Table Of ContentSurface Proofs for Nonsymmetric Linear Logic
LawrenceDunn JamieVicary
NorthFloridaCommunityCollege,USA UniversityofOxford,UK
[email protected] [email protected]
We show that a proof in multiplicative linear logic can be represented as a decorated surface, such
thattwoproofsarelogicallyequivalentjustwhentheirsurfacesaregeometricallyequivalent. Thisis
anextendedabstractforarXiv:1601.05372.
1 Introduction
Multiplicative linear logic [6, 7] is a formal calculus for reasoning about resources, which is similar to
traditional logic, except that resources cannot be duplicated or neglected in the way that propositions
can.
Acentralprobleminlogicisdeterminingwhentwoproofsshouldbeconsideredequivalent. Inthis
paper,wedescribeaschemeforinterpretingproofsinmultiplicativelinearlogicasgeometricalsurfaces
embedded in 3d space. We define two surfaces as equivalent just when one can be deformed into the
other,insensewemakeprecise. Ourmaintheoremthenreadsasfollows.
Theorem2.2. Twosequentproofsinnonsymmetricmultiplicativelinearlogichaveequalinterpretations
inthefree∗-autonomouscategoryjustwhentheirsurfacesareequivalent.
Thetheoryof∗-autonomouscategories[18,20]isastandardmathematicalmodelforlinearlogic,sothis
theoremsaysthatthenotionofproofequalityprovidedbythesurfacecalculusagreeswiththestandard
one.
Oursurfacesaresimilarinspirittoproofnets[5,7];however,wearguethatourschemehasseveral
advantages. Inparticular,correctnessislocal;anywell-typedcompositeproducesavalidproof-theoretic
object, with no global property, such as the long-trip criterion, to be verified. Also, our notion of
equivalenceisbroad,establishingsomeproofequivalencesinfewerstepsthanforproofnets;sometimes
in just a single step. Our scheme also certainly has disadvantages: in particular, we do not present a
decision procedure for equivalence of our 3d diagrams, although we expect such a procedure could be
described. Despitethesedifferences,theformalismsareintimatelyconnected,inthefollowingway: the
proof net is the 2d projection of the 3d surface geometry. From this perspective, we can make sense of
someofthefeaturesofproofnets: thelong-tripcriterioncanbeinterpretedasanon-localcheckthatthe
2dshadowisconsistentwithavalid3dgeometry,andthethinninglinkdecorationsindicatethedepthat
whichaunitisattachedinthe3dgeometry.
Theunderlyingtechnicalcontribution,whichwedonotdescribeinthisextendedabstract,isadirect
algebraic proof of the coherence theoremfor Frobenius pseudomonoids, which shows that all diagrams
of a certain sort commute. This is more appropriate for logical purposes, and more flexible, than an
existingtopologicalproofarisingfromMorsetheory[15,16].
The authors are grateful to Samson Abramsky, Nick Gurski, Sam Staton and the anonymous
reviewers for useful comments. 3d graphics have been written in TikZ, and 2d graphics have been
producedbytheproofassistantGlobular[2].
I.CervesatoandM.Ferna´ndez(Eds):FourthInternationalWorkshop
onLinearity2016(LINEARITY’16)
EPTCS238,2017,pp.33–43,doi:10.4204/EPTCS.238.4
34 SurfaceProofsforNonsymmetricLinearLogic
1.1 Relatedwork
It is well-recognized that ideas from topology are relevant for linear logic. The original proof nets of
Girard [7] are topological objects, and Mellie`s has shown how the topology of ribbons gives a decision
procedureforcorrectnessofproofnets[17]. Proofnetsallowreasoningaboutproofswithunits,butthe
formalism is complex, requiring a system of thinning links with moving connections [5]. Hughes [10]
givesavariantofproofnetswhichworkswellwithunits,andwhichhasgoodcompositionalproperties,
but which still requires a long-trip criterion, has non-local jumps, and requires successive individual
re-wirings. Our approach has a local flavour which is shared by the deep-inference model of proof
analysis[8]andtheaccesstomonoidalcoherencethatitallows[9];however,thecoherencepropertywe
makeuseofisstrictlymorepowerful,asitoperatesinafragmentthatcombinesthe⊗and`connectives.
WenotealsotheworkofSlavnov[21]onlinearlogicandsurfaces,whichinvolvessomesimilarideasto
thepresentarticle,butistechnicallyquiteunrelated.
