Syntax and Semantics of Abstract Binding Trees

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by  Jonathan Sterling

Syntax and Semantics of Abstract Binding Trees
JonathanSterlingandDarinMorrison
6
Abstractbindingtrees(abts)areageneralizationofabstractsyntaxtreeswhereoperators
1
0 mayexpressvariablebindingstructureaspartoftheirarities. Originallyformulatedby
2 PeterAczel[1],unisortedabtshavebeendeployedsuccessfullyastheuniformsyntactic

n frameworkforseveralimplementationsofConstructiveTypeTheory,includingNuprl[3],
a
MetaPRL[13]andJonPRL[21].
J
In Practical Foundations for Programming Languages [12], Robert Harper develops the
3
2 multi-sortedversionofabstractbindingtreesandproposesanextensiontoincludefamilies

ofoperatorsindexedbysymbols,whichare,unlikevariables,subjecttoonlydistinctness-
]
O preserving renaming and not substitution; furthermore, symbols do not appear in the
L syntax of abts, and are only introduced as parameters to operators. This extension to
. supportsymbolsisessentialforacorrecttreatmentofprogramminglanguageswithopen
s
c sums(e.g. ML’sexntype),aswellasassignablereferences.
[
Inparallel,M. Fioreandhiscollaboratorshavedevelopedthecategoricalsemantics

1 forseveralvariationsofsecond-orderalgebraictheories[8,10,6,7];ofthese,thesimply
v
sortedvariantsareequivalenttoHarper’sabstractbindingtreesaugmentedwithanotion
8
9 ofsecond-ordervariable(metavariable).
2
The contribution of this paper is the development of the syntax and semantics of
6
0 multi-sorted nominal abts, an extension of second order universal algebra to support
. symbol-indexedfamiliesofoperators. Additionally,wehavedevelopedthecategorical
1
0 semanticsforabtsformallyinConstructiveTypeTheoryusingtheAgdaproofassistant[18];
6 we have also developed an implementation for abstract binding trees in Standard ML,
1
whichweintendtointegrateintotheJonPRLproofassistant[21].
:
v
i
X
1 Categorical Preliminaries
r
a
FixasetSofsorts;wewillsayτsortwhenτ S. Inthissection,wewilldevelopatheory
∈
ofsetsvaryingovercollectionsofS-sortedsymbols.
Let I be the category of finite cardinals and their injective maps; then the comma
construction I S, is the category ofcontexts ofsymbols whose objects are finite sets of
↓
symbols U and sort-assignments U s S, and whose morphisms are sort-preserving
renamings;wewillwriteΥforasymbolcontext(U,s).
1
Remark1.1. Tobeprecise,I Sisanabuseofnotationforthecommacategory|−| ∆[S],
↓ I ↓
with|−| theforgetfulfunctorfromItoSet,and∆[S]( ) , Saconstantfunctorfrom1toSet.
I ∗
Formally,anobjectofI Sis,then,atriple U, ,s withUanobjectinI, theuniqueobject
↓ h ∗ i ∗
in 1, andsafunction |U| S; an arrowρ : U, ,s V, ,t isacommuting triangle
I h ∗ i h ∗ i
likethefollowing:
|ρ|
|U| I |V|
I I
s t
S
Becausetheobject : 1isunique,wewillalwaysomititfromthedataofanobjectin
∗
I S.
↓
CopresheavesonI S ApresheafonacategoryCisacontravariantfunctorCop Set;
↓
whenC Dop,wecallthisacopresheaf (covariantpresheaf)onD. Therefore,acopresheaf
≡
onI SisafunctorI S Set;colloquially,suchacopresheafcanbethoughtofasaset
↓ ↓
thatvariesoversymbolcontexts.
Thisleadstoaveryintuitivesettinginwhichtoconsidersetsthatvaryovercollections
ρ
ofS-sortedsymbols. Inparticular,foranymorphismΥ Υ0 inI S,anycopresheaf
X : SetI↓S hasasymbol-renamingactionX(ρ) : X(Υ) X(Υ0). When↓thecopresheafXis
clearfromcontext,wewillwritem ρforX(ρ)(m).
