Table Of ContentAbstract p-time proof nets for MALL:
Conflict nets
8 Dominic J. D. Hughes∗
0
0
2 StanfordUniversity
n January11,2007
a
J
6
Thispaperpresentsproof netsfor multiplicative-additivelinearlogic(MALL),calledconflict
1
nets.Theyareefficient,sincebothcorrectnessandtranslationfromaproofarep-time(polynomial
time),andabstract,sincetheyareinvariantundertransposingadjacent &-rules.
]
O A conflict net ona sequentis concise: axiomlinks withaconflict relation. Conflict netsare
a variant of (and were inspired by) combinatorial proofs introduced recently for classical logic:
L
each can be viewed as a maximal map (homomorphism) of contractible coherence spaces (P -
. 4
h freegraphs,orcographs),fromaxiomstosequent.
t Thepaperpresentsnewresultsforotherproof nets: (1)correctnessandcuteliminationfor
a
m slice nets (Hughes / van Glabbeek 2003) are p-time, and (2) the cut elimination proposed for
monomialnets(Girard1996)doesnotwork.Thesubtletieswhichbreakmonomialnetcutelimi-
[
nationalsoapplytoconflictnets: aswithmonomialnets,existenceofaconfluentcutelimination
1 remainsanopenquestion.
v
1
2 1 Introduction
4
2
. Jean-YvesGirard’sseminalpaper[Gir87]onlinearlogicintroducedanelegantabstractrepresenta-
1
tion of a proof called a proof net. These original proof nets used boxes [Gir87, p.45] to deal with
0
thesuperpositionassociatedwith &-connectives. Boxesmimicthesequent calculus &-rule almost
8
0 directly, so that the following two proofs,which differ only in the orderof adjacent &-rules, have
: distinctboxnets:
v
i
X
P,P P,P P,P P,P P,P P,P P,P P,P
r & & &′ &′
a P&P,P P&P,P P,P&′P P,P&′P
&′ &
P&P,P&′P P&P,P&′P
(Themarkedconnective &′ isfordistinction,weomitsequentturnstiles⊢,andP isthedualofP.)
The follow-up paper [Gir96] tried a different approach to superposition. Every & is given an
eigenvariable, and every node in the proof net has a list of possibly-negated eigenvariables, its
monomial. Monomialnetssuffertwomaindefectsrelativetoboxnets:
∗VisitingScholar,ComputerScienceDepartment,StanfordUniversity,CA94305.
1
• There is no canonical surjection from cut-free proofs to monomial nets.1 One can no longer
ask “Which proofs are identified upon translation to a proof net?”: monomial nets fail to
provideasemanticsforcut-freeproofs.2
• Unfortunatelythe cutelimination proposedformonomial nets[Gir87,p.24] doesnotwork:
Section10givesacounterexample. Existenceofaconfluentcuteliminationremainsanopen
question.3
Theslicenetsof[HG03,HG05] solve theseproblemsbytakingaproofnettobe aset ofaxiom
linkings,orslices.4 (Equivalently,aslicenetcanberepresentedasasetofboolean-weightedaxiom
links.) Thereis a canonical surjection from proofs. Forexample, the two proofsabove map to the
followingslicenet,comprisingfouraxiomlinkings,eachlinkingcontainingjustoneaxiomlink:5
P&P P&′P
Slice nets were shown to have a simple confluent cut elimination, and a hyper-elimination which
occurs independently slice-by-slice (by GoI-style path composition), yielding a category [HG03,
HG05]. Thepresentpaper(Section9)provesthatcorrectnessofslicenetsisp-time.
Butallisnotrosywithslicenets: therecanbeanexponentialblowupinsizewhentranslatinga
proof.67 Thisisaflawifwetakeseriouslythenotionthatasemanticsisastructurepreservingmap,
orsomekindofhomomorphismfromproofs: wearefailingtorespectcomputationalcomplexity. A
key insight of propositional proof complexity [CR79] is that complexity is important in decidable
logicssuchasMALL.8
1Thereisacanonicalnon-surjectivefunction: identifynoformulasduringtranslation[Gir96, p.7]. Theimageofthis
functionispreciselytheboxproofnets,disguisedinmonomialform.Soasasemanticsofcut-freeproofs,thisisexactlythe
boxnetsemantics. Sinceeveryboxproofnetisamonomialproofnet,thereareactuallymoremonomialproofnetsthan
boxproofnets.
2See[HG03,HG05]foradetailedexplanation,withexamples.
