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An Institution-Independent Proof of the Beth Definability Theorem
Marc Aiguier, Fabrice Barbier
To cite this version:
Marc Aiguier, Fabrice Barbier. An Institution-Independent Proof of the Beth Definability Theorem.
Studia Logica, Springer Verlag (Germany), 2007, 85 (3), pp.333–359. �hal-00341973�
Denability Theorem
Mar Aiguierand FabrieBarbier
1
Universite d'
Evry, LaMI CNRS UMR 8042,
523 pl. des Terrasses F-91000
Evry
faiguier,fbarbierglami.univ-e vry. fr
fax number: (+33) 1 60 87 37 89
June 30,2006
Abstrat
Afewresultsgeneralisingwell-knownonventionalmodeltheoryones
havebeenobtainedintheframeworkofinstitutionstheselasttwodeades
(e.g. Craig interpolation, ultraprodut, elementary diagrams). In this
paper, we propose a generalised institution-independent version of the
Bethdenabilitytheorem.
Keywords: Bethdenability,Craiginterpolation,institutions,institutionmorphisms
andomorphisms,inlusiveategory
1 Introdution
A few results generalising well-known onventional model theory ones have
been obtained in the framework of institutions [15℄ these last two deades
(e.g.[8,18,21,16℄). Inordertoontinuethisgeneralisationworkofimportant
resultsfrom onventionalmodeltheory, wepresentin thispaperageneralised
institution-independentversionoftheBethdenabilitytheorem.
The theory of institutions extends Barwise's abstrat model theory [3℄ to
omputersiene. Thisextensionismanifold:
Institutions inlude both notions of signature (related to the notion of
softwareinterfae)andsignaturemorphism(tostruturesoftwares).
Sentenesare onlydened asmembersof aset. This meanssets ofsen-
tenes are neitherneessarily losed under the lassilogial symbols in
f:;^;_;);9;8gnorrestritedtothem. Thisallowsforalargerfamilyof
logistobetakenintoaountsuhasHornlauselogiormodallogis.
signaturemorphismandalledsatisfation ondition.
The original goals of institutions were to generaliseresults both in omputer
sieneand modeltheory. However,it ismainly in omputersienethat this
taskhasbeenaomplished. Despiteofits importaneforspeiationtheory,
theproblemofgeneralisingonventionalmodeltheoryresultswithintheframe-
work ofinstitutions haveonly been takled bysomeisolated works. Asfar as
weknow,thesearetheones:
Tarleki's works [20,21℄ whih generalisein a partiularform of institu-
tionsome lassialgebrairesultssuh astheBirkho theorem (equiva-
lene betweenequationaltheories andvarieties),the MKinsleytheorem
(equivalenebetweenuniversalHorntheoriesandquasi-varieties)andthe
Mal'evtheorem(existeneofinitialtermmodelsinuniversalHorntheo-
ries),
SalibraandSollo's works[17, 18℄whih dealwithrelationshipsbetween
Craig-style interpolation, ompatnessand Loweinem-Skolem properties
inarelaxedformofinstitutionsalled pre-institutions,
Diaonesu'sreentworks[6,7,8,9℄whihthrowthebasisforarealstudy
ofmodel theorywithin theframeworkofinstitutions.
Among all the onventional model theory results whih are of interestfor
omputersieneistheCraiginterpolationtheorem. Restatedinordertobetter
suitomputersiene's needsasapropertyoversetsofformulaeandanykind
ofsignaturemorphisms,ithasbeenshowntobestronglylinkedtoompleteness
ofstruturedinferenesystems[4, 5℄andto someaspet ofmodularity(faith-
fulness)[10,11, 24, 25℄. It hasreentlybeendiretly provedin theframework
ofBirkhoinstitutions 1
byR.Diaonesu[8℄. UnlikeaformerresultbyA.Sal-
ibraand G.Sollo [18℄, R. Diaonesu'sresult doesn'trequirenegationand is
thussuitableforlogiswithoutitsuhasequationallogiandHornlauselogi.
