i)|i∈I] [(G i,b i,a i,D i;gi)|i∈I] [(G i,a i,a i;b i)|i∈I]
(W) (X) (C)
[(G i,a i;b i)|i∈I] [(G i,a i,b i,D i;gi)|i∈I] [(G i,a i;b i)|i∈I]
[(G ;b )]∪M M∪N
(Fus) (P)
[(G ;b ),(G ;b )]∪M M
LogicalRules:→,&areglobaland∩islocal
[(G i;a i)|i∈I] [(D i,b i;gi)|i∈I] [(G i,a i;b i)|i∈I]
(→L) (→R)
[(G i,D i,a i→b i;gi)|i∈I] [(G i;a i→b i)|i∈I]
[(G i,a iL,a iR;b i)|i∈I] [(G i;a i)|i∈I] [(D i;b i)|i∈I]
(&L) (&R)
[(G i,a iL&a iR;b i)|i∈I] [(G i,D i;a i&b i)|i∈I]
[(G ,a k;b )]∪M [(G ;a ),(G ;b )]∪M
[(G ,a L∩a R;b )]∪M (∩Lk), k=L,R [(G ;a ∩b )]∪M (∩R)
Figure3: ThesystemISC
20 IntersectionLogicinsequentcalculusstyle
have an additive presentation. By abuse of notation, we will call global the rules dealing with → and
& and local the rules dealing with ∩. The rules (P) and (Fus) are derivable; they will be useful in the
cut-elimination procedure.
Lemma2.4 Let p :⊢ M. There is a function clean, such that clean(p ) is a derivation proving M
ISC
whichdoesnotcontain applications oftherules(P)and(Fus).
Proof Itiseasytocheckthatboth(P)and(Fus)commuteupwardswithanyrulepreceding themand
thattheydisappear whenfollowinganaxiomrule. Forinstance:
D
[(G ;a ),(G ;b )]∪M
(Fus)
D [(G ;a ),(G ;b ),(G ;b )]∪M
(Fus)
[(G ;a ),(G ;b )]∪M [(G ;a ),(G ;b ),(G ;a ),(G ;b )]∪M
(∩R) (∩R)
[(G ;a ∩b )]∪M [(G ;a ),(G ;b ),(G ;a ∩b )]∪M
(Fus) (∩R)
[(G ;a ∩b ),(G ;a ∩b )]∪M −→ [(G ;a ∩b ),(G ;a ∩b )]∪M
(Ax)
[(b ;b )]∪[(a ;a )|i∈I]
i i
(Fus) (Ax)
[(b ;b ),(b ;b )]∪[(a ;a )|i∈I] −→ [(b ;b ),(b ;b )]∪[(a ;a )|i∈I]
i i i i
and
(Ax)
[(a ;a )|i∈I]∪[(a ;a )|i∈J]
i i i i
(P) (Ax)
[(a ;a )|i∈I] −→ [(a ;a )|i∈I]
i i i i
Wewillcallcleanaderivation withoutapplications ofrules(P)and(Fus).
Theorem2.5 ⊢ M ifandonlyif ⊢ M
ISL ISC
Proof
(onlyif). Byinductiononthenaturaldeductionstylederivation. Rules(Ax),(W),(X)arethesamein
both styles. Rule (P) is derivable in both styles (see previous lemma and Theorem 11 in [15]). In case
of the global rules, the proof is quite similar to the standard proof of the equivalence between natural
deduction and sequent calculus for intuitionistic case, just putting similar cases in parallel. So we will
show just the case of implication as example, and then the case for local conjunction elimination (the
introduction isthesameinboththesystems).
Case(→E):
(Ax)
[(G ;a )] [(b ;b )]
i i i i i i
(→L)
[(G ;a →b )] [(G ,a →b ;b )]
i i i i i i i i i
(cut)
[(G ,G ;b )]
i i i i
(XC)
[(G ;b )]
i i i
wherethedashedlinenamed(XC)denotesasequence ofapplications ofrules(X)and(C).