3Onealwayshasatrivialnon-confluentcuteliminationviasequentialization,whichisuninteresting.
4ThisunderlyingdatastructureismentionedinappendixA.1.6of[Gir96]. Theessentialcontributionof[HG03,HG05]
wastoprovidetheelusivegeometriccorrectnesscriterionandexhibitasimpleconfluentcutelimination.
5Note thatthis isnotasinglelinkingwithfour axiomlinks; itisfour linkingseachwitha singleaxiom link. Inthis
particularcase,thereisalsoacanonicalmonomialnet,butthatisnottrueingeneral.
6Considertheuniquecut-freeproofof⊗n(1&1),where⊗ndenotesiteratedtensor⊗withnargumentsassociatedto
theleft(e.g.⊗3A = (A⊗A)⊗A),inwhich⊗-rulesarebelow &-rules.Sincetherearen &-rules,translatingthisproof
Πntoaslicenetθnblowsupexponentially:θnhas2nslices(axiomlinkings).(Foranexamplewithoutthetensorunit1,
readeach1asa&a.)TheexponentialblowupwhenmappingtoasetofslicesismentionedinAppendixA.1.6of[Gir96].
7As remarked earlier, a set of slicescan just as well be representedas a set of weighted axiom links (arbitrary non-
monomialbooleanweights,e.g.p∪qforeigenvariablespandq). Thistrivialchangeofnotationdoesnoteliminatethe
exponentialblowup:withn &’sinthesequent,abooleanweightisasubsetofthe2nhypercube.
8Infirst-orderlogic,whichisundecidable,thevalueofaproofasacertificateoftheoremhoodisabsolutelyclear. But
inadecidable,propositionalsetting,whatisthepointofbeinghandedaproof? Todeterminetheoremhood,weonlyneed
theformula. Theideainpropositionalproofcomplexityistoreinstateandquantifythevalueofaproofcertificate: ifthe
correctnessofacertificatecanbecheckedinpolynomialtimeinitssize,andthecertificateisnot‘toobig’relativetothe
formula, checkingthe certificatewillbe fasterthan than decidingthe theoremhood of the formula. See[Urq95] for an
accessibleintroductiontopropositionalproofcomplexity.
2
Representationefficiency Abstraction Cutelimination
Raise
P-time P-time Raise ⊗-rule Confluent
Proofnet &/⊕/&-rule P-time
correctness translation over &-rule (unit-free)
over &-rule
Box X X 7 7 7? ?
[Gir87]
Monomial ? Xa 7a 7a 7? ?b
[Gir96]
Slice X 7 X X Xd X
[HG03,05]
Conflict X X X 7c 7? ?
X=yes 7=no ?=openquestion 7?=openquestion,probablyno
a Withrespecttothecanonical(non-surjective)prooftranslation[Gir96,p.7]. Seefootnote1.
b Thedefinitionproposedin[Gir96,p.24]doesnotwork: seeSection10.
c Seeminglythepriceofhavingap-timetranslationfromproofs.
d P-timesincenormalisationisslicewise.
Table1: Comparisonofproofnets.
Thispaperpresentsanewnotionofproofnet,calledaconflictnet,suchthat:
(1) Checkingcorrectnessisp-timeinthesizeoftheproofnet.
(2) Translationfromaproofisp-time(improvingonslicenets[HG03,HG05]).
(3) Translationisinvariantundertransposingadjacent &-rules,andraisinga&-or ⊕-ruleovera
&-rule(improvingonboxnets[Gir87]andmonomialnets[Gir96]9).
(4) Extractingasequentializationisp-time.
(5) Aconflictnetonasequentisconcise: axiomlinkswithaconflictrelation.
(6) Prooftranslationissimple: axiomsbecomeaxiom links,and twoaxiomlinksconflict iffthey
arefromoppositebranchesabovea &-rule.
Examples of conflict nets are shown in Figure 1. Figure 1 also illustrates how translation from
a proof to a conflict net is invariant with respect to raising a ⊕/&/&-rule over a &-rule. Table 1
comparesdifferentproofnets.
9Withrespecttothecanonical(non-surjective)prooftranslation[Gir96,p.7].Seefootnote1.