Inonventionalmodeltheory,animportantonsequeneoftheCraiginter-
polation theorem is theBeth denability theorem. This theorem providesan
answertothequestiontowhatextentimpliitdenitionsanbemadeexpliit.
Forexample,when onewantstoformalizeatheory,theveryrststepistox
thelanguage, that is deidingwhih notionsare primitives, theothers having
tobedenedfrom them. Buthowanonehekuselesssymbolshaven'tbeen
introdued? Thisisaproblemofruialimportaneinspeiationtheoryand
artiialintelligeneandtheBethdenabilitytheorem isatoolto solveit.
Considering the assets of the institutional framework for model theory (ab-
stratness,logi-independene)westudyinthispaperbothdenabilitynotions
(impliit and expliit) from aninstitution-independent point of view. We ad-
dresstheirgeneralisationfromtwodierentangles.
1
Birkhoinstitutionsareapartiularformofinstitutionsmodellassesofwhiharelosed
undersomealgebraioperationssuhasvarietiesandquasi-varietiesare.)
signaturemorphisms. Tothisend, wemakeuseof thenotionofstrongly
inlusive ategory [10℄ in order to give a ategorial denition of both
set-theoretialnotionsofinlusionanddierene.
2. Sinenon-injetivemorphismsareofgreatimportaneinspeiationthe-
ory,wethengeneralisebothdenabilitynotionsto anykindofsignature
morphisms.
WealsostudythepreservationofBethdenabilitytheoremthroughinstitution
morphismsandomorphisms. Thisallowsinheritaneofthistheoremfromone
logi to another, both ones being presented as institutions and linked by an
institutionmorphism.
This paper is strutured asfollows: Setion 2 reviews some onepts, no-
tationsand terminologyaboutinstitutionsand institution morphismsand o-
morphisms whih are used by this work. Setion 3 reviews the institution-
independentmodeltheoretioneptofCraiginterpolation. Takinginspiration
fromasimilarresultoninstitutiontransformations[18℄,weformulatepreserva-
tiontheorems for Craiginterpolation propertythroughinstitution morphisms
andomorphisms. Setion 4formulates ageneralinstitution-independentver-
sion of the Beth denability theorem and proves it as a onsequene of the
Craiginterpolationproperty. AsforCraiginterpolationwedevelopapreserva-
tiontheoremforBeth denabilitythroughinstitutionmorphisms.
2 Institutions
Intuitively, the theory of institutions abstrats the semantial part of logial
systems aording to the needs of software speiation in whih hanges of
signatures our frequently. In this setion wereview and dene someof the
basinotionsoninstitutionsin usein thispaper.
2.1 Basi denitions and examples
Aninstitution[15℄onsistsofaategoryofsignaturessuhthatassoiatedwith
eah signaturearesentenes,modelsand arelationshipof satisfationthat, in
a ertain sense, is invariant under hange of signature. More preisely, this
meansthat ahange of signature (by asignature morphism)indues \onsis-
tent"hangesinsentenesandmodelsinasensemadepreisebythe\Satisfa-
tionCondition"inDenition2.1below. ThisgoesastepbeyondTarski'slassi
\semantidenition of truth"[23℄ and alsogeneralises Barwise's\Translation
Axiom"[3℄. Moreover,it is fundamental that sentenes translatein the same
diretion as thehange of notation, whereas models translate in the opposite
diretion(thinkofsignatureenrihmentandmodelredution). Thisistherea-
sonfor the funtor Mod in Denition 2.1 below to beontravariant. For the
sake of generalisation, signatures are simply dened as objets of a ategory
ontingeniessuhasindutivedenitionofsentenesarenotonsidered. Simi-
larly,modelsaresimplyseenasobjetsofaategory,i.e.nopartiularstruture
isimposed onthem. Finally,propertiessatisedbyagivenlassofmodelsare
haraterizedthroughabinaryrelationbetweenmodelsandsentenesofagiven
signature. Moreformally,aninstitutionisdenedasfollows:
Denition2.1 (Institution) An institutionI =(Sig;Sen;Mod;j=)onsists
of
aategorySig,objetsof whih are alledsignatures,
a funtor Sen : Sig ! Set giving for eah signature a set, elements of
whih arealledsentenes,
a ontravariant funtor Mod : Sig op
! Cat giving for eah signature a
ategory,objetsandarrowsofwhiharealled-modelsand-morphisms
respetively, and
ajSigj-indexedfamily ofrelations j=
jMod()jSen() alledsatis-
fationrelation,
suhthatthe followingproperty holds:
8:! 0
; 8M 0
2jMod(
0
)j; 8'2Sen(),
M 0
j=
0
Sen()('),Mod()(M 0
)j=
'
Example2.2 The following examples of institutions are of partiular impor-
taneforomputersiene. Manyotherexamplesanbefoundintheliterature
(e.g. [15,22 ℄).