2 Surface calculus
In this section we develop the 2d string diagram calculus for sequents, and the 3d surface calculus for
proofs. We show how to translate a sequent calculus proof into the surface calculus, and we define the
equivalencerelationonsurfaces.
2.1 The2dcalculus
The 2d calculus, which we will use to represent individual sequents, is the Joyal-Street calculus for
monoidal categories [13], directed from left to right. We use the standard 2-sided sequent calculus for
nonsymmetricmultiplicativelinearlogicwithunits[5]: oursequentsarepairsΓ(cid:96)∆,whereΓand∆are
ordered lists (separated with “,”) of expressions in the following grammar, whereV ={A,B,C,...} is a
setofatomicvariables:
S::=I|⊥|V |S⊗S|S`S|S∗|∗S
Wehaveleftandrightnegation,andisomorphisms∗(S∗)(cid:39)S(cid:39)(∗S)∗areanativepartofthecalculus(see
Example3.2);forsimplicity,wesuppressthematthesyntacticlevel.
Our graphical language for sequents is as follows. Atomic variables are represented as black dots,
pointingindifferentdirectionsdependingontheirsideofthesequent:
A A
A(cid:96)··· ···(cid:96)A
Thesediagramshavenonemptyboundary,inthesensethatnotalltheendsareterminatedbynodes.
Thetwosidesofasequentarerepresentedgraphicallybytrees,whicharedrawnconnectedtogether
attheirroots. Thebasicconnective“,”isdenotedasabluevertexwithzeroormorebranchestotheleft
orright,asfollows:
C A
A A
D B
B B
E C
A,B(cid:96)C,D,E (cid:96)A,B A,B,C(cid:96)
L.DunnandJ.Vicary 35
The connectives ⊗ and `, which are always binary, are drawn in blue on their natural side (left for ⊗,
rightfor`),andinredontheotherside,asweshowwiththefollowingexamples:
A E
A
B F D
B
C G E
C
D H F
A,(B⊗C),D(cid:96)E,(F⊗G),H (A`B),C(cid:96)D`(E⊗F)
Note that a blue dot with a binary branching is therefore an overloaded notation; this is a deliberate
feature.
The units I and ⊥ are represented by blue dots on their natural side (left for I, right for ⊥), and red
dotsontheotherside,asshown:
A B
A
C
B
A,I,B(cid:96)C A,⊥(cid:96)B,I,⊥
Werepresent(−)∗ asturningrightbyahalf-turn,and∗(−)asturningleftbyahalf-turn,asshown:
A C
D
B
A∗,∗B(cid:96)∗∗C,D
Diagrams built from sequents in this way are of a simple kind; as graphs, they are all acyclic and
connected. Ingeneralwecanallowarbitrarywell-typedcompositesofthesecomponents;suchdiagrams
represent1-morphismsinthemonoidalbicategoryF(F∗),describedinthefullpaper.
2.2 The3dcalculus
Diagramsinthe3dcalculusaresurfacesembeddedinR3. Formallytheyareexpressionsinthegraphical
calculusforGraycategories,whichisbynowwell-developed[3,4,12,19]. However,the3dcalculusis
quiteintuitive,andwetakeadvantageofthistointroduceitinaninformalway.
Diagrams consist of sheets, bounded on the left and right by edges, which are bounded above and
below by vertices. (Sheets can also be bounded by the sides of the diagram, and edges can also be
boundedbythetoporbottomofthediagram.) Diagramsareimmersedin3dspace,meaningthatsheets
can exist in front or behind other sheets, and wires on sheets of different depths can cross; however,
36 SurfaceProofsforNonsymmetricLinearLogic
componentsneverintersect. Hereisanexample:
Here we have front and back sheets, each containing an edge, which contains a vertex. Towards the
bottomofthepicture,thewirescross: thisiscalledaninterchanger.
For our application to linear logic, we allow two types of vertex: coherent vertices and adjunction
vertices.
• Coherentvertices. Saythata2dcalculusdiagramissimplewhenitisconnectedandacyclicwith
nonempty boundary, and in the blue fragment of the calculus, not involving red nodes or black
atomic variable nodes. Then any two simple diagrams can be connected by a coherent vertex,
denotedasfollows:
(1)
On the left we give the surface representation, and on the right we give the 2d calculus
representation of the upper and lower boundaries. The coherent vertex is the point in the middle
ofthesurfacediagramwhere4edgesmeet.
