NATURAL DEDUCTION AND COHERENCE FOR
NON-SYMMETRIC LINEARLY DISTRIBUTIVE CATEGORIES
Dediated to Joahim Lambek
ROBERTR. SCHNECK
ABSTRACT. Inthispaperertainproof-theoretitehniquesof[BCST℄areapplied
to non-symmetri linearly distributive ategories, orresponding to non-ommutative
negation-free multipliative linear logi (mLL). First, the orretness riterion for the
two-sidedproof netsdeveloped in [BCST℄isadjusted to work in thenon-ommutative
setting. Seond,theseproofnetsareusedtorepresentmorphismsina(non-symmetri)
linearlydistributiveategory;anotionofproof-netequivaleneisdevelopedwhih
per-mits aonsiderablesharpeningofthepreviousohereneresultsonerningthese
ate-gories, inludingadeisionproedure for theequality ofmaps when there isaertain
restritionontheunits. Inpartiularadeisionproedureisobtainedfortheequivalene
ofproofsin non-ommutativenegation-freemLLwithoutnon-logialaxioms.
1. Introdution
Linearly distributive ategories were introdued (as \weakly distributive ategories") in
[CS91℄, where these ategories were shown to be a natural ategorial setting for
multi-pliative linear logi (mLL) without negation. The sequel [BCST℄ further introdued a
naturaldedutionsystemfortheprooftheoryofthis logi|thatis,aversionofproofnets
suitableforstudyingthenegation-freelogi. Withoutnegation,itwaseasiertoisolatethe
problemof the units inoherene results: using these proof nets, the authorssettled the
oherene problemfor symmetrilinearly distributiveategories, by providinga deision
proedure forequivalene of morphisms.
However, this result was obtained only for symmetri linearly distributive ategories
(orresponding toommutative mLL withoutnegation, and non-planar proof nets).
Cer-tain additionaldiÆulties arise with planar proof nets (non-symmetri linearly
distribu-tive ategories, non-ommutative mLL without negation), whih prevent an immediate
generalizationfromthe ommutativease. Furthermore,there isalsoaproblemwith the
generalization,to the non-ommutativease, of the orretness riterionfor proof nets.
This paper addresses these two problems. I desribe a working orretness riterion
for planar proof nets. (A riterionfor one-sided planar proof nets, without units or
non-logial axioms, was rst given in [A95℄, using the notion of trips.) However, a omplete
Thismaterial isbaseduponwork supportedin partunderaWinston ChurhillSholarship, andin
partunder aNational Siene FoundationGraduateResearh Fellowship. Theauthor wishes to thank
J.R.B.CokettandR.A.G. Seelyfortheirenouragementofthiswork.
Reeivedbytheeditors1999January30and, inrevisedform,1999September8.
Publishedon1999November30.
1991MathematisSubjetClassiation: 03B60,03F05,03F07,03G30,18D10,18D15.
Keywordsandphrases: ategorialprooftheory,linearlogi,monoidalategories.
deision proedure for oherene in non-symmetri linearly distributive ategories has
still not been ahieved. I present a modiation of the notion of net-equivalene whih
onsiderably narrows the problem;inpartiular, adeision proedureisahieved given a
ertain restrition onmorphisms involvingthe units.
The main ontent of the paper is a study of two rewrite systems: the
sequentializa-tion proedure whih provides the orretness riterion for planar proof nets, and the
net-equivalene rewrite system whih forms the bakbone of the oherene results.
Ap-pendix A desribesthe smallamountof neessary informationfrom rewriting theory.
This paper follows the pattern of [BCST℄. In the interest of being somewhat
self-ontainedI begin withadenition oflinearlydistributiveategories,and adesriptionof
the fragment of linear logi to whih they orrespond. Following a denition of iruits,
whiharetheunderlyingstruturesoftheseproofnets,Idevelopthenewsequentialization
proedure, andshowthatitisanormalizingrewritesystemforproofnets. NextIdevelop
a rewrite system whih denes iruit equivalene, modied for the non-ommutative
ase. I note the fat that the proof from [BCST ℄, that planar proof nets form a linearly
distributive ategory, arries over orretly given the modiations. Finally I show that
the iruit equivalene rewrite system is normalizing, and disuss how this relates to a
deisionproedure for the oherene of non-symmetri linearlydistributiveategories.
A reader already familiarwith the ideas of[CS91℄ and [BCST ℄ should onentrate on
setions 4{6 for the new material.
2. Preliminaries
2.1. Notation. We shall use symbols (tensor or times) and (otensor, plus, or
par) to refer to the two tensors of linearly distributive ategories, and of multipliative
linear logi; the units of the tensorsare respetively > (top, true, unit) and ? (bottom,
false, ounit). NotethatthisonitswithGirard'snotationin [G87℄(wherepar wouldbe
reverse-ampersand and the tensor unit 1), but is onsistent with the notationof [BCST℄.
2.2. Definition. A linearly distributive ategory C is a ategory with two tensors
and; eahtensor isequippedwithaunit objet,an assoiativity naturalisomorphism,
and left and right unit natural isomorphisms:
(;>;a
;u
L
;u
R
)
a
: (AB)C !A(BC)
u L
: >A!A
u R
: A>!A
(;?;a
;u
L
;u
R
)
a
: (AB)C !A(BC)
u L
: ?A!A
u R
The two tensorsare linked by two linear distribution natural transformations:
Æ
L
: A(BC)!(AB)C
Æ
R
: (BC)A!B(C A):
Furthermore the following oherene onditions must be satised:
a
;1u
L =u R 1 (1) a ;a =a 1;a ;1a
(2)
a
;1u
L =u R 1 (3) a ;a =a 1;a ;1a
(4) u L =Æ L ;u L 1 (5) u R =Æ R ;1u
R (6) u L
1=Æ
R ;u
L
(7)
1u R =Æ L ;u R (8) Æ L ;a
1=a
;1Æ
L ;Æ L (9) a ;Æ R =Æ R 1;Æ R ;1a
(10) Æ R ;a =a 1;Æ R ;1Æ
R
(11)
1a
;Æ L =Æ L ;Æ L 1;a (12) Æ L ;Æ R 1;a =Æ R ;1Æ
L
(13)
a
;1Æ
R ;Æ L =Æ L 1;Æ R : (14)
Also note that a symmetri linearly distributive ategory is one inwhih the tensors
are symmetri (in whih ase further oherene onditions need to be introdued). A
more detailedexposition an befound in[CS91℄.
2.3. mLL without negation. Figure 1 lists the rules for a sequent alulus
pre-sentation of non-ommutative mLL without negation. Notie, in partiular, the
non-ommutativerestritiononut: the usualutruleseparates intofourrules. Sinethereis
no negation we willbe working with two-sided sequents. This sequent alulus was rst
given by Abrusi in [A91℄.
A link between linearly distributive ategories and linear logi is indiated by the
\resoure-sensitive" harater of the linear distributivities, as ompared to an ordinary
distribution. In [CS91 ℄ and [BCST ℄ the onnetion is fully developed. To be more
pre-ise: there is anequivalene between (the 2-ategories of) linearlydistributiveategories
and two-tensor polyategories, whih are a onservative extension of polyategories, a
known presentation of the above sequent alulus; furthermore -autonomous ategories
(orresponding to full multipliativelinear logi) are a onservative extension of linearly
appro-identity axiom A`A units (>L) 1 ; 2 ` 3 1 ;>; 2 ` 3
(>R ) `>
(?L) ?` (?R )
1 ` 2 ; 3 1 ` 2 ;?; 3
tensor and par
(L) 1
;A;B;
2 `
3
1
;AB;
2 ` 3 (R ) 1 ` 2 ;A 1 `B; 2 1 ; 1 ` 2
;AB;
2
(L) 1
;A`
2 B; 1 ` 2 1
;AB;
1 ` 2 ; 2 (R ) 1 ` 2 ;A;B;
3
1 `
2
;AB;
3 CUT 1 `A 1 ;A; 2 ` 3 1 ; 1 ; 2 ` 3 1 ` 2 ;A; 3 A` 1 1 ` 2 ; 1 ; 3 1 ` 2
;A A;
1 ` 2 1 ; 1 ` 2 ; 2 1 ;A`
2 1 `A; 2 1 ; 1 ` 2 ; 2
Figure 1: Sequent alulus of mLL withoutnegation
2.4. Linear biategories. We an also onsider the eet of taking formulas to be
typed relations with domain and odomain in some lass. Tensor and par an then be
viewed asompositionson formulas, where the domainand odomain must math,e.g.:
A X!Y
B Y!Z
=(AB) X!Z
:
The logi of these generalized relations is naturally non-ommutative. The
ategory-theoreti presentation of suh data is a linear biategory, whih has 0-ells
(orre-sponding to types), 1-ells (formulas), and 2-ells (proofs), with two dierent horizontal
ompositionson 1-ells (tensor and par) in additionto a vertial omposition on 2-ells.
