A New Foundation for Finitary Corecursion

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by  Stefan Milius

A New Foundation for Finitary Corecursion
The Locally FiniteFixpointand itsProperties
StefanMilius1,⋆,DirkPattinson2,andThorstenWißmann1,⋆
1 Friedrich-Alexander-Universita¨tErlangen-Nu¨rnberg
2 TheAustralianNationalUniversity
Abstract. This paper contributes to a theory of the behaviour of “finite-state”
systemsthatisgenericinthesystemtype.Weproposethatsuchsystemsaremod-
eledascoalgebraswithafinitelygeneratedcarrierforanendofunctoronalocally
finitelypresentablecategory.Theirbehaviourgivesrisetoanewfixpointofthe
coalgebraictypefunctorcalledlocallyfinitefixpoint(LFF).Weprovethatifthe
givenendofunctorpreservesmonomorphismsthentheLFFalwaysexistsandisa
subcoalgebraofthefinalcoalgebra(unliketherationalfixpointpreviouslystudied
byAda´mek,MiliusandVelebil).Moreover,weshowthattheLFFischaracterized
by two universal properties: 1. as the final locally finitely generated coalgebra,
and2.astheinitialfg-iterativealgebra.AsinstancesoftheLFFwefirstobtainthe
knowninstancesoftherationalfixpoint,e.g.regularlanguages,rationalstreams
andformalpower-series,regulartreesetc.Andweobtainanumberofnewexam-
ples,e.g.(realtimedeterministicresp.non-deterministic)context-freelanguages,
constructivelyS-algebraicformalpower-series(andanyotherinstanceofthegen-
eralizedpowersetconstructionbySilva,Bonchi,Bonsangue,andRutten)andthe
monadofCourcelle’salgebraictrees.
1 Introduction
Coalgebrascapturemanytypesofstatebasedsystemwithinauniformandmathemati-
callyrichframework[39].Oneoutstandingfeatureofthegeneraltheoryisfinalseman-
ticswhichgivesafullyabstractaccountofsystembehaviour.Forexample,coalgebraic
modellingofdeterministicautomata(withoutafinitenessrestrictiononstatesets)yields
the set of all formallanguagesas a final model,and restricting to finite automataone
preciselyobtainstheregularlanguages[38].Thiscorrespondencehasbeengeneralized
tolocallyfinitelypresentablecategories[8,20],wherefinitelypresentableobjectsplay
the role of finite sets, leading to the notion of rational fixpoint that provides final se-
manticstoallmodelswithfinitelypresentablecarrier[31].Itisknownthattherational
fixpointisfullyabstract(identifiesallbehaviourallyequivalentstates)aslongasfinitely
presentableobjectsagreewithfinitelygeneratedobjectsinthebasecategory[12,Propo-
sition3.12].Whilethisisthecaseinsomecategories(e.g.sets, posets,graphs,vector
spaces, commutative monoids), it is currently unknown in other base categories that
are used in the construction of system models, for example in idempotent semirings
(usedinthetreatmentofcontext-freegrammars[43]),inalgebrasforthestackmonad
⋆SupportedbyDeutscheForschungsgemeinschaft(DFG)underprojectMI717/5-1
2 S.Milius,D.PattinsonandT.Wißmann
(usedformodellingconfigurationsofstackmachines[23]);oritevenfails,forexample
in the category of finitary monads on sets (used in the categorical study of algebraic
trees[7]),orEilenberg-Moorecategoriesforamonadingeneral(thetargetcategoryof
generalizeddeterminization[41],inwhichtheaboveexampleslive).Coalgebrasovera
categoryofEilenberg-MoorealgebrasoverSetinparticularprovideaparadigmaticset-
ting:automatathatdescribelanguagesbeyondtheregularlanguagesconsistofafinite
state set, buttheirtransitionsproducesideeffectssuchasthemanipulationof astack.
Thesecanbedescribedbyamonad,sothatthe(infinite)setofsystemstates(machine
states plusstackcontent)isdescribedbya freealgebra(forthatmonad)thatisgener-
atedbythefinitesetofmachinestates.Thisisformalizedbythegeneralizedpowerset
construction[41]andinteractsnicelywiththecoalgebraicframeworkwepresent.