• Adjunctionvertices. ListedinFigure1,theseintroduceandeliminateredandblackedgesinthe
surfacecalculus.
We now define equivalence in the graphical language, giving intuitive interpretations of each
generatingrelationinitalics.
Definition 2.1. Two surface diagrams are equivalent when they are related by the least equivalence
relationgeneratedbythefollowing:
• Coherence. LetP,Qbesurfacediagramsbuiltfromcoherentvertices,withequallowerboundaries
and equal upper boundaries, with all these boundaries being simple; then P = Q. (All acyclic
equationsofcoherentverticeshold.)
• Adjunction. TheequationslistedinFigure2hold. (Bentwirescanbepulledstraight.)
• Isotopy. Theequationsofamonoidalbicategoryhold. (Iftwodiagramsareambientisotopic,they
areequivalent.)
L.DunnandJ.Vicary 37
[empty]
[empty]
[empty]
Figure1: Thegeneratorsofthe3dcalculus
= =
= =
= =
= =
= =
Figure2: Theequationsofthe3dcalculus
• Locality. Suppose surface diagrams P,Q differ only with respect to subdiagrams P(cid:48),Q(cid:48), with
P(cid:48)=Q(cid:48). ThenP=Q. (Equivalenceapplieslocallyintheinteriorofadiagram.)
It is a fair summary of this definition to say that two diagrams are equivalent just when one can be
deformed into the other. We emphasize that our contribution here is the Coherence axiom, which has
notpreviouslybeennoted1;therestfollowsinprinciplefromtheworkofStreet[22],althoughithasnot
toourknowledgebeenexplicitlydescribedintheliterature,anditsimplicationsforlinearlogicunpacked.
However,wenotethatitistheCoherenceaxiomthatmakesthenotionofequivalencetractable.
Our presentation here is informal, but we emphasize that our definition of the surface calculus and
its equivalence relation can be made completely precise in terms of the formal development of the full
article: twodiagramsareequivalentjustwhentheyareequalas2-morphismsinthemonoidalbicategory
F(F∗).
1Inthefullversionofthispaperwegiveadirectcombinatorialproofofthecoherenceproperty,whichalsofollowsfrom
topologicalargumentsduetoKerlerandLyubashenko[15]asrefinedbyLauda[16].
38 SurfaceProofsforNonsymmetricLinearLogic
V V
A A
V V V V
W W W W
V V
AXIOM AXIOM AXIOM AXIOM AXIOM
A(cid:96)A ∗V(cid:96)∗V V∗(cid:96)V∗ V⊗W(cid:96)V⊗W V`W(cid:96)V`W
∆ Γ
Γ A A ∆ V V V V Γ VW VW ∆ Γ VW VW ∆
Γ ∆
Γ ∆ Γ ∆ Γ ∆
Γ ∆ Γ ∆
Γ(cid:96)AA(cid:96)∆ Γ(cid:96)∗V∗V(cid:96)∆ Γ(cid:96)V∗V∗(cid:96)∆ Γ(cid:96)V⊗WV⊗W(cid:96)∆ Γ(cid:96)V`WV`W(cid:96)∆
CUT CUT CUT CUT CUT
Γ(cid:96)∆ Γ(cid:96)∆ Γ(cid:96)∆ Γ(cid:96)∆ Γ(cid:96)∆
Figure3: AXIOMandCUTrulesforanatomicvariableA,andrecursivelyforvariablesV,W
2.3 Interpretingthesequentcalculus
We saw in Section 2.1 how individual sequents in multiplicative linear logic can be interpreted as 2d
diagrams. Wenowseehowproofscanbeinterpretedas3dsurfacediagrams. Weviewthesesurfacesas
directed from top to bottom, just like traditional sequent calculus proofs; so for a particular surface, its
hypothesisistheupperboundary,anditsconclusionisthelowerboundary.
Weuseabasisforthesequentcalculuswithasymmetrybetweenintroductionandeliminationfor⊗,
`,I and⊥;therules⊗-R,`-L,I-Rand⊥-Larederivable(seeExample3.1.) Furthermore,weinclude
only CUT rules with minimal overlapping contexts; the more general CUT rules are derivable using
negation. (These two features account for the differences between our presentations and others in the
literature[1].) TheinterpretationofAXIOMandCUTrulesaregivenrecursivelyinFigure3,withblack
wires standing for atomic variables and green wires standing for general variables; the interpretation of
theremainingrules,whichwecallthecorefragmentofthelogic,isgiveninFigure4. Wenowstateour
maintheorem.