Propositional Logi(PL) Signatures and signature morphisms are sets of
propositionalvariables andfuntionsbetween themrespetively.
Given asignature , the set of -sentenes is the least set of sentenes
nitely builtoverpropositional variables inand Boolean onnetivesin
f:;_g. Given a signature morphism : ! 0
, Sen() translates -
formulae to 0
-formulaeby renamingpropositionalvariables aording to
.
Givenasignature,the ategoryof-modelsistheategoryofmappings
: ! f0;1g 2
with identities as morphisms. Given a signature mor-
phism : ! 0
, the forgetful funtor Mod() maps a 0
-model 0
to
the -model = 0
Æ.
Finally, satisfation istheusualpropositional satisfation.
Many-sorted FirstOrder Logi (FOL) Signaturesaretriples(S;F;P)where
S isasetofsorts,andF andP aresetsoffuntion andprediate names
2
f0;1garetheusualtruth-values.
respetively, both with arities in S S and S respetively. Signature
morphisms :(S;F;P)!(S 0
;F 0
;P 0
)onsist of threefuntions between
sets of sorts,sets of funtionsandsets of prediates respetively, the last
twopreservingarities.
Given a signature =(S;F;P), the -atoms are of two possible forms:
t
1
= t
2
where t
1
;t
2 2 T
F (X)
s 4
(s 2 S), and p(t
1
;:::;t
n
) where p :
s
1
:::s
n
2 P and t
i 2 T
F (X)
si
(1 i n, s
i
2 S). The set of
-sentenes is the least set of formulae built over the set of -atoms by
nitely applying Booleanonnetives inf:;_g andthequantier 8.
Given asignature=(S;F;P), a-modelM isafamilyM=(M
s )
s2S
of sets (one for every s 2 S), eah one equipped with a funtion f M
:
M
s1
:::M
sn
! M
s
for every f : s
1
:::s
n
! s 2 F and with
a n-ary relation p M
M
s1
:::M
sn
for every p:s
1
:::s
n 2P.
Given asignaturemorphism :=(S;F;P)! 0
=(S 0
;F 0
;P 0
)and a
0
-modelM 0
,Mod()(M 0
)isthe -model Mdenedfor everys2S by
M
s
=M 0
s
,andfor everyfuntionnamef 2F andprediate namep2P,
byf M
=(f) M
0
andp M
=(p) M
0
.
Finally, satisfation istheusualrst-ordersatisfation.
HornClause Logi(HCL) An universalHornsenteneforasignaturein
FOL is a-senteneof the form ) where isa nite onjuntion
of-atomsandisa-atom. TheinstitutionofHornlauselogiisthe
sub-institution of FOL whose signatures and models are those of FOL
andsentenesare restritedtothe universalHorn sentenes.
EquationalLogi (EQL) Analgebraisignature(S;F)simplyisaFOLsig-
naturewithoutprediatesymbols. Theinstitutionofequationallogiisthe
sub-institution of FOL whose signatures andmodels are algebrai signa-
turesandalgebras respetively,andsentenesarerestritedtoequations.
RewritingLogi (RWL) Given an algebrai signature = (S;F), -
sentenes areformulae of the form': t
1
!t 0
1
^:::^t
n
!t 0
n
)t !t 0
where t
i
;t 0
i 2 T
F (X)
s
i
(1 i n, s
i
2S) andt;t 0
2 T
F (X)
s
(s 2 S).