Technically,the shortcomingof the rationalfixpointis due to the fact thatfinitely
presentableobjectsarenotclosedunderquotients,sothattherationalfixpointitselfmay
failtobeasubcoalgebraofthefinalcoalgebraandsoidentifiestoolittlebehaviour.The
mainconceptualcontributionofthispaperistheinsightthatalsoincaseswherefinitely
presentableandfinitelygenerateddonotagree,thelocallyfinitefixpointprovidesafully
abstract model of finitely generated behaviour. We give a construction of the locally
finite fixpoint, and support our claim both by general results and concrete examples:
we show that under mild assumptions, the locally finite fixpointalways exists, and is
indeed a subcoalgebraof the finalcoalgebra.Moreover,we give a characterizationof
thelocallyfinitefixpointastheinitialiterativealgebra.Wetheninstantiateourresults
toseveralscenariosstudiedintheliterature.
First, we show that the locally finite fixpoint is universal (and fully abstract) for
theclassofsystemsproducedbythegeneralizedpowersetconstructionoverSet:every
determinizedfinite-state system inducesa uniquehomomorphismto the locally finite
fixpoint,andthelattercontainspreciselythefinite-statebehaviours.
Appliedtothecoalgebraictreatmentofcontext-freelanguages,weshowthatthelo-
callyfinitefixpointyieldspreciselythecontext-freelanguages,andreal-timedetermin-
istic context-freelanguages,respectively,when modelled using algebrasfor the stack
monad of [23]. For context-free languages weighted in a semiring S, or equivalently
for constructively S-algebraic power series [36], the locally finite fixpoint comprises
preciselythose,byphrasingtheresultsofWinteretal.[44]intermsofthegeneralized
powersetconstruction.Ourlastexampleshowstheapplicabilityofourresultsbeyond
categories of Eilenberg-Moore algebras over Set, and we characterize the monad of
Courcelle’s algebraic trees over a signature [16, 7] as the locally finite fixpointof an
associatedfunctor(onacategoryofmonads),solvinganopenproblemof[7].
Theworkpresentedhereisbasedonthethirdauthor’smasterthesisin[45].Most
proofsareomitted;theycanbefoundintheappendix.
2 Preliminaries andNotation
Locallyfinitelypresentablecategories.A filteredcolimit isthecolimitofadiagram
D → C where D is filtered (everyfinite subdiagramhas a coconein D) anddirected
if D is additionally a poset. Finitary functors preserve filtered (equivalently directed)
colimits. Objects C ∈ C are finitely presentable (fp) if the hom-functorC(C,−) pre-
TheLocallyFiniteFixpointanditsProperties 3
serves filtered (equivalently directed) colimits, and finitely generated (fg) if C(C,−)
preservesdirectedcolimitsofmonos(i.e.colimitsofdirecteddiagramswhereallcon-
nectingmorphismsaremonic).Clearlyanyfpobjectisfg,butnotviceversa.Also,fg
objectsareclosedunderstrongepis(quotients)whichfailsforfpobjectsingeneral.A
cocomplete category is locally finitely presentable (lfp) if the full subcategory C of
fp
finitelypresentableobjectsisessentiallysmall,i.e.isuptoisomorphismonlyaset,and
every object C ∈ C is a filtered colimit of a diagram in C . We refer to [20, 8] for
fp
furtherdetails.
Itis wellknownthatthe categoriesofsets, posets andgraphsare lfp with finitely
presentableobjectspreciselythefinitesets,posets,graphs,respectively.Thecategoryof
vectorspacesislfpwithfinite-dimensionalspacesbeingfp.Everyfinitaryvarietyislfp
(i.e.anequationalclassofalgebrasinducedbyfinite-arityoperationsorequivalentlythe
Eilenberg-MoorecategoryforafinitarySet-Monad,seeSection4.1later).Thefinitely
generatedobjectsarethefinitelygeneratedalgebras,andfinitelypresentableobjectsare
algebrasspecifiedbyfinitelymanygeneratorsandrelations.Thisincludesthecategories
ofgroups,monoids,(idempotent)semirings,semi-modules,etc.Everylfpcategoryhas
mono/strongepifactorization[8,Proposition1.16],i.e.everyf factorsasf =m·ewith
mmono(denotedby֌),estrongepi(denotedby։),andwecallthedomainIm(f)
ofetheimageoff.Anystrongepiehasthediagonalfill-inproperty,i.e.m·g =h·e
withmmonoandestrongepigivesauniquedsuchthatm·d=handg =d·e.