Case(∩E ):
k
(Ax)
[(g ;g )],(a ;a )
i i i k k
[(G ;g )],(G ;a ∩a ) [(g ;g )],(a ∩a ;a ) (∩Lk)
i i i L R i i i L R k
(cut)
[(G ;g )],(G ;a )
i i i k
S.RonchiDellaRocca,A.Saurin,etal. 21
(if).
Byinduction onthesequentcalculus stylederivation.
Wewillshowjustthecaseofimplication, localconjunction elimination andcut.
Case(→L): LetZ =G ,D ,a →b . Then:
i i i i i
(Ax)
[(D ,b ;g )] [(a →b ;a →b )] [(G ;a )]
i i i i i i i i i i i i
(WX) (WX) (W)
[(Z,b ;g )] [(Z ;a →b )] [(Z ;a )]
i i i i i i i i i i i
(→I) (→E)
[(Z ;b →g )] [(Z ;b )]
i i i i i i i
(→E)
[(G ,D ,a →b ;g )]
|i i {zi }i i i
Zi
Case(∩L ): LetG =G ′,q andE =G ′,q ,q =G ,q . Then:
k i i i i i i i i i
(Ax)
[(G ;g )],(G ,a ;b ) [(q ;q )],(a ∩a ;a ∩a )
i i i k i i i L R L R
(WX) (WX)
[(E ;g )],(G ,a ∩a ,a ;b ) [(G ;q )],(G ,a ∩a ;a ∩a )
i i i L R k i i i L R L R
[(G ;q →g )],(G ,a ∩a ;a →b ) (→I) [(G ;q )],(G ,a ∩a ;a ) (∩Ek)
i i i i L R k i i i L R k
(→E)
[(G ;g )],(G ,a ∩a ;b )
i i i L R
Case(cut):
[(D ,a ;b )]
i i i i
(WX)
[(G ,D ,a ;b )] [(G ;a )]
i i i i i i i i
(→I) (W)
[(G ,D ;a →b )] [(G ,D ;a )]
i i i i i i i i i
(→E)
[(G ,D ;b )]
i i i i
3 ISC and LJ
AderivationinISCcorrespondstoasetofderivationsinIntuitionistic SequentCalculus(LJ),wherethe
twoconjunctions collapseinto∧.
Definition3.1 GivenaISCderivationp ,wedefinethesetp˜ =(p ) ofLJderivationsbyinductionon
i i∈I
thestructure ofp asfollows:
•if p :[(a ;a )|i∈I] (Ax) then p˜ =(p ) with p :a ⊢a (Ax)
i i i i∈I i i i
p 1:[(G ;a )|i∈I] p 2:[(D ,a ;b )|i∈I]
i i i i i
•if p :[(G ,D ;b )|i∈I] (cut) with p˜1=(p 1:G ⊢a )
i i i i i i i∈I
p 1:G ⊢a p 2:D ,a ⊢b
i i i i i i i
and p˜2=(p 2:D ,a ⊢b ) ,then p˜ =(p ) with p :G ,D ⊢b (cut)
i i i i i∈I i i∈I i i i i
p ′:[(G ;b )|i∈I] p ′:G ⊢b
i i i i i
(W) (W)
•if p :[(G ,a ;b )|i∈I] with p˜′=(p ′:G ⊢b ) ,then p˜ =(p ) with p :G ,a ⊢b
i i i i i i i∈I i i∈I i i i i
•p˜ isdefinedinthesamewayasabovefor(X),(C)and(→R).