Now, when m X(Υ) for a cop·resheaf X : SetI↓S, we can define the notion that m
∈
dependsonatmostasubcontextΥ ofΥ;thisiscalledsupport.
0
ρ
Definition 1.2 (Support). When Υ Υ , we say that Υ supports m Υ (written
0 0
∈
Υ I m)when,forallΥ ρ1,ρ2 Υ ,ifρ ρ = ρ ρthenm ρ = m ρ .
ρ 0 00 1 2 1 2
◦ ◦ · ·
Wecanalsoalwayscalculatetheleastsupportofatermm X(Υ ),writtensupp(m):
0
∈
supp(m) , {Υ | Υ I m}
ρ
ρ
Υ \ Υ
0
Intuitively,supp(m)istheexactsymbolcontextthatmdependson.
1.1 Sheavesontheatomicsite
ρ
When Υ Υ such that Υ I m for some m X(Υ ), we would expect that we can
0 ρ 0
∈
workbackwardtoauniquem : Υsuchthatm ρ = m;whilstitisnotingeneralthecase
0 0
·
forpresheavesX,thisispreciselythesheafconditionforcopresheavesonI Sunderthe
↓
atomictopology[16,p.126],whichwewilldefineinDefinition1.6.
Thesheafconditionensuresthatthenotionofsupportiswell-behavedinthefollowing
sense: ifsometermliesintwodifferentfibers,thenitmaybestrengtheneduniquelyintothe
fiberofitssupport.
2
Definition1.3(Sievesandmatchingfamilies). AsieveSonanobjectCisasubfunctorof
theYonedaembeddingy(C) , C[−,C],i.e. S y(C). ForapresheafXonthecategoryC,a
v
matchingfamilyforasieveSisanaturaltransformationφ : S X. Anamalgamationforφis
anobjectm X(C)suchthatm f = φ (f)foreachf S(D),forallobjectsD.
D
∈ · ∈
Definition1.4(Sitesandcoverages). Asiteisapair C,J ,withCacategoryof“worlds”
h i
andJaGrothendiecktopology(or“coverage”). AGrothendiecktopologyJ(orcoverage)is
afamilyofsetsofsieves,indexedbyC-objects;thatis,foreachobjectC : C,J(C)istheset
ofC-sieveswhichwillbesaidtocoverC. AtopologyJmustsatisfyseveralaxioms(see[16]
fordetails).
Definition1.5(Separationandsheafhood). Apresheafonasite C,J isapresheafonC;
h i
we saythatsucha presheafX is separated in case foreverysieve S J(C) andmatching
∈
familyφ : S X,thereisatmostoneamalgamationm X(C)forφ. Additionally,Xisa
∈
sheaf whenthereisexactlyonesuchamalgamation.
Definition 1.6 (The atomic topology). The atomic coverage Jatm is the one in which all
non-emptysievescover;inotherwords:
Jatm(C) , {S v y(C) | DS(D)non-empty}
Inthefuture,wewillwriteI[S]fortheGroSthendiecksite (I S)op,Jatm . Sheafhood
↓
ontheatomicsiteI[S]amountstothefollowingcondition,usingDefinition1.2:
(cid:10) (cid:11)
AcopresheafX : SetI↓S isasheafonI[S]justwhen,foralln X(Υ0),ifwehave
∈
ρ
Υ Υ andΥ I n,thenthereisauniquem X(Υ)suchthatm ρ = n.
0 ρ
∈ ·
Therefore,asheafonI[S]isasetthatvariesoverS-sortedsymbolcontexts,whichhasa
well-behavednotionofsupport. Anotherwaytoputitis,ifm X(Υ)only“uses”some
∈
ofthesymbolsavailableatworldΥ,there’sacanonicalwaytorestrictmtoaterminthe
worldwhichcontainsonlythesymbolsthatituses.
Constructionsonsheaves Everycategoryofsheavesonasitegivesrisetoatopos,which
equipsuswithanumberofstandardconstructions,including(amongotherthings)the
disjointunionofsheavesX Y,theproductofsheavesX Y,andtheterminalsheaf1.
⊕ ⊗
2 Operators and signatures
Avalence{~σ}[~τ].τspecifiesanexpressionofsortτwhichbindssymbolsin~σandvariables
in~τ.