Coalgebras. If H : C → C is an endofunctor, H-coalgebras are pairs (C,c) with
c : C → HC, and C is the carrier of (C,c). Homomorphismsf : (C,c) → (D,d)
are maps f : C → D such that Hf · c = d · f. This gives a category denoted
by CoalgH. If its final object exists then this final H-coalgebra (νH,τ) represents a
canonical domain of behaviours of H-typed systems, and induces for each (C,c) a
unique homomorphism,denoted by c†, giving semantics to the system (C,c). The fi-
nal coalgebra always exists provided C is lfp and H is finitary. The forgetfulfunctor
CoalgH →Ccreatescolimitsandreflectsmonosandepis.Amorphismf inCoalgH is
mono-carried(resp.epi-carried)iftheunderlyingmorphisminC ismonic(resp.epic).
Strong epi/mono factorizations lift from C to CoalgH whenever H preserves monos
yieldingepi-carried/mono-carriedfactorizations.Adirectedunionofcoalgebrasisthe
colimit of a directed diagram in CoalgH where all connecting morphisms are mono-
carried.
The Rational Fixpoint. For C lfp and H : C → C finitary let Coalg H denote the
fp
full subcategoryof CoalgH of coalgebraswith fp carrier, and Coalg H the full sub-
lfp
categoryof CoalgH of coalgebrasthatarise as filtered colimits of coalgebraswith fp
carrier[31,CorollaryIII.13].ThecoalgebrasinCoalg H arecalledlfpcoalgebrasand
lfp
forC =Setthosearepreciselythelocallyfinitecoalgebras(i.e.thosecoalgebraswhere
everyelementiscontainedinafinitesubcoalgebra).Thefinallfpcoalgebraexistsand
isthecolimitoftheinclusionCoalg H ֒→ CoalgH,anditisafixpointofH (see[6])
fp
calledtherationalfixpointofH.Herearesomeexamples:therationalfixpointofapoly-
nomialsetfunctorassociatedtoafinitarysignatureΣ isthesetofrationalΣ-trees[6],
i.e. finite and infinite Σ-treeshaving,up to isomorphism,finitely manysubtreesonly,
andoneobtainsrationalweightedlanguagesforNoetheriansemiringsS fora functor
onthecategoryofS-modules[12],andrationalλ-treesforafunctoronthecategoryof
4 S.Milius,D.PattinsonandT.Wißmann
presheavesonfinitesets[2]orforarelatedfunctoronnominalsets[34].Iftheclasses
of fp and fg objects coincide in C, then the rational fixpoint is a subcoalgebra of the
finalcoalgebra[12,Theorem3.12].Thisisthecase intheaboveexamples,butnotin
general,see[12,Example3.15]foraconcreteexamplewheretherationalfixpointdoes
not identifybehaviourallyequivalentstates. Conversely,even if the classes differ,the
rationalfixpointcanbeasubcoalgebra,e.g.foranyconstantfunctor.
Iterative Algebras. If H : C → C is an endofunctor,an H-algebra (A,a : HA →
A) is iterative if every flat equation morphism e : X → HX + A where X is an
fp object has a unique solution, i.e. if there exists a unique e† : X → A such that
e† = [a,id ]·(He†+id )·e.Therationalfixpointisalsocharacterizedastheinitial
A A
iterative algebra [6] and is the starting point of the coalgebraic approach to Elgot’s
iterativetheories[18]andtotheiterationtheoriesofBloomandE´sik[11,6,3,4].
3 The LocallyFiniteFixpoint
Thelocallyfinitefixpointcanbecharacterizedsimilarlytotherationalfixpoint,butwith
respecttocoalgebraswithfinitelygenerated(notfinitelypresentable)carrier.Weshow
thatthelocallyfinitefixpointalwaysexists,andisasubcoalgebraofthefinalcoalgebra,
i.e. identifies all behaviourally equivalent states. As a consequence, the locally finite
fixpointprovidesafullyabstractnotionoffinitelygeneratedbehaviour.Fromnowon,
werelyonthefollowing:
Assumption3.1. ThroughouttherestofthepaperweassumethatC isanlfpcategory
andthatH :C →C isfinitaryandpreservesmonomorphisms.
As forthe rationalfixpoint,we denotethefullsubcategoryofCoalgH comprisingall
coalgebraswith finitely generatedcarrier by Coalg H and have the followingnotion
fg
oflocallyfinitelygeneratedcoalgebra.
x
Definition3.2. AcoalgebraX −→ HX iscalled locallyfinitelygenerated(lfg)iffor
all f : S → X with S finitely generated, there exist a coalgebra p : P → HP in
Coalg H,acoalgebramorphismh:(P,p)→(X,x)andsomef′ :S →P suchthat
fg
h·f′ =f.Coalg H ⊆CoalgH denotesthefullsubcategoryoflfgcoalgebras.
lfg
Equivalently, one can characterize lfg coalgebras in terms of subobjects and subcoal-
gebras, making it a generalization of of local finiteness in Set, i.e. the property of a
coalgebrathateveryelementiscontainedinafinitesubcoalgebra.
x f
Lemma3.3. X −→HX isanlfgcoalgebraiffforallfgsubobjectsS X,thereexist
asubcoalgebrah:(P,p)֌(X,x)andamonof′ :S ֌P withh·f′ =f,i.e.S is
asubobjectofP.