3
a,a a,a a,a aa
& ⊕ ⊕
b⊕a a&a 2 2
a, a&a b⊕a, a b⊕a, a
⊕ &
2
b⊕a, a&a b⊕a, a&a
P,P P,P P,P P,P P,P P,P P,P P,P
& & P&P P&′P &′ &′
P&P,P P&P,P P,P&′P P,P&′P
&′ &
P&P,P&′P P&P,P&′P
c,c a,a c,c a,a c,c a,a c,c a,a
⊗ ⊗ ⊗ ⊗
a,c,c⊗a a,c,c⊗a a&c (c⊗a)&(c⊗a) a,c,c⊗a a,c,c⊗a
& & &
a, c, (c⊗a)&(c⊗a) a&c, c⊗a a&c, c⊗a
& &
a&c, (c⊗a)&(c⊗a) a&c, (c⊗a)&(c⊗a)
Q,Q Q,Q P,P Q,Q P,P Q,Q
& ⊗ ⊗
P,P Q,Q&Q P,P⊗Q,Q P,P⊗Q,Q
⊗ &
P,P⊗Q,Q&Q P,P⊗Q,Q&Q
a a⊗Q Q&Q a a⊗Q Q&Q
Figure 1: Illustrating the surjective translation function from proofs to conflict nets. The first three rows
showhowconflictnetsareinvariant withrespecttoraisinga⊕-, &-or&-ruleovera &-rule,respectively:
eachpairofproofs(leftandright)mapstothesameconflictnet(centre). Thelasttwotranslations(flowing
downwards) showthatraising a⊗-ruleovera &-rule changes theconflictnet; thisseemstobetheprice
of p-time proof translation. Conflicts between axiom links are shown asdotted edges. Axiom links which
overlap(shareanatominthesequent)conflictimplicitly.
4
Relatedwork. ThelastfewyearshaveseenarenaissanceofworkinvolvingMALLproofnets,in-
cluding[Ham04](extendingmonomialnetswithmix,analysing softness),[CP05](alanguage for
MALL proofs, viewed as processes), [CF05] (a ludics-based analysis of sequentiality/parallelism),
[BHS05] (a fully complete relational model for MALL), [Mai07] (extending Danos contractibility
[Dan90]toadditives,usingadistributivityrewrite),[Abr07](adomain-theoreticviewofunfolding
the &-rulesaswegoupaproof),tonamebutafew.10
In each case the underlying data structure involved are more complex than a conflict net, car-
rying additional machinery such as monomial weights on subformulas, subformula occurrences,
focalisation, contractionnodes,domains, partialleft/right resolutionsofthe &’sin asequent,and
soon. Likeboxnetsandmonomialnets,mostdealwithoccurrencesofsubformulas;thedatastruc-
ture ofaconflict net involves onlyatoms, trueto the spirit ofthe geometryofinteraction[Gir89].
By not dealing with internal nodes of subformula trees, which are sequential, conflict nets are in
somesensemaximallyparallel.
Current work for conflict nets includes arranging them into a category, possibly via a strongly
normalising cut elimination. A naive cut elimination can be obtained by emulating the elimina-
tion of box nets (copying empires around). One possible approach is to try and use pullbacks of
(contractible)coherencespacestoobtainacompletelyabstractformofcuthyper-elimination(com-
position) in a compact closed category. If it worked out, this would ensure a forgetful functor to
theunderlyingcompactclosedcompositionofslicenets.
Conflict nets are a variant of (and were inspired by) combinatorial proofs introduced recently
for classical logic [Hug06a, Hug06b]: each conflict net can be viewed as a maximal map (homo-
morphism)ofcontractiblecoherencespaces(P -freegraphs,orcographs),fromaxiomstosequent.
4
TherelationshipwithcombinatorialproofsissketchedinSection11.
Acknowledgement. I’mgratefultoRobinHoustonfordiscussionsaboutabstractcategoricalver-
sions of cut elimination, based on pullbacks of coherence spaces. In particular, Robin showed
me how to construct pullbacks in the category of coherence spaces. I’m also indebted to Roberto
Maieli, whose extension of Danos’ contractability criterion [Mai07] stimulated me to think about
MALLproofnetsagain.
2 Preliminaries
2.1 MALL
We work with cut-free, unit-free multiplicative-additive linear logic [Gir87], henceforth denoted
MALL.
Fix a set A = {a,b,c,...} of literals equipped with a function ( ) : A → A such that a 6= a
and a = a for all a ∈ A. MALL formulas are generated from literals by the binary connectives ⊗
(tensor) & (par) & (with) and ⊕ (plus). Define ⊗ = &, & = ⊗, ⊕ = & and & = ⊕. Define
negation (.)⊥ by a⊥ = a on literals, and (A(cid:3)B)⊥ = A⊥(cid:3)B⊥. Formulas A and A⊥ are dual. A
sequent is a list (finite sequence) A ,...,A of formulas (n > 0). Throughout this document we
1 n
takeP,Q,R,...torangeoverliterals,A,B,C,...overformulas,andΓ,∆,Σ,...oversequents.