τsort σi sort (i 6 m) τi sort (i 6 n)
{σ ,...,σ }[τ ,...,τ ].τvalence
0 m 0 n
3
Anarity(~v)τspecifiesanoperatorofsortτwithargumentsofvalences~v. Wewillcall
thesetofvalencesV ,andthesetofaritiesA .
S S
τsort vi valence (i 6 n)
(v ,...,v )τarity
0 n
FixafamilyofsheavesofoperatorsOa Sh(I[S]),indexedbyaritiesa AS;foreach
∈ ∈
Oa,arrowsinI Swilllifttorenamingsinoperators’symbolicparameters. Together,S
↓
andOaresaidtoformasignatureΣ , S,O .
h i
WewillwriteΥ(cid:13)ϑ : aincaseϑ O (Υ);notethatthisjudgmentenjoysthestructural
a
∈
propertiesofweakeningandexchangeviafunctorialityofO, andstrengtheningviathe
sheafcondition. Anoperatorsignatureisdefinedbyspecifying,foreacharitya,thesheaf
O ,whose“elements”aretheoperatorsofaritya.
a
Examples Forinstance,consideraλ-calculuswithasinglesort,exp;wegiveitssignature
Σ byassertingthefollowingaboutitsoperators:
λ
Υ(cid:13)lam : ([exp].exp)exp
Υ(cid:13)fix : ([exp].exp)exp
Υ(cid:13)ap : (.exp,.exp)exp
TheserulescorrespondtothefollowingdefinitionofO:
O([exp].exp)exp (cid:27) 1 1
⊕
O(.exp,.exp)exp (cid:27) 1
Sofar,wehavemadenouseofsymbolsandparameters;however,considertheextension
ofthecalculuswithassignables(references):
Υ(cid:13)decl : (.exp,{exp}.exp)exp
Υ,u : exp(cid:13)get[u] : ()exp
Υ,u : exp(cid:13)set[u] : (.exp)exp
LetS bethesheafofsymbolsofsortτ,asubobjectoftheYonedaembeddingofthe
τ
emptysymbolcontext,y( ):
·
S (Υ) , {u |Υ| | Υ(u) τ}
τ
∈ ≡
Thentheaboverulescorrespondtothefollowingfamilyofsheaves:
O(.exp,{exp}.exp)exp (cid:27) 1
O()exp (cid:27) Sexp
O(.exp)exp (cid:27) Sexp
4
Declaringanewassignableconsistsinprovidinganinitialvalue,andanexpressionbind-
ingasymbol(whichshallrepresenttheassignableinscope). NotethatthefunctorialityofO
guaranteesforanyrenamingρ : Υ Υ ,afamilyofliftedmapsO (ρ) : O (Υ) O (Υ )
0 a a a 0
natural in a, such that Υ (cid:13)O(ρ)(ϑ) : a when Υ(cid:13)ϑ : a. In particular, the renaming
0
Υ,u Υ,vshalltakeget[u]toget[v]. Inthefuture,wewillwriteϑ ρforO (ρ)(ϑ).
a
·
3 Contexts
Ingeneral,wewillhavethreekindsofcontext: symbolic(parameter)contexts,variable
contexts,andmetavariablecontexts. Thesymbolcontextshavealreadybeendefinedvia
thecommaconstructionI S,buttheyalsoadmitasyntacticcharacterization,
↓
Υsctx τsort u < |Υ|
sctx Υ,u : τsctx
·
Because,modulonotation,wehaveΥ I SjustwhenΥsctx,wewillusethesyntactic
∈ ↓
viewwhenitisconvenient.