Models of rewriting logi are preorder models, i.e. given a signature
= (S;F), Mod() is the ategory of -algebras A suh that for ev-
ery s2S,A
s
isequippedwithapreorder. Hene, Aj='if andonlyif
for every variable interpretation :X !A, ifeah (t
i )
A
(t 0
i )
A
then
(t) A
(t 0
) A
where A
:T
F
(A)!A isthe mapping indutivelydened
by: f(t
1
;:::;t
n )
A
=f A
(t A
1
;:::;t A
n ).
Modal FirstOrder Logi with global satisfation (MFOL) 5
The at-
egory ofsignatures istheategory ofFOL signatures.
Given a FOL signature = (S;F;P), -axioms are of the form
3
S +
isthe setof allnon-emptysequenesofelementsinSand S
=S +
[fgwhere
denotestheemptysequene.
4
T
F (X)
s
isthetermalgebraofsortsbuiltoverF withsortedvariablesinagivensetX.
5
aka.quantiedmodallogiK.
1 n
overthesetof-axiomsbynitelyapplying Booleanonnetivesinf:;_g
andthe quantier8 andthe modality.
Given asignature =(S;F;P), a-model (W;R ), alledKripke frame,
onsists of a family W = (W i
)
i2I
of -models in FOL (the possi-
ble worlds) suh that 6
W i
s
= W j
s
for every i;j 2 I and s 2 S,
and an\aessibility" relation R I I. Given a signature mor-
phism : (S;F;P) ! (S 0
;F 0
;P 0
) and a (S 0
;F 0
;P 0
)-model (W 0
;R 0
),
Mod()((W 0
;R 0
)) is the (S;F;P)-model (W;R ) dened for every i 2 I
by W i
= Mod()(W i
) and by R = R 0
. A -sentene ' is said to be
satised by a-model (W;R ), noted(W;R )j=
', if for every i2 I we
have (W;R )j= i
', wherej= i
isindutively denedon the strutureof '
asfollows:
atoms,Boolean onnetives andquantierarehandledasinFOL,
(W;R )j= i
' when(W;R )j= j
'for every j2I suhthat iR j.
Modal propositional logi (MPL)is the sub-institutionof MFOL whose
signatures are restrited to empty sets of sorts and funtion names and
only 0-aryprediate names.
Modal FirstOrder Logi with loal satisfation (LMFOL) Signatures
and sentenes are MFOL signatures and MFOL sentenes. Given
a signature = (S;F;P), a -model is a pointed Kripke frame
(W = (W i
)
i2I
;R ;W j
) where j 2 I. The satisfation of a -sentene
' by a -model (W;R ;W j
), noted (W;R ;W j
) j=
', is dened by:
(W;R ;W j
)j=
',(W;R )j= j
'.
LMFOL with innitedisjuntion and onjuntion (LIMFOL) This
institutionextendsLMFOLtosentenesoftheform V
and W
where
is a set (possibly innite) of -sentenes. Given a pointed Kripke
frame(W;R ;W j
),
(W;R ;W j
)j=
V
()8'2;(W;R ;W j
)j=
'
(W;R ;W j
)j=
W
()9'2;(W;R ;W j
)j=
'
2.2 Theories in institutions
Letus nowonsideraxedbut arbitraryinstitutionI=(Sig;Sen;Mod;j=).
Notation2.3 Let2jSigj beasignatureand T beasetof -sentenes.
Mod(T)isthefullsub-ategoryofMod()whoseobjetsaremodelsofT,
T
= f' 2 Sen()=8M 2 jMod(T)j; M j=
'g is the set of so-alled
semantionsequenesofT.
6
Intheliterature,Kripkeframessatisfyingsuhapropertyaresaidwithonstantdomains.
onlyif T =T
.
Denition2.5 (Theategory of theories) A theory morphism from a -
theory T to a 0
-theory T 0
is any signature morphism : ! 0
suh that
Sen()(T)T 0
.
Let us note Th
I
the ategory whose objets and morphisms are theories and
theory morphisms respetively.
Thefollowingpropositionisadiret onsequeneofthesatisfationondition.