Cohereneonditions similartothose oflinearly distributiveategoriesmustbesatised;
see [CKS℄ for a fullexposition.
Lookingahead toplanar iruits,we an introdue the 0-ells by labelingthe regions
of the plane in whih the iruit is loated (where the input and output wires are
on-sidered to extend to the edge of the planar area ontaining the iruit). Wires are then
viewed as 1-ells, mapping from the region on the left to the region on the right. The
iruit equivalenes are unhanged. Thus, all of the development below, onerning
pla-nar iruitsand linearly distributiveategories, applieswithouthangetoplanar iruits
with typed regions and linear biategories.
3. Ciruit diagrams
non-two-sided nets. Pitorially the proof strutures are presented as graphs in whih edges
(wires)representformulas,and nodes(omponents,links,orboxes)representlogialrules
and additionalaxioms. The non-ommutative nature of the logi leads to the restrition
to planar proof strutures: that is, wires are not allowed to ross. To dierentiate from
the usual proof strutures (as in [G87℄) we shall refer to our strutures suggestively as
iruit diagrams.
Similar proof nets (without units ornon-logialaxioms) were developed in [A91℄.
Although in this paper we shall primarily work with the graphial representation
of iruit diagrams, it should be kept in mind that there is an underlying formalism
supporting the proof strutures. In partiular the iruits are more than just graphs;
there is a spei order to the omponents in a giveniruit, althoughwe willwork up to
equivalene moduloexhanging the order ofnon-interating subiruits (see below). The
underlyingformalsyntax (whih ats asa term alulus)is explainedin [BCST ℄.
3.1. Formulasand omponents. Beginwithasetof atomiformulasAand
om-ponents C. From the atomi formulas build the positive linear formulas indutively
as follows:
A2A is aformula;
> and ? are formulas;
if A and B are formulas, then AB and AB are formulas.
The typesof the wires ina iruit are taken fromthese formulas.
Eah omponent f 2 C is equipped with two lists of types, one for the input ports
and one for the output ports. We represent the omponent in a iruit as a box with
wires of the orret types attahed tothe ports:
f
Logially, the omponents in C orrespond to non-logialaxioms suh that the input
formulas entail the output formulas; ategorially, to morphisms from the tensor of the
inputformulastotheparoftheoutputformulas(theformulasaretheategorialobjets).
Therearealsopolymorphiomponents,orrespondingtologialrules;alltheselinks
to distinguish them from the additional omponents in C. The links are represented in
Figure2.
Note that the tensor-elimination and par-introdution links are drawn with a small
ar; this is a reminder that the links are swithing in the sense of Girard [G87℄, and
the net riterion of Danos-Regnier [DR89℄ (disussed below). Also note the thinning
links usedfor top-eliminationandbottom-introdution. Formallythe dotted lineshould
j
>
(>E)
L >
j
>
(>E)
R
>
j
? (?E) j
?
(?I) R
? j
?
?
(?I) L
j
(E)
A B
AB j
AB
(E)
A B
j
(I)
AB
A B
j
(I)
AB
A B
j
>
(>I)
Figure2: Links
of the speial role of thinning links in the proof-net equivalene introdued later in the
paper.
3.2. Juxtaposition and exhange. Components (and links) with orretly typed
wires as inputs and outputs are the simplest iruits. Larger iruits are built
indu-tively by a proess of juxtaposition. Simply attah (some of) the output wires of one
omponent/subiruit to (some of) the input wires of another, in suh a way that the
types of the wires math. The extra input and output wires extend past the other
om-ponent/subiruit to beome additionalinputs oroutputs of the new juxtaposed iruit.
The operation isobvious ina piture:
It is permissible to juxtapose without any attahments; then for deniteness, one needs
tospeify the order of the inputs and of the outputs of the juxtaposed iruit.
It isatthis pointthatthe non-ommutativenatureofthe systemisintrodued,inthe
requirement that juxtaposition be planar; simply, wires are not allowed to ross. Also
note that iruit juxtaposition isan assoiativeoperation.
We also allow the operation of iruit exhange 1
; two non-interating omponents
1
(and by generalization subiruits) an oat past eah other. For example:
=) f
h
f
h g
g
Ciruits are onsidered equivalent up to juxtaposition reassoiation and exhange of
non-interating subiruits. This equivaleneis deidable.
3.3. Rules of surgery. The ore onern of this paper is working with rewrite
sys-tems on iruits. Rewrites are speied asredex/redut pairs, whih are alled rules of
surgery. When applying a rule of surgery one must be able to ollet the omponents
of the redex by iruit exhange. It is at this point that the order of omponents an
have reperussions. As an example, onsider the following possible rule of surgery (a
non-planar ut):
=)
A B
C D
A
B
C D
In the followingiruit, the rule annot be applied to omponentsf and g, sine the
loationofomponenthpreventstheolletionoftheredex, even thoughthe appropriate
graph-adjaenies are there:
f
h
g
A more omplete disussion of iruitsan befound in[BCST ℄.
4. Sequentialization
Theiruitsdesribedsofararetheunderlyingproofstruturesforournaturaldedution
system. Onlyertainiruitdiagramsatuallyrepresent proofs. These iruitsare alled
proof nets. We hek the orretness of airuit by a proess of sequentialization.
EXPANSION: A
=)
A`A
REDUCTIONS: > j =) `> A j > =)
>;A`A
A > j =)
A;>`A
? j =) ?` A ? j =)
A`?;A
A j ? =)
A`A;?
B A j =) B A
A;B`AB
B A j =) B A
AB `A;B
f
=)
`
foreah additional
omponent f
2 1
1
;A;B;
2 ` AB
j
=)
1
;AB;
2 ` AB
1 2 2 1 ` 1
;A;B;
2
AB j
=)
`
1
;AB;
2
AB 1 2 Cut: A 1 ;A; 2 ` 3 1 `A 3 2 =) 1 ; ; 2 ` 3 1 3 2 A 1 ` 2 ;A; 3 1 3 2 A` =) 1 ` 2 ;; 3 2 1 3 J J 1 1 ` 2 ;A A; 1 ` 2 2 1 2 A =) 1 ; 1 ` 2 ; 2 1 1 2 2 1 ;A`
2 1 `A; 2 1 2 2 1 A =) 1 ; 1 ` 2 ; 2 1 1 2 2
Figure 3: Sequentialization rewrites
exhange and juxtaposition reassoiation) and replaed with a sequent box representing
the sequent in whih the input formulas of the redex entail the output formulas. If the
entire iruit an be olleted into a single sequent, the iruit is sequentializable and
thus aproof net.
Note that proof nets atually orrespond, not just to sequents, but to entire sequent
alulus proofs in the fragment of linear logi we are onsidering. One an easily
reon-strut a sequent alulus proof from the steps of the sequentializationproess.