Proof. (⇒) Given some mono f : S ֌ X, factor the induced h into some strong
epi-carriedandmono-carriedhomomorphismsandusethatfgobjectsareclosedunder
strongepis.(⇐)Factorf :S →X intoanepiandamonog :Im(f)֌X andusethe
diagonalfill-inpropertyforg. ⊓⊔
TheLocallyFiniteFixpointanditsProperties 5
Evidentlyall coalgebraswith finitely generatedcarriersare lfg. Moreover,lfg coalge-
brasarepreciselythefilteredcolimitsofcoalgebrasfromCoalg H.
fg
Proposition3.4. EveryfilteredcolimitofcoalgebrasfromCoalg H islfg.
fg
Proof (Sketch;forthefullproofseetheappendix).Onefirstprovesthatdirectedunions
ofcoalgebrasfromCoalg H arelfg.Nowgivenafilteredcolimitc : X → C where
fg i i
X arecoalgebrasinCoalg H,oneepi-monofactorizeseverycolimitinjection:c =
i fg i
( Xi ei Ti mi C ).UsingthediagonalizationofthefactorizationoneseesthattheTi
formadirecteddiagramofsubobjectsofC.FurthermoreC isthedirectedunionofthe
T andthereforeanlfgcoalgebraasdesired. ⊓⊔
i
Proposition3.5. Everylfgcoalgebra(X,x)isadirectedcolimitofitssubcoalgebras
fromCoalg H.
fg
Proof. Recallfrom[8,ProofIofTheorem1.70]thatX isthecolimitofthediagramof
allitsfinitelygeneratedsubobjects.Nowthesubdiagramgivenbyallsubcoalgebrasof
X iscofinal.Indeed,thisfollowsdirectlyfromthefactthat(X,x)isanlfgcoalgebra:
for every subobject S ֌ X, S fg, we have a subcoalgebra of (X,x) in Coalg H
fg
containingS. ⊓⊔
Corollary3.6. The lfg coalgebras are precisely the filtered colimits, or equivalently
directedunions,ofcoalgebraswithfgcarrier.
Asaconsequence,acoalgebraisfinalinCoalg F ifthereisauniquemorphismfrom
lfg
everycoalgebrawithfinitelygeneratedcarrier.
Proposition3.7. AnlfgcoalgebraLisfinalinCoalg H iffforeveryforeverycoalge-
lfg
braX inCoalg H thereexistsauniquecoalgebramorphismfromX toL.
fg
The proof is analogousto [31, Theorem 3.14];the full argumentcan be found in the
Appendix.CocompletenessofC ensuresthatthefinallfgcoalgebraalwaysexists.
Theorem3.8. ThecategoryCoalg H hasafinalobject,andthefinallfgcoalgebrais
lfg
thecolimitoftheinclusionCoalg H ֒→Coalg H.
fg lfg
Proof. By Corollary3.6, the colimit of the inclusion Coalg H ֒→ Coalg H is the
fg lfg
sameasthecolimitoftheentireCoalg H.Andthelatterisclearlythefinalobjectof
lfg
Coalg H. ⊓⊔
lfg
Thistheoremprovidesaconstructionofthefinallfgcoalgebracollectingpreciselythe
behavioursofthe coalgebraswithfgcarriers.Inthefollowingweshallshowthatthis
constructiondoesindeedidentifypreciselybehaviourallyequivalentstates,i.e.thefinal
lfg coalgebra is always a subcoalgebra of the final coalgebra. Just like fg objects are
closed under quotients – in contrast to fp objects – we have a similar property of lfg
coalgebras:
Lemma3.9. Lfgcoalgebrasareclosedunderstrongquotients,i.e.foreverystrongepi
carriedcoalgebrahomomorphismsX ։Y,ifX islfgthensoisY.
6 S.Milius,D.PattinsonandT.Wißmann
The failure of this property for lfp coalgebras is the reason why the rational fixpoint
is not necessarily a subcoalgebra of the final coalgebra and in particular the rational
fixpointin[12,Example3.15]isanlfpcoalgebraforwhichthepropertyfails.