We identify a formula with its parse tree: a tree with leaves labelled with literals and internal
verticeslabelledwithconnectives,equippedwithalinearorderonleaves. Edgesareorientedaway
10Withpolarization,proofnetsbecomemucheasier:see[LdF04].
5
Γ
ax perm
σ
P,P σΓ
Γ, A, B Γ, A
i
& ⊕
i
Γ, A&B Γ, A ⊕A
o 1
Γ,A B,∆ Γ,A Γ,B
⊗ &
Γ,A⊗B,∆ Γ,A&B
Figure 2: MALL proof rules. Here σ is any permutation on n, the number of formulas in the sequent Γ
abovethe perm-rule.
from the leaves. We identify a sequent with its parse forest: the disjoint union of its formulas
(formula parse trees), with a linear order on leaves. For example, we identify the three-formula
sequent P, P⊗Q, (Q&Q)⊗P withthefollowingparseforest:
P P Q Q Q P
⊗ &
⊗
The linear order on leaves is given by the left-to-right order on the page. Two leaves are dual if
theirliterallabelsaredual.
IfΓ = A ,...,A , and σbeapermutationonn(i.e., abijection{1,...,n} → {1,...,n}),write
1 n
σΓ for the sequent A ,...,A . Proofs are generated using the rules in Figure 2. As a technical
σ1 σn
convenience,weshalloftensupresspermutation(perm)rules,forexample,writing
Γ, A, B,∆
&
Γ, A&B,∆
whichleavesimplicitapermutationruleaboveandbelowthe&-rule,if∆isnon-empty.
2.2 Coherencespaces
⌢
We write for strict coherenceand # forstrict incoherenceof coherence spaces[Gir87, §3]. We
⌢
call adjacency and # conflict. The elements of the web |X| of a coherence space X are tokens
of X. Recall that a map X → Y between coherence spaces is a binary relation R ⊆ |X|×|Y| which
preservesstrict coherence and reflectsstrict incoherence: y Ropx ⌢ x Ry implies y ⌢ y , and
1 1 2 2 1 2
x Ry #y Ropx impliesx #x . (WewritexRyoryRopxforhx,yi∈R.)
1 1 2 2 1 2
3 Conflict linkings
Informal definition. A link on a sequent Γ is an edge between dual leaves. A linking on Γ is a
finite set L oflinks on Γ equipped with asymmetric, irreflexivebinaryconflict relation # ⊆ L×L
6
λ λ λ
L Γ L Γ, A, B L Γ, Ai
permσ & ⊕i
L λ σΓ L λ Γ, A&B L λ Γ, A ⊕A
1 2
λ µ λ µ
L Γ,A M B,∆ L Γ,A M Γ,B
⊗ &
λ∪µ λ∪µ
L×M Γ,A⊗B,∆ L+M Γ,A&B
Figure3: Inductivetranslationfromaprooftoaconflictlinking.
such that overlap implies conflict: if distinct links l and m share an atom, then l#m. Links may
be parallel (betweenthe same pair of leaves). Examples of linkings are shown in Figure 1. When
drawinglinkings,weleaveimplicittheconflictsimpliedbyoverlap.
Formalisation. AdualpairinΓ isapair{x,y}ofdualleavesinΓ.
DEFINITION 1 AlinkingonΓ isabinaryrelationλ:L→|Γ|fromafinitecoherencespaceL,whose
tokensarecalledlinksonΓ,totheset|Γ|ofleavesinΓ,satisfying:
• Dualpair. ForeverylinklinLthedirectimageλ[l]={x∈|Γ|:hl,xi∈λ}isadualpair.
• Overlap. Ifhl,xi∈λandhl′,xi∈λwithl6=l′ (landl′ overlapatx)thenl#l′ [modL].
λ
Weabbreviatealinkingλ:L→|Γ|toλ:L→Γ orL Γ.
4 P-time proof translation function from proofs
Informal definition. A MALL proof of Γ translates to a linking on Γ by viewing each axiom rule
as a link onΓ (bytracing its two leaves downthe proofinto Γ),and defining l#miff l and mare
inoppositebranchesabovea &-rule. Figure1showsexamplesofprooftranslation.
Formalisation. Thefollowingformalisationisbyinductiononthenumberofrulesinaproof.