4.1. Original rewrite system. Figure 3 portrays the rewrites given in [BCST ℄ for
the sequentialization proedure. Againnote the planar restritionon ut. We willuse X
and outputs of the boxes be multi-sets rather than lists. In partiular this removes any
restrition onthe ut rule.)
In the non-planar ase, this system was proved to be onuent on sequentializable
iruits. That is, if a iruit an be sequentialized, any order of sequentialization steps
willsueed; if not, there may be several dierent ways for the proess tobeomestuk.
It was thought that the proof of onuene transferred orretly to planar iruits. In
fat this is not the ase, as the example inFigure 4 shows. In this example, the redued
iruit on the right will sequentialize fully with the rules we have; but no rules apply to
the iruit on the left. Beause of the ompliations this introdues, partiularly with
respet tothe notion of iruit equivalene tobe disussed below, it seems preferable to
regard this apparently non-sequentializable iruitas a proof net. Thus we need to have
more sequentialization rules.
a
b
j
ut a . & ut b
j
j
Figure4: Sequentializationnot onuent
4.2. New rewritesystem. The problemevidentlyarisesdue tothenon-ommutative
restritiononut|workingwithnon-planarrules,weouldsimplyuttheoendingwires
in the above situation. It seems also that the assoiation of the inputs and outputs of
a box are more important in the absene of exhange: reall that from one perspetive
a box maps the tensor of its inputs to the par of its outputs, but how assoiated? This
helps tomotivate the rules wewilladd tothe system. Theserules an alsobeonsidered
asways topass toanequivalentiruitfromwhih weansequentializefurther usingthe
originalrules; this willbe madepreise in Lemma5.7.
The rules in Figure 5 are equations or E-moves; they may be performed in either
diretion. This inreases the omplexity of the rewrite system: a normal form an only
1 ;A;B;
2 ` 1
A B
2
=
1 2
1
;AB;
2 ` A
B
j
1 `
1
;A;B;
2
A B
2 =
2 `
1
;AB;
2
1
A B
j
Figure5: New sequentialization equations
A
B; ` B j
=)
AB; `A; AB
A
;A` A
B j
=)
;AB`;B
B
AB
A
`B;
B
j
=)
A; `AB; A
AB
A
`;A
B
j
=)
;B`;AB
B
AB
Figure6: New sequentialization redutions
These rulesmodelreassoiationby allowingaboxtoextend and retrattimesorplus
linksbetween any twoadjaent inputsor output. Wemarkthese plus and timeslinks to
distinguishthemfromthelinksoriginallyintheiruit. Inpartiularanoriginallinkmay
not be pulled into a box by an E-move, and a marked link (alled a reassoiation link)
may not be replaed with a box by one of the original R-moves 2
. However, note that
this distintion is tobeonsidered a temporarydevie used only inestablishing whether
airuitis sequentializable;wewillnot have twokindsof tensororpar linksinanyother
setting.
Naturallywewanttobeabletodosomethingwiththesereassoiationlinks;motivated
bythe problematiexampleabove,addthenew redutionrulesinFigure6(whih,again,
2
Thisdistintionmayseemunneessary. Afterall,theeetofapplyinganE-movetoanoriginallink
ouldbedupliatedbyusingaredutiontoreplaeitwithabox,andthenuttingtheinterveningwire;
and the eet of replaing a reassoiation link with abox ould be dupliated by expanding the wire
behindthereassoiationlink,then pullingit intothenew box withanE-move. The distintionshould
probablyberegardedasapurelytehnialdevietoensurethatthesystem'sredutions,expansions,and
equationshaveertainappropriatepropertieswhihimplyonuene.
Forsuppose that theE-movesapplied to theoriginal links. Thenthesystem would notbe redution
E-terminating,beauseasjustnoteditwouldbepossibletodupliateonediretionofanequationwitha
seriesofredutions. Onemightattempttoavoidthisproblembyforbiddingtheredutionruleswhihbox
non-swithingtensorandpar,sinethoserulesouldbedupliatedwithanexpansionandanequation.
However,withthishangethesystemwouldnotbeX-reduing,sineoneouldndadivergeneoftwo
redutionswhere anexpansionisrequiredin anyonvergene.
In pratie, keeping the two kinds of links apart is never an issue; the sensible way of starting a
apply only tothe reassoiation plus and times links,not the originalones).
This ertainly solves the problem example. In a loose sense, these rules allow us to
undo the wrong ut and redo it inthe right way:
j
=
j
j
=)
j
It will nowbe shown that these rules are enoughfor onuene.
4.3. Confluene. First observe that one a plus or times link is extended by one of
theequations,therewillalwaysbeaboxintowhihtoretratit(possibly afterretrating
other reassoiation linksthat were later extended), regardless of intervening appliations
of rules. It is thus easy to see that an E-equivalene lass of iruits ontains a unique
fully retratedform, inwhihallthereassoiationplus andtimeslinksarepulledbak
into boxes.
In terms of retrated forms, any appliation of the new redution rules will have the
form of Figure 7, or the orresponding operation for plus on the right, or times on the
left or right. Figure 8 shows the derivation of this transformation; note that the desired
iruit exhange will always be possible in a planar iruit. Call this derived rule jump,
and say that the wire of type A in Figure 8 (the jump wire) jumps along the wire of
typeB (thealong wire). The jumpwire must beadjaent tothe alongwire (asinputs or
outputsof a box) inany appliation of the jump rule.
Observethat the jumprule isanon-loaloperation;itannotbeformulatedinterms
ofolletingaredexandsubstitutingaredut,sineinterveningomponentsmayprevent
suhaolletion. Thus, thesystem isformulatedusingtheloalequationsandredutions
instead. Nonetheless, sine in proving the onuene of the system we have to work up
to E-equivalene, it is onvenient to refer to the retrated forms, and to what amounts
`
1
;A;B;
2
B;`
A B
1
2
=)
`
1
;AB;
2
AB;`A;
A
1
AB
`
1
;A;B;
2
B;`
A B
1 2
=
2
B `
1
;AB;
2
B;` A
1
j
.
(iruit exhange) .
2
B `
1
;AB;
2
B;`
A
1
j
=)
`
1
;AB;
2
AB;`A;
A
1
AB
2
Figure 8: Derivation of the jumprule
toappliationsof these non-loalrules. (The non-loalharater of the rulemay make it
seem that the rule is non-planar, but we are indeed stillrequiring planarity.)
Nextobservethatthe rewritesystem anbesimpliedbyremovingmostkindsofut.
Allow only the most simple uts, where there are no inputs or outputs adjaent to the
ut wire:
`A
A`
A
=)
`
The full power of non-ommutativeut an bereprodued, usinga series of jumps along
the ut wire to remove any inputs or outputs adjaent to it, followed by an appliation
of the restrited ut. It is onvenient, in proving onuene, to leave the full ut out of
the system. However, as wehave any non-ommutative ut asaderived rule,we willnot
hesitate touse the full form when needed.
Weare now ready toprove that wean hek the sequentializabilityof airuit inan
eetive way.
4.4. Theorem. Given any sequentializable iruit, one an rewrite in any order to
ahieve sequentialization, with a guaranteed bound on the number of redutions and
sequential-Proof. First note that the theorem allows that given a non-sequentializable iruit, it is
possiblethattherearemultiplewaysfortheproesstoend(noneofthemsinglesequents).
FollowingAppendix A, forthe rst statementit suÆes toshow:
R is E-terminating;
X [R is loally E-onuent and X-reduing; and
X/R is expansionE-terminating;
where we are onsidering the rewritesystem onlyon sequentializableiruits.
R is E-terminating. Assign to eah sequent box a height and a depth. The height
ofa box isthe maximum,underiruitexhange, of thenumberofboxesthat omeafter
it in the iruit; depth is the maximum number that an ome before it. Roughly, the
heightof a box B is the greatest number of boxes we an manage to draw belowB in a
graphial representation of a iruit; similarly, the depth is the greatest number we an
drawabove B.