Theorem 2.2. Two sequent proofs in multiplicative linear logic have equal interpretations in the free
∗-autonomouscategoryjustwhentheirsurfacediagramsareequivalent.
It is interesting to analyze the different contributions to proof equivalence made by each part of
Definition 2.1 of surface equivalence. Coherence tells us that any two proofs built in the core part
of the logic given in Figure 4 are equal. Adjunction tells us that AXIOM and CUT cancel each other
out, both for atomic and compound variables. Isotopy tells us that that ‘commutative conversion’ is
possible, where by exchanging heights of disconnected parts of the diagram, we exchange the order of
separatesequentcalculusproofsteps. Localitytellsusthatwecanapplyourequationsinthecontextof
alargerproof,inthemannerofdeepinference[8].
We give a formal statement of coherence for the core part of the logic, since it is a result of
independent interest. Note that this is not a theorem about the surface calculus, although its proof uses
thesurfacecalculus.
L.DunnandJ.Vicary 39
I ⊥
I ⊥
Γ1,I,Γ2(cid:96)∆ Γ1,Γ2(cid:96)∆ Γ(cid:96)∆1,⊥,∆2 Γ(cid:96)∆1,∆2
I-INT I-ELIM ⊥-ELIM ⊥-INT
Γ1,Γ2(cid:96)∆ Γ1,I,Γ2(cid:96)∆ Γ(cid:96)∆1,∆2 Γ(cid:96)∆1,⊥,∆2
A A A A
B B B B
A A A A
B B B B
Γ1,A⊗B,Γ2(cid:96)∆ ⊗-INT Γ1,A,B,Γ2(cid:96)∆ ⊗-ELIM Γ(cid:96)∆1,A`B,∆2 `-ELIM Γ(cid:96)∆1,A,B,∆2 `-INT
Γ1,A,B,Γ2(cid:96)∆ Γ1,A⊗B,Γ2(cid:96)∆ Γ(cid:96)∆1,A,B,∆2 Γ(cid:96)∆1,A`B,∆2
∆ Γ
Γ ∆ A A
A A Γ ∆
∆ Γ
A A
Γ ∆
∆ Γ
∆ Γ
Γ ∆
A A
Γ(cid:96)∆,A Γ,A(cid:96)∆ Γ(cid:96)A,∆ A,Γ(cid:96)∆
∗(−)-R (−)∗-L (−)∗-R ∗(−)-L
Γ,∗A(cid:96)∆ Γ(cid:96)∆,A∗ A∗,Γ(cid:96)∆ Γ(cid:96)∗A,∆
Figure4: Surfaceinterpretationsofthecorelogicalrules
Corollary 2.3. If two sequent proofs in the core fragment of the logic given in Figure 4 have the same
hypothesesandconclusion,thentheyareequalinthefree∗-autonomouscategory.
Wecommentonsomeinterestingfeaturesofthetranslationbetweenthesequentcalculusandthesurface
calculus. Thefundamentalsimplicityofthesurfacecalculusisclear,fromtheminimalityofthedatain
Figure 1, as compared to Figures 3 and 4. Partly this is achieved by the greater degree of locality: for
example, the cut rules for ∗V andV∗ are both interpreted using the same surface generators, composed
in different ways. But more significantly, the entirecore fragment of the sequentcalculus is interpreted
inthetrivialpartofthesurfacecalculus,significantlyreducingthebureaucracyofproofanalysis,touse
Girard’s phrasing [7]. To make the most of these advantages, we suggest that the surface calculus can
servedirectlyasatoolkitforlogic,notjustasawaytovisualizesequentcalculusproofs.
40 SurfaceProofsforNonsymmetricLinearLogic
Γ1Γ2 A B ∆1 ∆2 Γ1 ∆1 A B
A⊗B(cid:96)A⊗B AXIOM Γ2 ∆2
⊗-INTRO
A,B(cid:96)A⊗B
Γ2(cid:96)B,∆2 =
CUT
A,Γ2(cid:96)A⊗B,∆2
Γ1(cid:96)∆1,A
CUT
Γ1,Γ2(cid:96)∆1,A⊗B,∆2 ∆1
∆1 Γ1 ⊗ A
Γ1 ⊗ A Γ2 B
Γ2 B ∆2
∆2
Figure5: Derivationofthesurfacerepresentationofthe⊗-Rrule
AXIOM AXIOM
B(cid:96)B A(cid:96)A
⊥-INT
AXIOM A(cid:96)⊥,A
A(cid:96)A
AXIOM
⊗-R B(cid:96)B
A,B(cid:96)A⊗B
=
⊥-INT ⊗-R
A,B(cid:96)⊥,A⊗B A,B(cid:96)⊥,A⊗B
B B
B ⊗ B ⊗
A A
A A
⊥ ⊥
Figure6: Geometricalequivalenceoftwoproofsininvolvingtheunit
3 Examples
In this Section, we look in detail at a number of examples: we derive the surface form of the missing
⊗-R rule; we analyze equivalence of a proof involving units; and we investigate the classic triple-unit
problem.