Theorem3.10. ThefinallfgH-coalgebraisasubcoalgebraofthefinalH-coalgebra.
Proof. Let(L,ℓ)be the finallfg coalgebra.Considerthe uniquecoalgebramorphism
L→νH andtakeitsfactorization:
id (L,ℓ) e (I,i) m (νH,τ) , withestrongepiinC.
i†
By Lemma3.9, I is an lfg coalgebra and so by finality of L we have the coalgebra
morphismi†suchthatid =i†·e.Itfollowsthateismonicandthereforeaniso. ⊓⊔
L
In other words, the final lfg H-coalgebra collects precisely the finitely generated be-
haviours from the final H-coalgebra. We now show that the final lfg coalgebra is a
fixpointofH whichhingesonthefollowing:
c Hc
Lemma3.11. ForanylfgcoalgebraC −→HC,thecoalgebraHC −−→HHC islfg.
Proof. Consider f : S → HC with S finitely generated. As C is lfp we know that
HC is the colimit of its fg subobjects, and so f : S → HC factors through some
subobject in : Q ֌ HC with Q fg and f = in · f′. On the other hand, (C,c)
q q
is lfg, i.e. the directed union of its subcoalgebras from Coalg H. Then, since H is
fg
c
finitaryandmono-preserving,HC −→HHCisalsoadirectedunionandthemorphism
Hp
in : Q → HC factors through some HP −−→ HHP with (P,p) ∈ Coalg H via
q fg
in :(P,p)֌(C,c),i.e.Hin ·q =in .Finally,wecanconstructacoalgebrawithfg
p p q
carrier
[q,p] Hinr
Q+P −−−→HP −−−→H(Q+P)
andacoalgebrahomomorphismHin ·[q,p]:Q+P →HC.Inthediagram
p
f Hc
S HC HHC
Hinp HHinp
f′ inq HP Hp HHP H(Hinp·[q,p])
q
[q,p] H[q,p]
inl [q,p] Hinr
Q Q+P HP H(Q+P)
everyparttriviallycommutes,soHin ·[q,p]isthedesiredhomomorphism. ⊓⊔
p
SowithaproofinvirtuetoLambek’sLemma[28,Lemma2.2],weobtainthedesired
fixpoint:
Theorem3.12. ThecarrierofthefinallfgH-coalgebraisafixpointofH.
We denote the above fixpoint by (ϑH,ℓ) and call it the locally finite fixpoint (LFF)
of H. In particular,the LFF always exists under Assumption3.1, providinga finitary
corecursionprinciple.
TheLocallyFiniteFixpointanditsProperties 7
3.1 IterativeAlgebras
Recall from [6, 31] that the rational fixpoint of a functor H has a universal property
bothasacoalgebraandasanalgebraforH.Thissituationiscompletelyanalogousfor
the LFF. We already established its universalpropertyas a coalgebrain Theorem3.8.
NowweturntostudytheLFFasanalgebraforH.
Definition3.13. AnequationmorphismeinanobjectAisamorphismX →HX+A,
whereX isafinitelygeneratedobject.IfAisthecarrierofanalgebraα : HA → A,
we callthe C-morphisme† : X → Aasolutionofe if[α,id ]·He† +id ·e = e†.
A A
An H-algebra A is called fg-iterative if every equation morphism in A has a unique
solution.
Example3.14 (see [30, Example 2.5 (iii)]). The final H-coalgebra (considered as an
algebraforH)isfg-iterative.Infact,inthisalgebraevenmorphismsX →HX+νH
whereX isnotnecessarilyanfgobjecthaveauniquesolution.
Definition3.15. For fg-iterative algebrasA andC, an equationmorphism e : X →
HX +A anda morphism h : A → C of C defineanequationmorphism h•e in C
as X e HX +AHX+hHX+C. Wesaythathpreservesthesolutione† of
eifh·e† = (h•e)†.Themorphismhiscalledsolutionpreservingifitpreservesthe
solutionofanyequationmorphisme.
Similarlyto[6],thealgebrahomomorphismsarepreciselythesolutionpreservingmor-
phismsbetweeniterativealgebras,theproofisalsoverysimilar.
Proposition3.16. Thelocallyfinitefixpointisfg-iterative.
Proof (Sketch).Onecantransforme:X →HX+ϑHintoanlfgcoalgebraonX+ϑH.