It is easy tosee that the jumprule an only derease the heightor depth of any box.
Any ordering on the boxes after the jump ould be reprodued in the iruit before the
jump, e.g.:
after jump before jump
If we an iruit-exhange n boxes past B after the jump, we an ertainly do the same
(or more) before the jump.
Moreover, onsidering the iruits infully retrated form,a par jump (a jump
down-ward) sends an output wire from a box of greater height to a box of stritly less height,
sine the along wire keeps the ending box after the starting box in any onguration.
Similarlyatensorjump(ajumpupward) sendsaninputwirefromabox ofgreaterdepth
toa box of stritly less depth.
Thus, anappliation of the jumprule always redues the quantity
X
boxes
[depthnumber of inputs+heightnumber ofoutputs ℄
where inputsand outputs are ounted as for the fully retrated formof eah iruit.
Nowobserve that any redution redues the lexiographi order on
boxes
[depthnumberof inputs+heightnumberof outputs ℄,
using fully retrated values.
The third value is redued by any jump (whih leaves the rst two values unhanged);
the seond value is redued by any ut (whihleaves the rst value unhanged); the rst
value is redued by any other redution.
Moreoveritis lear(sine allrelevantvalues are ounted infullyretrated form)that
thisquantity isnothangedbyE-moves. Itfollowsthattheredutions areE-terminating.
Wewillusethisvalueagaintoobtainonuene;refertotheabovetripleastheiruit
value.
Theexpansion. TehniallyX-reduing andexpansionE-terminating onlymakesense
after loal onuene is established. However it willsimplify the presentation toget the
loneexpansionout ofthewayfromthe beginning;wean dothisby notingthefollowing.
In a sequentializable iruit ontaining any links or omponents, the expansion is
always reduible. For, if the expanded wire is next to a box, the expansion box an be
ut into that box; if itnext to a non-swithing link, the link an be boxed and then the
intervening wire an be ut. If the expansion is next to a swithing link, we know by
sequentializabilitythat theswithinglink willeventuallybeabsorbed intoabox,and the
expansionbox an beut atthat time. It iseasyto see thatin eah ase the sameresult
ould be obtained using only redutions. (To be preise, this shows only that the last
expansion of the sequentialization proess is reduible; an immediate indution on the
number of expansions ompletes the argument.)
Thus the expansion is only irreduible when applied to a iruit whih is a single
wire (and at that point, sequentialization is omplete). So if we an prove that R is
loallyE-onuent, then itwillfollowimmediatelythat X [R isloallyE-onuentand
X-reduing,and that X=R isexpansion E-terminating.
R is loally E-onuent. To deal with the E-moves in determining possible
diver-genes, simplytreatalliruitsasifinfullyretratedform,andallowjumpasanon-loal
move.
Conuene is proved by indution on the iruit value dened above. Clearly
onu-eneholdsonsolesequentboxes(iruitvalue(0;1;0)). Assumethat onueneholdson
allsequentializableiruitsofvalueless than ;nowtakeairuitofvalue andonsider
the possibledivergenes.
Firstobservethatboxingalink/omponentanneverleadtoadivergene. Thisleaves
1. absorbing a swithing link;
2. (restrited) ut;
3. (non-loal) jump.
appli-a graph), it isalways possible to dothe iruit exhanges neessary to ollet the redex.
(For the jump rule equations will also be neessary.) It follows that a divergene an
our only between two redexes sharing boxes or wires. This makes listing the ritial
divergenes muh easier.
Now we proeed by exhibiting examples of the various ritial divergenes whih an
arise,and howtoresolvethem. Theeasyases,wherethetwomovesanbedone,though
not simultaneously, in either order for the same result, are omitted. Also note that a
onvergene an be onsidered valid as long as the ongurations of boxes and wires are
thesame,even ifsomeofthetypesoftheinternalwiresaredierent(perhapsA(BC)
ratherthan (AB)C). Sinethe iruit isassumed tobesequentializable,onuene
up to hanging the typesof internalwires doesnot reallygive anything away, sine suh
wires willeventually be ut (thus givingonuene ina stritersense as well).
For most ases, nding a onvergene willnot require the indution hypothesis.
Fig-ures 9{13 portray onvergenes for allbut one of the non-trivialritial divergenes.
TheonediÆultaseiswhenthereareompetingjumpstothesamebox. Anexample
of suh adivergene would resemblethe following:
C
C
B A
(= =)
C C
C
A
B
C
C A
B
D
Given a divergene of this sort, all the two possible iruits so obtained C and D.
We may assume that one of C or D is sequentializable. (If not, there would be another
ritial divergene between C and the rst step of the \orret" sequentialization, the
resolution of whihwould show C tobe sequentializable.)
Withoutlossofgenerality,suppose C issequentializable. SineC isaredutionofthe
original iruit, it has a lower iruit value. Then, by the indution hypothesis, we an
sequentialize inany order onC.
OurstrategywillbetofollowasequentializationofC,mimikingeahstepalsoonthe
iruitD,untilaonvergeneours. Thismaygosofarasshowingthereisaonvergene
by showing that both iruits an be sequentialized all the way to single sequent boxes!
Wemust ensure that the two iruits remain similarenough that sequentializationsteps
ononeanbesuessfullymimikedonthe other. Infat,weshallmaintainthefollowing
three invariants:
C B A
j
)
AB C
)
(AB)C
AB
+
B BC A
j
)
A B
A(BC)
j
)
A(BC)
AB
Figure 9: Swithing link absorption broken by jump
A
B
C )
A
B
)
AB
A
+
AB
C A
)
AB
A AC
)
AB
A
C
C
C B
A C
)
C
C
C
B A BC
)
C
C
C
A(BC)
B BC
+
C
C
C
C AB
B
)
C
C
C
(AB)C
B
C
)
C
C
C
(AB)C
BC
B
Figure 11: Twojumps, same jumpwire
A
C B
)
C AB
A
)
(AB)C
A AB
+
A
B BC
)
A(BC)
A
B )
A(BC)
A AB
B
B
A B
)
B
B
A AB
+
+
+
(utA)
B
B B AB
))
(utB)
AB
Figure13: Mutual jump
stak (whih onnet to the non-divergent parts of the iruits) side wires. (Note
that the inputs atthe top of the stak and the outputs at the bottom of the stak
arenot divergent. Alsonotethatthe numberofboxesinthestak isnot invariant.)
2. Call two wires adjaent if they are onseutive inputs or outputs of a box. Then
we maintain that two side wires adjaent in one iruit are adjaent in the other,
andthat aside wireadjaenttoaentralwire inoneiruit isadjaenttoaentral
wire in the other; and further, that the left-to-right ordering of the adjaenies is
the same in both iruits. (Here weare referringto the side wire of C and the side
wireof D,whihonnettothe non-divergentpartsof theiruitsinthe sameway,
as if they were the same wire.)
3. View the side wires as if olleted into sets, by grouping the adjaent inputs or
outputs of eah box. We will have distint sets on the left and right sides of the
entralwires. We maintain that the top-to-bottom orderingof the sets ona given
side of the entralwires will bethe same inboth iruits.
Observethatatthebeginningofthisproessthetwoiruitsareunlikelytosatisfythe
adjaenyonditionoftheseondinvariant;aseriesofjumpsofthesidewiresisneessary.
ReturningtothepartiularCandDportrayedintheexampledivergeneabove,wewould
need tomakejumps asfollows (where only the divergent stak isportrayed):
C =) D =)
per-Beause sequentializationisonuent onC we an add these jumps if neessary without
losing sequentializability.
To motivate that there should be a onvergene, observe that the divergent staks of
the two iruits dier essentially only the relative position of the side wires to the left
of the entralwires, with respet to those tothe right of the entralwires. But beause
of planarity, there is no interation aross the entral wires. In partiular, any rewrite
depends on (at most) two adjaent wires, so as long as the adjaenies are maintained,
we willontinue to be able tomimi sequentialization steps inboth iruits.