22 IntersectionLogicinsequentcalculusstyle
M′
(R)
•if p :M where(R)=(Fus),(P) then p˜ =clea˜n(p )
p 1:[(G ;a )|i∈I] p 2:[(D ,b ;g )|i∈I]
i i i i i
•if p :[(G ,D ,a →b ;g )|i∈I] (→L) with p˜1=(p 1:G ⊢a )
i i i i i i i i i∈I
p 1:G ⊢a p 2:D ,b ⊢g
i i i i i i i
and p˜2=(p 2:D ,b ⊢g ) ,then p˜ =(p ) with p :G ,D ,a →b ⊢g (→L)
i i i i i∈I i i∈I i i i i i i
p ′:[(G ,a 0,a 1;b )|i∈I]
i i i i
(&L)
•if p :[(G ,a 0&a 1;b )|i∈I] with p˜′=(p ′:G ,a 0,a 1⊢b ) ,then p˜ =(p )
i i i i i i i i i i∈I i i∈I
p ′:G ,a 0,a 1⊢b
i i i i i
(∧L)
with p :G ,a 0∧a 1⊢b
i i i i i
p 1:[(G ;a )|i∈I] p 2:[(D ;b )|i∈I]
i i i i
•if p :[(G ,D ;a &b )|i∈I] (&R) with p˜1=(p 1 :G ⊢a )
i i i i i i i i∈I
p 1:G ⊢a p 2:D ⊢b
i i i i i i
and p˜2=(p 2:D ⊢b ) ,then p˜ =(p ) with p :G ,D ⊢a ∧b (∧R)
i i i i∈I i i∈I i i i i i
p ′:[(G ,a ;b )]∪[(G ;g )|i∈I]
j i i
•if p :[(G ,a ∩a ;b )]∪[(G ;g )|i∈I] (∩Lj) with p˜′={p ′ :G ,a ⊢b }∪(p ′:G ⊢g ) ,
0 1 i i ⋆ j i i i i∈I
p ′ :G ,a ⊢b
⋆ j (W),(X)
G ,a ,a ⊢b
0 1
(∧L)
then p˜ ={p }∪(p ′) with p :G ,a ∧a ⊢b
⋆ i i∈I ⋆ 0 1
p ′:[(G ;a ),(G ;b )]∪[(G ;g )|i∈I]
i i
•if p :[(G ;a ∩b )]∪[(G i;gi)|i∈I] (∩R) with p˜′={p a′ :G ⊢a }∪{p b′ :G ⊢b }∪(p i′:G i⊢gi)i∈I,
p a′ :G ⊢a p b′ :G ⊢b
(∧R)
G ,G ⊢a ∧b
(X),(C)
then p˜ ={p }∪(p ′) with p :G ⊢a ∧b
⋆ i i∈I ⋆
Letusstress thefact that, by Definition 3.1, each p inp˜ isaderivation in LJ. Thetranslation from
i
ISCtoLJisalmoststandard, butfortherules(Fus),(P),wheretheresultofLemma2.4hasbeenused.
4 Cut-elimination in LJ: a short survey
AnalogouslytoLJ,ISCenjoyscut-elimination, i.e.,thecutruleisadmissible. Inordertogiveaproofof
thisfactthemostnaturalideaistomimicthecuteliminationprocedureofLJ,inaparallelway. Wewill
useDefinition3.1,whichassociatestoeveryderivationinISCasetofderivationsinLJ. Therearediffer-
entversionsofsuchaproofintheliterature;wesuggesttheversionsin[9,22,13]. Letusbrieflyrecallit.
S.RonchiDellaRocca,A.Saurin,etal. 23
The cut-elimination algorithm in LJ is based on a definition of some elementary cut-elimination
steps, eachonedepending onthepremises ofthecut-rule, andonacertain orderofapplications ofsuch
steps. Iftheelementarystepsareappliedinthecorrectorder,thentheprocedureeventuallystopsandthe
resultisacut-free proofofthesamesequent.