• Basecase. Theaxiomrule P, P translatestotheuniquesingle-link linkingonP,P.11
• Inductivestep. Everyinstanceofaruleinducesaninclusionfunctionfromtheleavesofeach
sequent above the line to the sequent below the line.12 Via these leaf inclusions, Figure 3
interpretseach rule as an operationon linkings. The sum L+M in the interpretationof the
&-rule is the disjoint union (categorical sum/coproduct) of the coherence spaces L and M,
denotedL⊕Min[Gir87]. Withoutlossofgenerality,weassumethecanonicalinjectionsfrom
the tokensetsof L and M into the tokenset of L+M are inclusions. The product L×M in
theinterpretationofthe ⊗-rule isL+Mtogetherwithstrictcoherencebetweeneverytoken
inLandeverytokeninM. Thisiscategoricalproduct,denotedL&Min[Gir87].
11Ifxandx′arethetwoleaves,thelinkingis λ:I→P,P whereIhasasingletoken•andλ={h•,xi,h•,x′i}.
12Eachsequent(parseforest)abovethelineisasubgraphofthesequentbelowtheline.
7
• • •
P P⊗Q Q&Q P P Q Q Q
Figure4: Anexampleofaslicingλ:L→Γ withΓ =P,P⊗Q,Q,Q.Theunderlyingmaximalmapλ:L→
Γ#betweencontractiblecoherencespacesisshownontheright. Thefivestokensofthecoherencespace
Γ# are shown withtheir literal labels. Thethree links(tokens) ofLareshown as•. Note thatλisindeed
maximal: werewetoaddanyedgetothebinaryrelationλ,itwouldnolongerbeacoherencespacemap.
EachruleinterpretationpreservestheDualpairandOverlapconditionsinthedefinitionofalinking.
Thusthetranslationofaproofisawell-definedlinking.
Alinking issequentializable ifitisthetranslationofaproof;anysuch aproofisasequential-
izationofthelinking.
5 Slicings
Thissectiondefinesaslicingasarefinementofalinking,asteppingstonetowardsthedefinitionof
conflictnet.
A coherence space is contractible if its web is finite and P -free (no induced four-vertex path
4
[Sei74]): whenever x #x #x #x for distinct x then x #x or x #x or x #x [Hu99]. Define
1 2 3 4 i 1 3 2 4 1 4
Γ# as the coherence space whose tokens are the leaves of Γ with x#y iff x 6= y and the smallest
subformulacontainingxandyisadditive.13 IfΓ isnon-empty,itscoherencespaceΓ#iscontractible
(asimpleinduction).
DEFINITION 2 AslicingonΓ isamaximalmapλ:L→Γ#fromacontractiblecoherencespaceL.
MaximalityiswithrespecttoinclusionamongmapsL→Γ#.14 Anexampleofaslicingisshownin
Figure4withitsunderlyingmaximalmapclarified.
PROPOSITION 1 Checkingthatalinkingλ:L→Γ isaslicingisp-timeinthesizesofLandΓ.
⌢
Proof. Checking that λ is a map (preserving and reflecting #) is clearly polynomial. Checking
contractibility (P -freeness) is linear [CPS85]. Checking direct images are dual pairs is obviously
4
polynomial. Checkingmaximalityispolynomial: foreveryedgee6∈λwecheckthatλ∪{e}isnota
map.15 (cid:3)
Aslice ofaslicing λ :L →Γ# isamaximalclique inL.16 ThetwoslicesoftheexampleinFigure4
13Inotherwords,x#yiffxandyareinthesameformulaA,andthefirstcommonvertexalongthepathsfromxand
ytotherootofAislabelled& or⊕.Equivalently,thejoin(leastupperbound)zofxandyexistswhenweinterpretΓ as
apartialorderwithleavesmaximalandrootsminimal,andzislabelled& or⊕.
14ThusλismaximaliffitisamaximalcliqueinL⊸Γ#.
15ItsufficestotestwithsingleextraedgesesinceamapR:X→Yismaximaliffitisamaximalcliqueinthecoherence
spaceX⊸Y.
16AcliqueCisasetofpairwisecoherenttokens:ifx,y∈Candx6=ythenx⌢y.
8
areillustratedbelow.
P P⊗Q Q&Q P P⊗Q Q&Q
An additive resolution of Γ is a maximal clique in Γ# [HG03, HG05]. The image of a set Z ⊆ X
under a binary relation R ⊆ X×Y is {y ∈ Y : zRyforsomez ∈ Z}. The following proposition
formalisesthesenseinwhich“everysliceisanMLLlinking”(cf.[Gir87,Gir96,HG03,HG05]).