If the rst invariant above an be maintained throughout a sequentialization of C,
theneventuallyaonvergene willbereahed. For,atsomepointinthe sequentialization
ofC,allthe entralwiresofthedivergentstakwillbeut. Thismeansthattherewillbe
nosidewiresinC,and henenoneinDastheside wiresallonnettothenon-divergent
part of eah iruit. Thus all the entral wires of D an be ut, and the non-divergent
part willbeall of the iruit.
It remainsonly tosee that asequentializationof C an always proeed (and be
mim-iked on D) in a way whih preserves these invariants. Here again it helps to impose a
restrited ut. There are several ases:
Sequentialization steps applied in the non-divergent part (these inlude all
redu-tionswhihonlyputaboxaroundlinks)anbemimikeddiretly,anddonotaet
the invariants.
Absorbingaswithinglinkintoaboxofthedivergentstakanbemimikeddiretly
(sinethetwowiresleadingtothe swithing linkmust beadjaentinbothiruits),
and does not aet the invariants.
Jumpingasidewirealonganothersidewireanbemimikeddiretly(byonsidering
adjaeny) and doesnot aet the invariants.
Jumping a side wire along a entral wire, or reating a new side wire by jumping
a wire intothe divergent stak, an always bemimiked diretly (by the adjaeny
ondition and also the third invariant onerning the top-to-bottom order of sets
of side wires). However, in order to maintain the invariants it may be neessary
to further jump the wire in one of the iruits. In partiular, if in one iruit the
wire is jumped into an established set of side wires, it must also reah that set in
the other; and if in one iruit the wire is jumped out of the divergent setion, it
mustbejumped outinthe other. Thatthisisalwayspossiblefollowsfromthethird
invariant.
An example of what this might look like, at some stage of attempting to nd a
onvergene for somepartiular C 0
and D 0
, isgiven by Figure 14. The step leading
C 0
=)
A
D 0
=)
A
=) A
Figure 14: Maintaining adjaenies
C 0
=)
1
y
1 z
1
1
2 x
2 2
2
=)
x 1
2
y
1 1
1
z
2 2
2
D 0
=)
1
y
1 1
2
1
z 2
2 x
2
=)
2
2
1 1
1
1
2
2
z y
x
Figure15: Jumping aentralwire
Cuts of entralwires may not always be possible to mimi. However, not
mimik-ing these uts doesn't aet the invariants. It is best to simply ut entral wires
whenever possible in eitheriruit.
Finally, onsider jumps of the entral wires along side wires. These jumps an be
mimiked diretly,but thedivergent stak willgrowby onelevelafterthe jump|it
will now inlude the box jumped to, and the along wire of the jump as a entral
wire. An example of what this might look like, at some stage of attempting to
nd a onvergene for some partiular C 0
and D 0
, is given by Figure15. Only the
divergentstaks are portrayed.
This ompletes the proof of onuene.
There remainsthe nalstatementof the theorem,i.e., thatwe anbound thenumber
number of ways to fully determine the assoiations of the inputs and outputs of eah
box. It follows that wean hek whethera redutionan beappliedtoany E-equivalent
form, with a bound on the number of equations it is neessary to apply. The proof that
R is E-terminating applies to all iruits (not just sequentializable ones); thus we an
alwaysreahairuit,wherenoredutionsan beappliedtoany E-equivalentform,with
a bound onthe numberof moves needed. The result follows.
4.5. The net riterion. A typial riterion for a iruit to be a proof net follows
along the lines of the following (see [DR89℄):
4.6. Definition. Regardeahswithingomponentinairuitashavingaswithwhih
hoosesbetweenthenon-tensoredports: theinputsofthe(I)or theoutputs ofthe(E).
One port is onsidered to be onneted, the other disonneted. Then the iruit satises
the net riterion if for any hoie of swith settings, the undireted graph determined
by the onneted wires is ayli and onneted.
It is proved in [BCST ℄ that, for non-planar iruits, sequentializability is equivalent
to satisfyingthe net riterion. However the net riterion is not a suÆient ondition for
a planariruit tobesequentializable,as the followingexample shows.
j
(Theexamplegivenin[BCST℄tosupportthislaimturnsouttobesequentializablegiven
the modiations ofthis paper.) In fat,aplanar iruitissequentializableif and onlyit
it satisesthe ommutativenet riterionand alsothe following region riterion:
4.7. Definition. Aplanariruitdividestheplanarareaontainingitintoregions;that
is, if the iruit is viewed as a graph embedded in a planar area, the edges of the graph
(and the border of the area ontaining it) divide thatarea into regionsin the usual sense.
Calla region losedif it issurroundedbythe iruit, and open if itreahes theborder
of the area ontaining the iruit. Further saythat aswithing link opens onto the region
between its non-tensored ports: the region above andbetween the inputs of a (I), or the
region below and between the outputs of a (E).
Then a iruit satises the region riterion if every swithing link opens onto a
losed region, and every losed region has one and only one swithing link whih opens
onto it.
5. Ciruit equivalene
Now we introdue rules of surgery whih dene an equivalene relation on proof nets.
Proof nets modulo this iruit equivalene will dene a (non-symmetri) linearly
dis-tributive ategory|in fat, they will dene the free suh ategory on the polygraph of
omponents on whih the iruits are based. Equivalent proof nets will thus orrespond
toa singlemorphismof a non-symmetrilinearly distributiveategory,and by extension
toa single proof innon-ommutativemLL withoutnegation.
5.1. Rules of surgery. First we onsider the originalset of iruit-equivalene rules
of surgery given in[BCST ℄. There is animportantassumption underlying these rules: a
rule of surgery an only be applied to a proof net if it preserves sequentializability. This
is essentially an element of non-loalness in the apparently loal rules. However, this
non-loalness an be isolated into a very few ases: namely, the equations whih rewire
thinning links past swithing omponents. It is immediate to hek that all the other
rules automatially preserve sequentializability.
The redutions are listed in Figure 16; the expansions (whih an be thought of as
\expressing the type of the wire") are listed in Figure17.
The equationrulesdealwith themanipulationofthinninglinks. Thismanipulationis
alled rewiring; it is the ore of this approah to oherene. To motivate this, note that
the thinninglinksrepresent wherea unithas beenintroduedintothe proofby thinning.
There are typially many steps in a proof at whih this might our, without (so one
would desire) essentially hanging the proof.
The rst set of equation rules (Figure 18) deals with rewiring past omponents and
all non-swithing links (tensor-introdution, otensor-elimination and all the unit links).
These are alled \box rewiring" rules. Note the dierene in behavior between the top
and the bottom: onlythe top an berewiredaround the top ofa omponent,and only a
bottom around the bottom of aomponent.
Finally there is a set of equation rules (Figure19) dealingwith rewiring past
swith-ing links. Note the important fat that neither unit an be rewired diretly aross the
\opening"ofa swithing link(thatis,between the link'snon-tensored ports). Alsoreall
thattheseare theonlyrulesthatrequireone tohek thatsequentializabilityispreserved
B A
j
j
=)
A B A B
j
j
=)
A B
A
j
> >
j
=)
A A
j
> >
j =)
A A
j
? ?
j
=)
A A
j
? ?
j =)
AB
=)
B A
j
j
AB
=)
B A
j
j
>
=)
> j j
> >
=)
> j j
>
?
=)
? j
j
? ?
=)
? j
j
?
Figure17: Expansions
j
>
=
j
>
j
>
=
j
>
j
>
=
j
>
j
>
=
j
>
j
?
=
j
?
j
?
=
j
?
j
? =
j
?
j
?
=
j
?
j >
j
=
j
>
j
j
>
j
=
j
>
j
j
j
>
=
j
j
>
j
j
>
=
j
j
>
j ?
j
=
j
?
j
j
? j
=
j
?
j
j
? j
=
j
? j
j
? j
=
j
? j
Figure19: Equations for rewiringpast swithing links
by the rewrite.