Example 3.1 (Additional rules). Presentations of multiplicative linear logic usually include the rules
⊗-R, `-L, I-R, ⊥-L, which are missing from Figure 3 and Figure 4; however, they are derivable. We
analyze ⊗-R in detail in Figure 5. On the left-hand side, we derive the rule in our chosen basis for the
the sequent calculus. In the middle image, we interpret it in the surface calculus, using the rules we
have described. In the third image, we simplify the surface calculus interpretation using the rules in
Figure2. Fromthissimplifieddiagram,weseethatitdoesnotinfactinvolvethevariables,thenontrivial
generators being applied in the central part of diagram only. Elegant interpretations of the other 3 rules
canbederivedsimilarly.
Example 3.2 (Triple-dual problem). Starting with the identity A(cid:40)X →A(cid:40)X, we can uncurry
on the left to obtain a morphism A⊗(A(cid:40)X)→X, and curry on the right to obtain a morphism
L.DunnandJ.Vicary 41
AXIOM
A(cid:96)A
⊥-INT
A(cid:96)⊥,A ⊥
AXIOM
B(cid:96)B
⊗-R
A,B(cid:96)⊥,A⊗B
⊗
B
B ⊗ B A⊗B
A
A A
⊥ ⊥
Figure7: Adeductioninthesequentcalculus,anditssurfacecalculusandproofnetrepresentations.
p : A → X (cid:18) (A (cid:40) X); in a similar way, we can also define a morphism q : A → (X (cid:18) A) (cid:40) X.
A A
Then the triple-dual problem, originally due to Kelly and Mac Lane [14] and generalized here to the
non-symmetricsetting,istodeterminewhetherthefollowingequationholds:
X (cid:18)q
X (cid:18)((X (cid:18)A)(cid:40)X) A X (cid:18)A
pX(cid:18)A (2)
id
X (cid:18)((X (cid:18)A)(cid:40)X)
ChoosingX =⊥,thenq and p aretheisomorphisms∗(A∗)(cid:39)A(cid:39)(∗A)∗. Wegivethesurfacesforthe
A A
clockwiseandanticlockwisepathsof(2):
A A
= (3)
A A
We conclude that the proofs are equivalent by a single application of the Coherence rule. Contrast this
with the treatment of Blute et al [5, Section 4.2] and Hughes [11, Example 2] in terms of proof nets,
wheretheproofsrequireseveralrewiringsteps.
Example 3.3 (Equivalence of two proofs with units). The example is given in Figure 6. We present
two distinct sequent proofs of the tautology A,B(cid:96)⊥`(A⊗B), along with their corresponding surface
proofs. The heights are aligned to help understand how the surface proofs have been constructed. We
makeuseofthe⊗-Rrulederivedinthepreviousexample.
42 SurfaceProofsforNonsymmetricLinearLogic
It can be seen by inspection that the surface proofs are equivalent, as follows. Starting with the
surface on the left, we allow the ⊥-introduction vertex to move up and to the left; this is an application
of Coherence and Locality. We also allow the B-introduction vertex at the top of the diagram to move
down,behindboththeA-introductionand⊥-introductionvertices;thisisanapplicationofIsotopy.
Example 3.4 (Proof net projection). We argued in the introduction that proof nets can be considered
the 2d ‘shadow’ of the full 3d geometry, with the correctness criterion and thinning links arising to
compensate for the fact that this ‘shadow’ has lost some essential geometrical data. We illustrate this
inFigure7,whichgivesasequentproofalongsideitssurfacediagramandproofnetrepresentations. In
particular,intheproof-netpicture,thebottomattachmentmustbe‘hooked’ontoasolidwire,indicated
bythedottedline,afeaturewhichisabsentfromthe3dimage,whichcontainsconnectivitydatadirectly
aspartofthe3dgeometry.
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