PROPOSITION 2 Letλ : L → Γ beanon-emptyslicing.Theimageofeverysliceof λisanadditive
resolutionofΓ.
Proof. Acorollaryof[Hu99,Prop.2.2]: anon-emptymapbetweencontractiblecoherencespacesis
maximaliffitpreservesmaximalcliques,i.e.,theimageofanymaximalcliqueisamaximalclique.
(cid:3)
Note that the proposition holds for the two slices depicted above. The proposition is somewhat
surprising,sincecheckingeverysliceappearsexponential-time(becauseasliceisasubset).
6 Introducing erasure: Boxless nets
In Section 7 we define a conflict net as a slicing which is erasable under a confluent, terminating
(stronglynormalising) erasurerewrite;. Erasabilityischeckableinp-time inthenumberoflinks
andinthenumberofleavesinthesequent. Aformoferasurewillalsoyieldp-timecorrectnessfor
the slice nets of [HG03, HG05]. For didactic purposes, we begin by defining erasure in a simple
settingrelatedtoboxnets[Gir87],sincethatisthemostlikelytobefamiliartothereader. However,
thereadercansafelyskiptoSection7withoutlossofcontinuity.
Weshalldescribeavariantofboxnetsinwhichthecircumscribingboxesarenotdrawnexplic-
itly. Accordingly, we shall refer to them as boxless nets. The translation from a proof to a boxless
net is exactly the same as the translation to a box net — only one forgets to draw the boxes. For
example,thetwoproofsonpage1translate(respectively)tothefollowingpairofboxnets:
P P P P P P P P P P P P P P P P
P&P P P&P P P P&′P P P&′P
P&P P&′P P&P P&′P
Nowemphasisethesuperposition/contractionoftheseformulas,anddropthesurroundingboxes:
P P P P P P P P P P P P P P P P
P&P P P&P P P P&′P P P&′P
P&P P&′P P&P P&′P
9
Finally, draw nodes instead of formulas, to remove some redundancy, and where two formulas
merge,makethatexplicitbydrawingacontractionnode(C-node):
P P P P P P P P P P P P P P P P
& C & C C &′ C &′
C &′ & C
6.1 Circuits
Acircuitcomprises:
• Afinite,non-emptysetofnodes.
• Afinitesetofwires. Eachwireislabelledwithaformula,andisassignedasourcenodeand,
possibly,atargetnode. Ifatargetnodeispresent,itisdistinctfromthesourcenode. Awire
withnotargetisanexit.
• Eachnodehasoneofthefollowingforms:
– Axiom. Thesource of two wires and the target of none. Thewires are labelled bydual
literals.17
– Contraction. The target of two wires and the source of one. All three wires have the
sameformula.
– Binary. The target of two wires and the source of one. The incoming wires are distin-
guished as a left wire and a right wire. A binary node is typed as one of ⊗, & or &. If
theformulaoftheleftwireisA,theformulaoftherightwireisB,andthenodetypeis
(cid:3),theformulaoftheoutgoingwireisA(cid:3)B.
– Plus. The target of one wire and the source of one wire. The incoming wire is distin-
guishedasleftorright. LetAbetheformulaoftheincomingwire. Iftheincomingwire
isleft(resp.right)thentheformulaoftheoutgoingwireisA⊕B(resp.B⊕A)forsome
formulaB.
• The graph is connected: for any two nodes N and N′ there exists a sequence of nodes
N ...N withN =NandN =N′ (k>1)suchthatforalli∈{1,...,k−1}thenodesN
1 k 1 k i
andN arejoinedbyawire,i.e.,thereexistsawirewhosesourceisN andtargetisN ,
i+1 i i+1
orviceversa.18
• Theexitsareequippedwithalinearorder. Thesequentcomprisingtheformulasoftheexits,
inorder,istheconclusionofthecircuit.
An example of a circuit with concluding sequent P&P,P&P is drawn in Figure 5, formalising the
lastgraphinourmotivatingdiscusionabove. Anaxiomnodeisdrawnasahorizontallinesegment.
17Ifwewishtoincludecuts,wedefineacutnodeasthetargetoftwowires,labelledbydualformulas,andthesourceof
nowire.
18Bydroppingconnectedness,andslightlymodifyingthedefinitionoferasurebelow,onecouldchoosetovalidatethemix
rule.
10