(Additionalrewirings must beaddedto the systemfor the ommutative ase. In
par-tiular,in non-planarnets, tops an be rewiredaroundthe bottomedges of omponents,
and bottoms around top edges; furthermore there are rules to reverse the left/right
ori-entation of a thinninglink. The result is that possible rewirings in non-planar nets are
not restritedbythe handednessof the thinninglinkorthetypeof unit involved; infat,
rewiring is not even restrited by the order in whih several thinning links are attahed
toa single wire. The situationis quitedierentin the non-ommutativease.)
5.2. Planar modifiations: negation links. The rewritesystem from[BCST℄
de-veloped aboveisnot onuent onplanar proofnets, asthe authorswere aware. Consider
the seriesof iruits,equivalentmodulo rewiring,portrayed inFigure20. It iseasytosee
that if the obvious redution were applied to eliminate the \oating unit barbell" from
the rst or lastiruit inthe series, no further rewrites would apply tothe iruit.
In the ommutative senario these two iruits would in fat be equivalent. But as
planar iruits there is nothing more to be done with them. This kind of undesirable
behavior seems to be a diret result of the fat that, inthe planar ase, ertain rewiring
movesan bemade by tops but not by bottomsand vieversa.
In [BCST ℄ this problem was avoided by admitting the unit redution rules as
j ? j ? j j ? j ? ? j > j > j = j j ? ? j ? j ? j > j > j ? j = j ? j ? j j ? j ? > j > j ? j . j ? j ? j ? ? j ? > j > j j j = j ? j ? j ? ? j ? > j > j j j . > j j j ? j ? j ? j ? j ? > j = > j j j ? j ? j ? j ? j ? > j = j j > j > j ? j ? j ? j ? j ?
>
j =
>
: j
j ?
>
j =
>
: j
j ?
? j
=
?
:
j j
>
? j
=
?
:
j j
>
Figure21: Negationlink equationrules
>
=
?
> >
:
j
j :
>
=
>
>
?
: j
j :
?
=
>
? ?
:
j
j :
?
=
?
>
?
: j
j :
Figure 22: Negationlink equation rules (doublenegation)
unsatisfying solution, beause the possibility of an arbitrary number of oating barbells
makes adeision proedurefor oherene seem unlikely.
A more manageable means of interhanging the units is reated by introduing a
metalogial link that an only be introdued in a ontrolled way. This link should, in a
ertainsense, beviewed asanabbreviationforaunitbarbell(thisismademorepreisein
Lemma5.6below). Duetothe mannerinwhihthislinkinterhangesthe dualunits,all
thelinkanegation link. AddtheequationsinFigure21totherewritesystemtodealwith
it. Bymeans of these equations, the top and bottom linkshave idential mobility. This
resolves problems like the non-onuent example above. For onveniene, add further
equations, listed inFigure22, toadd or remove doublenegations.
Refer to a series of negationlinks ending in a thinning link as a negation link hain.
A negation link hain forms a kind of extended thinning link whih is more exible. It
is onvenient to restrit the negation links in a iruit to those ourring in negation
link hains. The equations for negationlinksnext tothinning linkslearly maintain this
restrition; the double negation rules an be appropriately restrited by requiring that
they only be appliedwithin a negationlink hain. 3
By imagininga trip in the iruit along a negationlink hain, we an assign a
hand-3
Intermsoftheredex/redutpresentation,thisanbedonebyreplaingeahoftherulesinFigure22
with six rules, one for eah of the various ways that either anegation link or a thinning link an be
attahed,at eithertheinputortheoutput,tobothredexandredut. Theseare: anegationlinkat the
edness to eah link in the hain (inluding its nal thinninglink), aording to whether
the trip makes a left or right turn at the link. Using the equations, we an remove any
negationlinks whih failto reverse the handedness of the next onnetion. The negation
links leftafter suhan operationwillbethose that produe a winding in a negationlink
hain. A negation link hain is said to be tightly wound when all the negation links
reverse the handedness of the next onnetion in the hain.
Consider a negation link hain ending in a top-introdution or bottom-elimination
link,whih islike aunit barbellthat somehowmanagesto get wound up, e.g.:
: j
? j :
j
: j >
j
In keeping with the idea that negationlink hains should behave likehains of unit
bar-bells,weshouldbeabletoreduesuhstrutures. Tothisend,extendtheunitredutions
to innite series of redutions to remove any suh hains. Note that we need series for
top and bottom units, left and right side of wire, and lokwise and ounterlokwise
winding,but due tothe equations we may assume that the hain is tightlywound.
Observethatnomatterhowsuhan\extendedbarbell"iswoundaroundother
ompo-nentsintheiruit,oneanalwaysuseiruitexhangetoolletitintoasinglewinding,
beause it is only attahed to the rest of the iruit at the end. Thus the redex/redut
presentation for the rules is retained.
Forthepurposesofsequentializationandrewiring,thenegationlinkisanon-swithing
link. Add thefollowingrules tothe sequentializationrewrite system (learlythat rewrite
system's importantpropertiesare unhanged):
?
> :
j
=)
`>;?
> ?
: j
=)
`?;>
?
>
: j
=)
>;?`
> ?
: j
=)
?;>`
This expands our notion of proof net to inlude negation links. Again, it is onvenient
torestrit proof nets tobe onlythose sequentializableiruits inwhih allnegationlinks
our within negationlink hains.
5.3. Sequent box rewiring. Next it is proven that the \box rewiring" rules an
be extended so that the boxes an represent, not just omponents and non-swithing
links,butsequent boxes whihan beolleted duringthesequentializationproess. This
5.4. Proposition. Theboxrewiringrulesapplytosequentboxes,in thefollowingsense.
Consider a iruit C, and suppose a iruit D an be reahed from C by applying the
sequentialization rules. If a thinning link an be rewired between two positions in D, by
allowingthe box rewiring rulesto applyto thesequentboxes,then thethinning linkanbe
rewired between the orresponding positions 4
in the original iruit C.
Proof. ItsuÆestohekthatforeahsequentializationrule,allrewirings(inludingbox
rewirings aroundsequent boxes) whih are possibleafter the rule are alsopossible before
therule. Thisisimmediatefortherulesthatboxnon-swithinglinks,andstraightforward
for most of the other rules. (It is helpful to heavily restrit the ut rule as was done in
the proof that sequentializationis onuent.)
The only triky ase is a jump when the box involved has either no inputs or no
outputs, for example:
j >
j
=
j
j
>
=
j
>
j
=
j
> j
In[BCST℄ 5
this washandled by addinga oatingbarbell;insteadnegationlinksan now
be used as follows:
j >
j
=
j
?
j
: j
=
j j
? j
:
=
j j
? j
:
=
j
> j
This ompletes the proof.
4
Making preise the required notion of orrespondene is not diÆult: for eah possible position
for a thinning link to onnet to eah sequentialization redut, assign an appropriate position in the
orrespondingredex.Thehoieisobviousinallasesexeptperhapsfortheexpansionandtheequations,
wherenewwires arereated. For theexpansionboththeinputwireandtheoutputwireorrespondto
theoriginalunexpandedwire;fortheequations,theleftsideandtherightsideofthenewwireorrespond
to theleft side of theleftmost, andthe right sideof therightmost,of thetwodistint wires whihare
involvedintheequation.
5.5. Proofnetsaslinearly distributiveategories. Planarproofnetswithone
inputandone outputan beregardedasaategory,whereeahsequentializableiruitis
amorphismfromthetypeoftheinputwiretothetypeoftheoutputwire. 6
Compositionis
simplyiruitjuxtaposition,whihasnotedisanassoiativeoperation;sequentializability
is preserved under juxtaposition sine if eah subiruit an be sequentialized, then the
two sequent boxes so obtained an be ut together to sequentialize the full iruit. The
identities are the single wires.