Contexts of variables are similar to contexts of symbols, except that they admit any
renamings,notjusttheinjectiveones. Assuch,whenFisthecategoryoffinitecardinals
andallfunctionsbetweenthem,thecommaconstructionF Sisthecategoryofvariable
↓
contexts. Asabove,wecangivethemanequivalentsyntactictreatment:
Γ vctx τsort x < |Γ|
vctx Γ,x : τvctx
·
A metavariable context consists of bindings of S-valences to metavariables; let VS ,
{v | vvalence} be the set of valences. Then, the category of metavariable contexts is the
commaconstructionF VS,whichlikewiseadmitsanequivalentinductivedefinition:
↓
Θmctx vvalence m < |Θ|
mctx Θ,m : vmctx
·
4 Nominal Abstract Binding Trees
LetthejudgmentΘ.Υ Γ M : τpresuppose1 Θmctx,Υsctx,Γ vctxandτsort,meaning
k ‘
thatMisanabstractbindingtreeofsorts,withmetavariablesinΘ,parametersinΥ,and
1Inthejudgmentalmethod, aspioneeredbyPerMartin-Lo¨f, aformofjudgmentispropoundedfirstby
specifyingitsrangeofsignificance(i.e. thecircumstancesunderwhichitshallhaveameaning),andthen,
ameaningexplanation(definition)forthejudgmentisgiven. Therangeofsignificanceofajudgmentis
calleditspresupposition,andthemeaningofthejudgmentmayproceedbyinductionontheevidencethat
thepresuppositionobtains.See[20]foramoredetailedexplanationofpresuppositionsandthejudgmental
method.
5
variablesinΓ. LetthejudgmentΘ.Υ Γ E : vpresupposevvalence. Then,thesyntaxof
k ‘
abstractbindingtreesisinductivelydefinedinfourrules:
Γ x : τ
3 var
Θ.Υ Γ x : τ ‘
k ‘
Θ m : {σ ,...,σ }[τ ,...,τ ].τ
0 m 0 n
3
Υ ui : σi (i 6 m)
3
Θ.Υ Γ Mi : τi (i 6 n)
k ‘
Θ.Υ Γ m{u ,...,u }(M ,...,M ) : τ ‘mvar
0 m 0 n
k ‘
Υ(cid:13)ϑ : (v ,...,v )τ
1 n
Θ.Υ Γ Ei : vi (i 6 n)
k ‘
app
Θ.Υ Γ ϑ(E ,...,E ) : τ ‘
0 n
k ‘
Θ.Υ,~u :~σ Γ,~x :~τ M : τ
k ‘
Θ.Υ Γ λ{~u}[~x].M : {~σ}[~τ].τ ‘abs
k ‘
Abtsareidentifieduptoα-equivalence.
4.1 Calculatingfreevariables
Wecaneasilycalculatethevariablesfreeinanexpressionbyrecursiononitsstructure:
FV
FV(x) { {x} var
FV(Mi) {~xi (i 6 n)
FV
mvar
FV(m{~u}(M0,...,Mn)) { i6n~xi
FV(Ei) {~xi (i 6 nS)
FV
app
FV(ϑ(E0,...,En)) { i6n~xi
FV(M) {~xS
FV
FV(λ{~u}[~y].M) {~x\~y abs
Becausethisisatotalrelation,henceforthwewillwriteFV(M)for~xwhenFV(M) {~x.
4.2 Calculatingfreesymbols
WhereasthecalculationoffreevariablespivotedontheFV rule,thecalculationoffree
var
symbolswillpivotontheFS andFS rules,becausetheonlywayasymbolcanbe
app mvar
introducedisasaparametertoanoperatororasaparametertoametavariable.
6
FS
FS(x) { { } var
FS(Mi) {~ui (i 6 n)
FS
FS(m{~u}(M0,...,Mn)) {~u∪ i6n~ui mvar
FS(Ei) {~ui (i 6 n)S
FS
app
FS(ϑ(E0,...,En)) { |supp(ϑ)|∪ i6n~ui
FS(M) {~u S
FS
FS(λ{~v}[~x].M) {~u\~v abs
Becausethisisatotalrelation,henceforthwewillwriteFS(M)for~uwhenFS(M) {~u.
4.3 Renamingofsymbols
Theonlyplacethatsymbolsappearinourcalculusisasparameterstooperators(unlike
variables, symbols are not terms). Therefore, the functorial action of the operatorsheaf
canbeliftedintotermsbyrecursionontheirstructure,viaapairofjudgmentsM ρ { N
·
and E ρ { F, presupposing ρ : Υ Υ0, and Θ.Υ Γ M : τ and Θ.Υ Γ E : v
· k ‘ k ‘
respectively:
x ρ { x •var
·
ρ(~u) ~v Mi ρ { Ni (i 6 n)
≡ · mvar
m{~u}(M0,...,Mn) ρ { m{~v}(N0,...,Nn) •
·
Ei ρ { Fi (i 6 n)
· app
ϑ(E0,...,En) ρ { ϑ ρ(F0,...,Fn) •
· ·
M ρ\~u { N
· abs
λ{~u}[~x].M ρ { λ{~u}[~x].N •
·
Above,thenotationρ\~umeanstheomissionofthevariables~ufromtherenamingρ.