Theprincipal problem for designing this algorithm isindefining theelementary stepwhen thecon-
traction ruleisinvolved. Infactthenaturalwayofdefiningthecontraction stepisthefollowing:
G ⊢a D ,a ,a ⊢b
(cut)
D ,a ,a ⊢b G ⊢a G ,D ,a ⊢b
(C) (cut)
G ⊢a D ,a ⊢b G ,G ,D ⊢b
(cut) (CX)
G ,D ⊢b → G ,D ⊢b
where(CX)representsasuitablenumberofcontractionandexchangerules. Thisstepgeneratestwocuts,
such that the last one appears of the same “size” and also of the same (or maybe greater) “height” as
the original one, according to any reasonable notions of size and height. A standard way to solve this
problemistostrenghenthecutrule,allowingittoeliminateatthesametimemorethanoneoccurrences
ofaformula,inthefollowingway:
m
z }| {
G ⊢a D ,a ,...,a ⊢b m≥1
(multicut)
G ,D ⊢b
It is easy to check that replacing the cut-rule with the multicut-rule produces an equivalent system.
Abusing thenaming, inwhatfollows wewillusethe name“cut” for multicut and wewillcall“LJ”the
system obtainedbyreplacing cutbymulticut.
Assuming, forsimplicity, thatG andD donotcontainanyoccurrence ofa ,thecontraction stepnow
becomes:
D ,a ,a ⊢b
(C)
G ⊢a D ,a ⊢b G ⊢a D ,a ,a ⊢b
(cut) (cut)
G ,D ⊢b → G ,D ⊢b
and the new cut-rule which has been generated has been moved up in the derivation with respect to the
original one. Thecutrulecanbeeliminated and,inordertodesignthealgorithm doingit,thefollowing
measuresareneeded.
Definition4.1 i) Thesizeofaformulas (denoted by|s |)isthenumberofsymbolsinit;
ii) Theheightofaderivationisthenumberofruleapplicationsinitsderivationtree. Leth(p )denote
theheightofp ;
iii) The measure of a cut p , denoted by m(p ), is a pair (s,h), where s is the size of the formula
eliminatedbyitandhisthesumsoftheheightsofitspremises.
Weconsider measuresorderedaccording toarestriction oflexicographic order,namely:
(s,h)<(s′,h′)ifeithers<s′ andh≤h′ ors≤s′ andh<h′.
Thenthefollowinglemmaholds:
24 IntersectionLogicinsequentcalculusstyle
Lemma4.2 Let p : G ⊢ s be a derivation in LJ, with some cut rule applications. Let p ′ : G ⊢ s be
the derivation obtained from p by applying an elementary cut-elimination step to a cut closest to the
axioms, let c. Then p ′ does not contain c anymore, and, if it contains some new cuts with respect to p ,
their measure are less than the measure ofc. Moreover the measures of the cuts different from c do not
increase.
Thecutelimination property cannowbestated.
Theorem4.3 Let p be a derivation in LJ. Then, there is a derivation p ′ proving the same judgement
whichdoesnotusethecut-rule.
Proof From our definition of measure, a topmost cut can be eliminated in a finite number of steps.
Sincethenumberofcutsisfinite,theproperty follows.
5 Cut-elimination in ISC: the elementary cut elimination steps
Firstofallwestrenghten thecut-ruleofISCanalogously toLJ,inthefollowingway:
m
z }| {
[(G ;a )|i∈I] [(D ,a ,...,a ;b )|i∈I]
i i i i i i
(cut)
[(G ,D ;b )|i∈I]
i i i
Itiseasytocheckthatweobtainanequivalentsystem,sowewillabusethenotation,andcallrespectively
cut thenewcut,andISCthenewsystem.
Then,wecandividethemostsignificantoccurrences ofcutinISCintwocases: theglobalandlocal
ones,depending onwhetherglobalorlocalconnectives areintroduced oncutformulasinthepremises.