Now,equivaleneofiruitsundertherewritesystemdenedaboveformsaongruene
on the ategory desribed so far. Denote by PNet (C) the ategory of planar proof nets
based onthe set of omponents C, quotiented by iruit equivalene.
It is nowshown that the new equivalenes using the negationlinksdonot ollapse or
reate equivalenelasses of nets.
5.6. Lemma. Any netinvolving anegation linkisequivalentto anetnotinvolving
nega-tion. Furthermoretwo nets, whih donot involveany negation linksand whih are
equiv-alent under the above rules of surgery (inluding the negation rules), are also equivalent
undertheoriginalrules(notinludingthenegationrules,butinludingtheunitredutions
as bidiretional equations).
Proof. Consider the following translation:
? >
: j
=)
>
j
? j
> ?
: j
=)
> j
j
?
?
>
: j
=)
? j
>
j
> ?
: j
=)
? j
j
>
By this translation all the negation links an be removed from a iruit. In fat,
beause negation links are always adjaent to other negation links or to thinning links,
eah negation link will be replaed by a thinning link attahed to a unit barbell. This
new iruit is equivalent to the original: for every link attahed to a unit barbell in the
new iruit, where the original iruit has a negation link, simply introdue a negation,
movethe new dual unit o the barbell, and then eliminatethe barbell.
Furthermore, referring to the translation, it is easily seen that the equations whih
introdue and eliminate negation links an be simulated by introduing and eliminating
unit barbells; and the negation link hain redutions an be simulated by a series of
barbell eliminations. Thus it is possible to show the equivalene of two equivalent nets,
whihlak negation links,withoutusing the negationrules.
6
We ould view proof nets with multiple inputs or outputs as the morphisms of a polyategory as
dened in [CS91℄. Howeverthat paper alsoshowedthat no informationwill begained bydoing so. In
fat,simplyonsideringtheinputsto betensoredtogetherand theoutputsparredtogether resultsin a
We have also hanged the sequentialization system; by adding the jump rule, more
iruits are proofnets. However, the followinglemma shows that the jump ruledoesnot
add equivalene lasses of nets.
5.7. Lemma. Any proof netisequivalent to anet whihis sequentializable withoutusing
the jump rule.
Proof. Itisnothardtoshowthat itsuÆes toestablishthefollowingresult: thatifanet
C sequentializes to a net D in one appliation of the jumprule, then C is equivalent to
a net C 0
,whih sequentializes to Dwithout usingthe jumprule. Suppose that the jump
is a par jump (a jump downward). Then obtain C 0
from C by running a par redution
bakwards betweenthejumpwireandthealongwire;thatis,jointhejumpwireandalong
wire with a par-introdution link, followed immediately by a par-eliminationlink. Note
that C 0
is sequentializable if C is. Now C an beobtained fromC 0
by one appliation of
apar redution. It islear howtoemulatethe eetof the jumponC 0
withoutusingthe
jumprule. The ase of atensor jumpuses a(bakwards) tensor redution similarly.
Given these fats, the results of [BCST ℄ willarry through, namely
5.8. Theorem. PNet (C) is the free non-symmetri linearly distributive ategory
gener-ated by the polygraph of omponents C.
Refer to[BCST℄ forthe proof.
6. Coherene
We now ome to the main result, whih addresses the equivalene of morphisms in
non-symmetri linearly distributive ategories, and by extension proofs in non-ommutative
negation-freemLL. Theorem5.8abovestates thatsuh aategory,free onapolygraphof
morphisms/omponentsC,isjustPNet(C),planarproofnetsundertheiruitequivalene
dened in the previous setion. To approah the problem of oherene, it is rst proved
thatnormalformsoftheiruitequivalenerewritesystemanbeahieved inaneetive
way.
6.1. Theorem. EahequivalenelassofiruitsinPNet(C)ontainsauniqueexpanded
normal form up to E-equivalene; one an reah the normal form of a iruitby applying
rewrites,inanyorder,withaguaranteedboundonthenumberofredutionsandirreduible
expansions. Furthermore there is an proedure to exeute the normalizationin a bounded
number of moves inluding equations.
Proof. Thisproof follows the patternofthe ohereneresultfor non-planarproofnets in
[BCST ℄. Following Appendix A, for the rst statement it suÆes to show
R is E-terminating;
R isE-terminating. Thisholdsbeauseeveryredutiondereases thenumberoflinks
other than negation links in the iruit, whereas the equations hange only the number
of negationlinks.
X [R is loally E-onuent and X-reduing. Consider the following notion:
6.2. Definition. The skeleton of a net is the graph obtained from the net by: rst,
disonneting all thinning links; and seond, replaing eah negation link hain with a
single wire, by removingeah negation link in turn from the thinning-link end of the now
disonneted hain, and reversing the unit type of the (disonneted) thinning link with
eah suh removal.
Clearly anytwoE-equivalentnetshavethe sameskeleton. Redutionsand expansions
an be applied to the skeleton of a net; for any suh move on the skeleton, dene a
transformationsk[℄ontheoriginalnet,whihleadstoanetwiththereduedorexpanded
skeleton, but alsomoves the thinning links in an appropriate way. This means that any
thinning links, whih prevent a diret appliation of the skeletal move to the original
net, need to be rewiredout of the way rst.
Askeletalexpansionontheunitwireofathinninglinkmayleadtoambiguity,beause
the thinning link ould orrespond to a negation link hain in the original net, and the
orresponding sk[℄ move ould thus be an expansion on any of several dierent wires.
However, itiseasytoseethatany expansioninanegationlinkhainanbefollowedby a
sequene ofredutionsandequationsthatreturnthenettoitsstatebeforetheexpansion.
Finding a onvergene for a divergene involving suh an expansion is thus trivial. We
an therefore safely negletsk[℄ forthese expansions.
Dening sk[℄ for other expansions is easy: just put all the thinning links on the
expanded wire diretly belowthe expansion, for instane:
?
j
>
sk[℄
!
j
?
j
?
j
>
(Note that there is some arbitrary hoie in the way sk[℄ is dened; we ould just as
easily put all the thinning links above the expansion. It will not matter exatlyhow we
doit as long asit isdenite.)
Forthetensorandotensorredutions,simplyrewireanythinninglinksontheredued
wire out of the way (whether up or down, they end up in the same plae after the
redution):
B A
j
>
j
j
sk[℄
!
A B
j
The remaining ases are the unit redutions. Here, a simple unit redution on the
skeletonmay orrespond toan ompliatednegationlink hain redution on the original
net. All the thinning links whih are attahed to the hain must be pushed o of it.
This needs to be done in a denite way so that we an show (shortly) that sk[℄ moves
ommute with rewirings. Dene sk[℄ sothat eahthinning link pushed o \opies" the
negationlinkstruture of the hain tobe redued. Applythe followingtwo steps toeah
link attahed to the hain, starting with the one losest to the thinning-link end of the
hain:
1. Ensure that the type of the unit of the attahed thinning link mathes the type of
thewiretowhihitisattahed;ifahangeisrequired,thismeansaddinganegation
link atthe end of the attahed thinning link.
2. Rewire the attahed thinning link past the next negation link of the hain. This
may mean adding another negation link to the attahed link, in order to give it
enough mobility. (If this is the ase, the types will already math, as required by
the rst step, during the next stage.)
Repeat these steps until allthinninglinks are rewired o of the hain. An example:
j >
?
j
: ?
j j
=)
> j
j
: j >
j
?
:
j
=)
j
:
:
j j
? >
j
: j
j ?