Becausetermsareidentifieduptoα-equivalence,therenamingjudgmentisfunctionalin
itsinput,andsowearejustifiedinwritingM ρforNwhenM ρ { N.
· ·
4.4 Substitutionofvariables
Variablesubstitutioninabtsisdefinedinductivelybyapairofjudgments,[N/x]M { M0
and[N/x]E { F:
7
x = y x # y
/ /
[N/x]y { N var1 [N/x]y { y var2
[N/x]Mi { Mi0 (i 6 n) /
[N/x]m{~u}(M0,...,Mn) { m{~u}(M00,...,Mn0 ) mvar
[N/x]Ei { Fi (i 6 n)
/
app
[N/x]ϑ(E0,...,En) { ϑ(F0,...,Fn)
x <~y ~u # FS(N) ~y # FV(N) [N/x]M { M0
/
[N/x]λ{~u}[~y].M { λ{~u}[~y].M0 abs1
x ~y ~u # FS(N) ~y # FV(N)
∈ /
[N/x]λ{~u}[~y].M { λ{~u}[~y].M abs2
Intheremainder,wewillwrite[N/x]MforM0 when[N/x]M { M0,and N~ /~x M
forthesimultaneoussubstitutionofN~ for~xinM. h i
4.5 Substitutionofmetavariables
Metavariablesaresubstitutedbyabstractions; sinceametavariablemayonlyappearin
anapplicationexpressionm{ }( ),wewillinstantiatetheabstractionatthesupplied
··· ···
parametersandarguments. Substitutionformetavariablesisdefinedinductivelybythe
judgments[E/m]M { Nand[E/m]F { F0:
/meta
[E/m]x { x var
m # n [E/n]Mi { Ni (i 6 n) /meta
[E/m]n{~u}(M0,...,Mn) { n{~u}(N0,...,Nn) mvar1
m = n M~ /~x N { N0 N0 ~u ~v { N00
· /meta
[λh{~u}[~x].iN/m]n{~v}(M~ ) { N00 mvar2
[E/m]Fi { Fi0 (i 6 n) /meta
app
[E/m]ϑ(F0,...,Fn) { ϑ(F00,...,Fn0 )
~u # FS(E) ~x # FV(E) [E/m]M { N
/meta
[E/m]λ{~u}[~x].M { λ{~u}[~x].N abs
8
Therulesformetavariablesubstitutionarereminiscentofthe“hereditarysubstitutions”
firstdevisedfortheConcurrentLogicalFrameworkbyWatkinsin[22]. Asusual,wewill
write[E/m]MforNwhen[E/m]M { N.