In the case of global cut-rules, the elementary cut elimination steps act as for LJ, but in parallel on
alltheatomsoftheinvolved molecules. Asanexample,letusshowthecaseofthe(&R),(&L)cut:
[(G ;a )] [(D ;b )] [(Z,a ,b ;g )]
i i i i i i i i i i i
(&R) (&L)
[(G ,D ;a &b )] [(Z,a &b ;g )]
i i i i i i i i i i
(cut)
[(G ,D ,Z ;g )]
i i I i i
↓
[(D ;b )] [(Z,a ,b ;g )]
i i i i i i i i
(cut)
[(G ;a )] [(D ,Z,a ;g )]
i i i i i i i i
(cut)
[(G ,D ,Z;g )]
i i i i i
The definition of the local cut-elimination steps inIUSCposes some problems. Letus consider the
followingexample:
Letp andp berespectively thefollowingderivations:
1 2
(Ax) (Ax)
[(a ;a ),(a ;a ),(m ;m )] [(a ;a ),(a ;a ),(m ∩n ;m ∩n )]
(→L)
[(a ,a →a ;a ),(a ,a →a ;a ),(m ,m →(m ∩n );m ∩n )]
(∩R)
p :[(a ,a →a ;a ∩a ),(m ,m →(m ∩n );m ∩n )]
1
(Ax)
[(a ∩a ;a ∩a ),(m ;m )]
(∩L)
p :[(a ∩a ;a ∩a ),(m ∩n ;m )]
2
S.RonchiDellaRocca,A.Saurin,etal. 25
Andletp be:
p p
1 2
(cut)
p :[(a ,a →a ;a ∩a ),(m ,m →(m ∩n );m )]
Notice that the right and left introduction of the connective ∩ create an asymmetry because they are
applied on different atoms, so it looks quite impossible to perform a cut-elimination step. In fact, the
derivation p corresponds (modulosomestructural rules)tothefollowingderivationinISL:
(Ax) (Ax)
[(a →a ;a →a ),(a →a ;a →a ),(m →(m ∩n );m →(m ∩n ))] [(a ;a ),(a ;a ),(m ;m )]
(→E)
[(a ,a →a ;a ),(a ,a →a ;a ),(m ,m →(m ∩n );m ∩n )]
(∩E)
[(a ,a →a ;a ),(a ,a →a ;a ),(m ,m →(m ∩n );m )]
(∩I)
[(a ,a →a ;a ∩a ),(m ,m →(m ∩n );m )]
Thisderivationisinnormalformaccordingtothedefinitiongivenin[15];roughlyspeaking,aderivation
inISLisinnormal form ifitdoesnot contain anintroduction ofaconnective immediately followed by
anelimination ofthesameconnective, modulosomestructural rules.
Sincetheproblemisduetothepresenceofthelocalconnective, inparticularwhenthetwopremises
ofacutaretherightandleftintroductionof∩ondifferentatoms,thesolutionistorestricttheuseofthis
connective, inparticular forbidding atomswhereitisinprincipal position.
Definition5.1 i) Aformulaiscanonical ifitsprincipal connective isnot∩.
i) AnISCderivation iscanonical ifalltheformulasintroduced byaxiomsand(W)arecanonical.
Lemma5.2 Letp :⊢ M. Then,thereisacanonical derivation p ′:⊢ M.
ISC ISC
Proof Byinduction onp .
Thefollowingexampleillustrates thebase-case.
Example5.3 Considerthederivation:
(Ax)
[(s ∩t ;s ∩t )]
Thecorresponding canonical derivation can be obtained byintroducing the molecule [(s ;s ),(t ;t )]by
anaxiomandthenbyapplying asequence oftwo(∩L)rulesfollowedbya(∩R)rule.
Fromnowon,wewillonlyconsidercanonical derivationstodefinethecut-elimination steps. More-
over we assume that the derivations are also clean. As said before, the global rules are completely
standard, since they act like in LJ, but in parallel on all the atoms of the involved molecules. Thus, we
will expose only some characteristic cases dealing with ∩. Note the use of the structural rules (P) and
(Fus).
Definition5.4 Acut-elimination stepinISCisdefinedbycases. Weassumedonothaveapplications of
rules(P)and(Fus),andweshowthemostinteresting structural andlocalcases.
Commutationsteps:
• caseof(∩R)