=)
: j :
j ?
j
> j
: j
? j
Next itisneessary toshow that,with appropriaterewirings, wean obtainthe same
resultfromaskeletalredutionorexpansioneven afterrewiringonthe originalnet. Thus
for any transformationsk[℄ and any equation e, we dene a series of equations sk[℄(e)
suh that the following diagramommutes:
n
1
sk[℄
qqqqq qqqq qqqq qqqqq
qqqqq
qqqq
qqqq
qqqqq
n 0
1
e
qqqqqqqqqqqqqqqqqq
qqqqq qqq qqqqq
qqqqq qqqqqqqqqqqqqqqqqq
qq qqqqq qqqq qqqq qqq
sk[℄(e)
n
2
sk[℄
qqqqq qqqq qqqq qqqqq
qqqqq
qqqq qqqq
qqqqq
n 0
2
(15)
If e is arewiring of a thinning link not aeted by sk[℄ (that is, not onthe redued
orexpanded wire beforeoraftere,and not arossalinkremoved by aredution),learly
j
>
j
j
sk[℄
!
j
>
e# =#sk[℄(e)
j
>
j
j
sk[℄
!
j
>
Figure 23: sk[℄(e)for tensor orotensor redution
j >
j
j
sk[℄
!
j
>
e# #sk[℄(e)
j >
j
j
sk[℄
!
j
>
Figure 24: sk[℄(e)for tensor orotensor redution
e adds or removes a double negation along the negation link hain of a thinning link
attahed toa unit redution.
If e moves a thinning link onto or o of the primary wire of a tensor or otensor
redution, sk[℄(e)is the identity. See Figure23.
If e moves a thinning link around the inside of the non-swithing link in a tensor
or otensor redution, sk[℄(e) is more ompliated. We use the fat that we have a
sequentializable iruit toassert that a sequent box an be formed between the adjaent
wires of the swithing link. Then sk[℄(e) is a \sequent box rewiring" (Proposition 5.4)
tomove the thinninglink aross this box. See Figure24.
If e moves a thinning link onto or o of an expanded wire, sk[℄(e) is typially the
j
?
sk[℄
!
j
? >
j >
j
e# #sk[℄(e)
j
?
sk[℄
!
j
? >
j >
j
Figure 25: sk[℄(e) for expansions
We have now taken are of all ases exept where is a unit redution. Here there
are two possibilities: either eapplies toathinninglink whih isattahed tothe negation
linkhainto beremoved, ore appliestothat negationlinkhain itself. Reall that sk[℄
opies the negation link onguration of the redued hain onto eah link pushed o of
the hain. It is not diÆult to see that, using this fat, we an just have sk[℄(e) insert
whateverequationsareneessarytoemulatethenegationlinkongurationfortheiruit
aftere. The next few paragraphsspellout the variousases indetail.
Consider the ase where e applies to a link whih is attahed to the struture to be
redued. If e pushes the link onto or o of the hain, then either sk[℄ would have the
same result before or after e, or the type of unit would be hanged by the addition of a
negationlink before being pushed o the hain. Sosk[℄(e) is either the identity, or the
additionor removalof a negationlink atthe aeted thinninglink.
Similarly, if e moves the link along the negation link hain in either diretion, or if
e adds or removes a negation link, then sk[℄ would either have the same result, or else
the result will dier only by a double negation. So sk[℄(e) is either the identity, or the
additionor removalof two opposing negationlinks atthe appropriatepoint.
If e moves a link, whih is attahed tothe \far end" of an wound barbell, around to
the otherside of the barbell,then the end resultwillbethe same (aopy ofthe negation
link struture willend up the same read along eitherside).
In the other ase, if e applies to the thinning link whih is to be removed by a unit
redution, then sk[℄(e) willbe the same rewiringapplied to eah link pushed o of the
redued hain; in the ase of a double negationalong the negationlink hain, sk[℄(e) is
the same rewiring applied only to the links that will be pushed past that setion. This
Now note that sk[℄(e) an be extended to apply to any series of equations e :
de-ne sk[℄(e
1 ;e
2
) to be sk[℄(e
1
);sk[℄(e
2
), and similarly for any series. These
transfor-mations an also be extended to apply to series of skeletal redutions and expansions:
dene sk[
1 ;
2
℄ to be sk[
1 ℄;sk[
2
℄ and then sk[
1 ;
2 ℄(e
) to be sk[
2 ℄(sk[ 1 ℄(e )). That
these denitions \work" is easily seen by plugging together opies of the ommutative
diagram(15).
Finally,weuse these transformationsto obtainonuene. Consider any divergene
n 0 1 1 n 1 e =n 2 2 !n 0 2 :
Observe that the onlyases where
1 and
2
annothappen onseutively in the skeleton
are whenone isaredution, andthe otherisanexpansionofthe wiretoberedued. But
learly all suhexpansions have the property that they an be followed by asequene of
redutions and equations whih returns the net to its state before the expansion. Thus
ndingaonvergene istrivial. As mentionedbefore, divergenes involvingaunit
expan-sion along a negationlinkhain an be taken are of similarly.
For any other divergene onstrut the followingdiagram:
n 1 e n 2 qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqq 1 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 qqqqqqqqqqqqqqqqqq qqq qqq qq qqq qqq qqq q 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 qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqq q qqq qq qqq qqq qqq qq q 2 n 0 1 sk[ 1 ℄(e ) n 12 n 21 sk[ 2 ℄(e ) n 0 2 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 qqqqqqqqqqqqqqqqqq qq qqq qqq qq qqq qqq qq qqqqqqqqqqqqqqqqqq qqq qqq qq qqq qqq qqq q 2 qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqq 1 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 qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqq n 0 21 sk[ 1 ; 2 ℄(e ) n 0 12
The dotted arrows represent sk[℄ moves. The solid arrows provide the required
on-vergene. This is immediatelyX-reduing. 7
7
Therearetwoslightsimpliationshere. First,weareapplyingtheommutativediagram(15)asif
itappliedintheasewhereoneofthehorizontalarrowsisatually ratherthansk[℄. Indeed,in most
aseswhere isavalidexpansionorredution, =sk[℄. However,ouronventionforsk[℄with an
X/R is expansion E-terminating. Call a wire in a iruit unexpandable if it is
onneted to the output portof an introdution link orthe input port of an elimination
link, and expandable otherwise. Clearly any expansion on an unexpandable wire willbe
reduible. For eah iruit C let v(C) be the total number of logial onnetives (, ,
>, ?)in the typesof all the expandable wires inC. Furtherlet v 0
(C)bev(D), whereD
is the skeleton of a redued E-normal form of C (whih exists by the preeding parts of
the proof). Clearly v 0
(C)isunhangedby redutions,equations,orreduible expansions;
thus to prove the desired result it suÆes to show that v 0
(C) is stritly redued by any
irreduible expansion.
Let C 0
be the resultof applying anirreduibleexpansionto C. Wewish toshowthat
thisexpansionouldbe\postponed"toD,inthesensethatthereisairuitD 0
,obtained
from D by a single (skeletal) expansion, whih is the skeleton of a (perhaps not fully)
redued form of C 0
. Observe that it is possible to set up a orrespondene between the
wires in the skeleton of C and the wires in the skeleton of its redued E-normal form,
D, by xinga redutionsequene, andonsidering eahredution todestroy the redued
wireand, inthe aseofthe tensorandotensorredutions, tojoinotherwireswhihwere
previously distint. Then any expansion inC ona wire whih is eventually destroyed is
reduible: the expansion doesn't prevent the appliation of any of the redutions in the
sequene up to the one whih would have destroyed its wire, but at that point a pair of
redutions learly has the same eet. So any irreduible expansion in C is on a wire
whihispreserved inD;itisnoweasytosee that thatthe iruitC 0
redues toairuit,
whoseskeletonisobtained by applyingthe orrespondingexpansiontoD. LetD 0
bethis
skeletaliruit.
Clearly v 0
(C 0
) = v 0
(D 0
) < v(D 0
), so it suÆes to show that v(D 0
) < v(D) = v 0
(C).
SineD 0
isobtainedfromDbyoneexpansion,itlearlyisenoughtoshowthatv(redut ) <
v(redex ) for any expansion, whih is indeed the ase (note that the wires in the redut,
whihhave the same type asthe expanded wire, have beome unexpandable).
To see that the nal statement of