5 Model Theory
Let H , (I S F S)op; then we fix the functor category HS as our semantic universe,
usingtheno↓tat×iono↓ftheFrenchschool,C , SetCop. LetV (Υ Γ) , {x |Γ| | Γ(x) τ}be
τ
k ∈ ≡
calledthepresheafofvariables;additionally,wehaveaprebsheafofsymbolsS (Υ Γ) ,
τ
k
{u |Υ| | Υ(u) τ}. Lastly,wehavethebpresheafofoperatorswitharitya,O (Υ Γ) ,
a
∈ ≡ k
O (Υ)
a
5.1 Substitutionmonoidalstructures
ForanobjectP : HS,wewillusethenotationP[Γ]tomean P ;likewise,S{Υ}shall
x |Γ| Γ(x)
∈
mean S . ForapresheafA : Handasort-indexedfamilyofpresheavesP : HS,
u |Υ| Υ(u) Q
wehavea∈noperbationA P,definedasacoendinthefollowingway:
Q •
b b
(A P)(Υ Γ) , (Υ0k∆)∈HA(Υ0 ∆) S{Υ0}(Υ Γ) P[∆](Υ Γ)
• k k × k × k
Z
Intuitively,thecoendconstructionA P providesthedataofasuspendedsubstitution
•
ofanA-term’svariablesbyP-terms. Usingthis,wecandefineatensorP QforP,Q : HS
(cid:12)
asfollows:
(P Q) , P Q (τ S) b
(cid:12) τ τ• ∈
Then,V istheunittothistensor. WewillsaythatanobjectP : HS isaΣ-monoidincase
itisequippedwiththefollowingnaturaltransformationswhereνembedsvariablesintoP
andς equipsP withanoperationforsimultaneoussubstitutionsofbvariables. Furthermore,
νandς inducemapsν andςτ :
Γ Υ Γ
k
ν ς
V P P P
(cid:12)
V[Γ] νΓ P[Γ] Pτy(ΥkΓ) S{Υ} P[Γ] ςΥτkΓ Pτ
× ×
5.2 Thesignatureendofunctoranditsalgebras
ForeachsignatureΣ S,O ,wehaveanendofunctorF : HS HS,whichisdefinedas
Σ
≡ h i
follows:
b b
FΣ(X)τ , Xyτi(σ~kτ~)
ϑ∈aO(~v)τ{σ~}[Yτ~].τi∈~v
9
Then, a Σ-model is a Σ-monoid P which is equipped with an algebra α : F (P) P,
Σ
whichshallinterpretapplicationsofeachoperator.
5.3 Interpretationofterms
The metavariable, symbol and variable contexts are interpreted for a model P as an
environmentpresheafinthefollowingway:
Θ.Υ k Γ P ,  Pτy(σ~kτ~)×S{Υ}×V[Γ]
(m:{σ~}Y[τ~].τ)∈Θ
 
Then,theinte(cid:18)rpretation(cid:19)ofaterminamodelP isamapfromitsenvironmenttoP:
Θ.Υ Γ M : τ : Θ.Υ Γ P
k ‘ P k P τ
Variablesareinterpretedbythemap Θ.Υ Γ x : τ whichprojectsthemfromthe
k ‘ P
environmentandembe(cid:18)ds them into the m(cid:19)ode(cid:18)l. Metava(cid:19)riables are resolvedby the map
Θ.Υ Γ m{~u}(M~ ) : τ (where Θ m : {~σ}[~τ].τ) which projects their interpretation
(cid:18) (cid:19)
k ‘ P 3
fromtheenvironmentandinstantiatesitviasubstitution:
(cid:20) (cid:21)
Θ.Υ Γ m{u~}(M~ ):τ ~Θ.Υ Γ x:τ(cid:127)
P k ‘ P Θ.Υ Γ k ‘ P P
τ k P τ
Pτy(σ~kτ~ς)σ~τkτ~ S{σ~} P[τ~(cid:20)] hπmπ1,φπ(cid:21)2,ψi (cid:18) (cid:19) νΓπ3 P[Γπ]x
× ×
P[τ~] ψ,h~Θ.ΥkΓ‘M:τ(cid:127)Pi(M,τ)∈(M~,τ~) Θ.Υ Γ S{Υ} φ,hπuiu∈u~ S{σ~}
k P
Interpretationofoperatorapplicationsisthemostcomplicated. Recallthat,unlikein
standardtreatmentsofuniversalalgebra,ourope(cid:18)ratorsare(cid:19)indexedbysymbolcollections;
therefore, operators must pass through suitable renamings in order to be used in the
interpretation. LetusbeginbyconstructingforeachoperatorΥ(cid:13)ϑ : athemorphism~ϑ(cid:127)
P
whichshallrenametheparametersoftheoperatorusingtheenvironment:
~ϑ(cid:127)P
Θ.Υ Γ S{Υ} O
k P π2 Oa(π(−))(ϑ) a
WewillproceedusingtheF -algebraα,asfollows,bycomposingitwithamorphism
(cid:18) (cid:19)Σ
β from the environment into the signature endofunctor, which interprets the syntax of
operatorapplications:
Θ.Υ Γ β F (P) ατ P
k P Σ τ τ
10
(cid:18) (cid:19)