Proposition 2.6 Given a -theory T and a 0
-theory T 0
suhthat there is a
theory morphism:T !T 0
,the funtorMod():Mod(
0
)!Mod() anbe
restritedtoMod():Mod(T 0
)!Mod(T).
Corollary2.7 The4-uple I
T
=(Sig
T
;Sen
T
;Mod
T
;j=
T
)where:
Sig
T
=Th
I ,
foreverytheoryT overasignature,Sen
T
(T)=Sen()andMod
T (T)=
Mod(T),and
j=
T
=j=.
isaninstitution.
2.3 Institution morphisms and omorphisms
Manydierent kindofmorphism anbe dened denotingdierentkindof re-
lationshipbetweentwoinstitutions. Theoriginaloneintroduedin[15℄denes
aforgetfuloperationfrom a\riher"institution I to a\poorer"one I 0
. Intu-
itively,itshowshowI isbuiltoverI 0
.
Denition2.8 (Institutionmorphism) LetI =(Sig;Sen;Mod;j=)andI 0
=
(Sig 0
;Sen 0
;Mod 0
;j= 0
)betwoinstitutions. Aninstitution-morphism=(;;):
I!I 0
onsistsof
afuntor:Sig!Sig 0
,
a natural transformation :Sen 0
Æ )Sen, i.e. for every 2jSigj a
funtion
: Sen
0
(()) !Sen() suh that for every signature mor-
phism :
1
!
2
inSig the followingdiagramommutes,
Sen 0
((
1 ))
1
! Sen(
1 )
Sen 0
(())
?
?
y
?
?
y Sen()
Sen 0
((
2
)) !
2
Sen(
2 )
a natural transformation : Mod ) Mod Æ , i.e. for every 2
jSigjafuntor
:Mod()!Mod 0
(())suhthat foreverysignature
morphism :
1
!
2
inSig the following diagramommutes,
Mod(
2 )
2
! Mod 0
((
2 ))
Mod()
?
?
y
?
?
y Mod
0
( op
())
Mod(
1
) !
1
Mod 0
((
1 ))
suhthatthe followingsatisfation property holds:
82jSigj; 8M2jMod()j; 8' 0
2Sen 0
(())
Mj=
('
0
)()
(M)j=
() '
0
Example2.9 Theinstitutionmorphism=(;;)fromFOLtoEQLmaps
anyFOLsignature(S;F;P)tothe orresponding algebraione (S;F), regards
anyset of equationsas asetof rst-order sentenes over(S;F;;), andregards
any(S;F;P)-modelasa(S;F)-algebrabyforgettingtheinterpretationsofpred-
iate names in P. Itis easy toshow that the satisfation property holds. The
institution morphism from HCL to EQL is dened as the previous one ex-
ept that it regards any equation as a onditional equational formula without
premises. Finally, the institution morphism from FOL to HCL is obvious.
Indeed,itmaps any FOL signature(S;F;P)andany (S;F;P)-model tothem-
selves, and regards Horn sentenesover (S;F;P) as rst-order sentenesover
the samesignature.
7
No institution morphism an denote a forgetful operation from FOL to
either LMFOL nor MFOL. However, we an show how both LMFOL and
MFOL are \embedded" in FOL. Indeed, eah LMFOL signature (S;F;P)
anbetransformedintotheFOL signature(S;F;P)denedby:
S=S[findg
F =ff:inds
1
:::s
n
!s=f :s
1
:::s
n
!s2Fg[fi:!indg
P =fr:inds
1
:::s
n
=r:s
1
:::s
n
2Pg[fR:indindg
LetX beaset ofvariables over(S;F;P),and letx2X
ind
[fig. Wean
dene FO
x : Sen
LMFOL
()T
F
(X) ! Sen
FOL
( )T
F
(X) indutivelyon
termsandformulaestrutureasfollows:
f(t
1
;:::;t
n
)7!f(x;t
1
;:::;t
n )
r(t
1
;:::;t
n
)7!r(x;t
1
;:::;t
n )
7
AHornsentenean beseen as a nitedisjuntion ofrst-order formulaeallof them
beingnegationsofatomsexeptforthelastonewhihisanatom.