Then one shows that there is a one-to-one correspondence between homomorphisms
intoϑH andsolutionsofeinthealgebraℓ−1 :H(ϑH)→ϑH bydiagramchasing.
⊓⊔
Theorem3.17. For an fg-iterative algebra α : HA → A and an lfg coalgebra e :
X →HX thereisauniqueC-morphismu :X →Asuchthatu =α·Hu ·e.
e e e
Corollary3.18. Thelocallyfinitefixpointistheinitialfg-iterativealgebra.
3.2 RelationtotheRationalFixpoint
Thereareexamples,wheretherationalfixpointisnotasubcoalgebraofthefinalcoal-
gebra.Incategories,wherefpandfgobjectscoincide,therationalfixpointandtheLFF
coincideaswell(cf.therespectivecolimit-constructioninSection2andTheorem3.8).
Inthissectionwewillsee,underslightlystrongerassumptions,thatfg-carriedcoalge-
brasarequotientsoffp-carriedcoalgebras,andinparticularthelocallyfinitefixpointis
aquotientoftherationalfixpoint:namelyitsimageinthefinalcoalgebra.
8 S.Milius,D.PattinsonandT.Wißmann
Assumption3.19. In additiontoAssumption3.1, assume thatin the basecategoryC,
everyfinitelypresentableobjectisastrongquotientofafinitelypresentablestrongepi
projectiveobjectandthattheendofunctorH alsopreservesstrongepis.
Theconditionthateveryfgobjectisthestrongquotientofastrongepiprojectiveoften
isphrasedashavingenoughstrongepiprojectives[14].Thisassumptionisapparently
verystrongbutstillismetinmanysituations:
Example3.20. – Incategoriesinwhichall(strong)episaresplit,everyobjectispro-
jectiveandanyendofunctorpreservesepis,e.g.inSetorVec .
K
– InthecategoryoffinitaryendofunctorsFun(Set),allpolynomialfunctorsarepro-
f
jective.ThefinitelypresentablefunctorsarequotientsofpolynomialfunctorsH ,
Σ
whereΣ isafinitesignature.
– IntheEilenberg-MoorecategorySetT forafinitarymonadT,strongepisaresur-
jectiveT-algebrahomomorphisms,andthuspreservedbyanyendofunctor.InSetT,
everyfreealgebraTX isprojective;thisiseasytoseeusingtheprojectivityofX
in Set. Every finitely generated object of SetT is a strong quotient of some free
algebraTX withX finite.Formoreprecisedefinitions,seeSection4.1later.
Proposition3.21. Every coalgebra in Coalg H is a strong quotient of a coalgebra
fg
withfinitelypresentablecarrier.
Theorem3.22. ϑH istheimageoftherationalfixpoint̺H inthefinalcoalgebra.
Proof. Consider the factorization (̺H,r) ։e (B,b) ֌m (νH,τ). Since ̺H is the
colimit of all fp carried H-coalgebras it is an lfg coalgebra by Proposition3.4 us-
ing that fp objects are also fg. Hence, by Lemma3.9 the coalgebra B is lfg, too. By
Proposition3.7itnowsufficestoshowthatfromevery(X,x)∈Coalg H thereexists
fg
auniquecoalgebramorphisminto(B,b).Given(X,x)inCoalg H,itisthequotient
fg
q : (P,p) ։ (X,x) of an fp-carried coalgebra by Proposition3.21. Hence, we ob-
tainauniquecoalgebramorphismp† : (P,p) → (̺H,r). ByfinalityofνH,wehave
m·e·p† =x†·q(withx† :(X,x)→(νH,τ)).Sothediagonalfill-inpropertyinduces
ahomomorphism(X,x)→(B,b),beingtheonlyhomomorphism(X,x)→(B,b)by
thefinalityofνH andbecausemismonic. ⊓⊔
4 Instances ofthe LocallyFiniteFixpoint
We will now present a numberof instances of the LFF. First note, that all the known
instances of the rational fixpoint(see e.g. [6, 31, 12] are also instances of the locally
finitefixpoint,becauseinallthosecasesthefpandfgobjectscoincide.Forexample,the
classofregularlanguagesistherationalfixpointof2×(−)ΣonSet.Inthissection,we
willstudyfurtherinstancesoftheLFFthataremostlikelynotinstancesoftherational
fixpointandwhich–to thebestofourknowledge–havenotbeencharacterizedbya
universalpropertyyet:
1. Behavioursoffinite-statemachineswithside-effectsasconsideredbythegeneral-
izedpowersetconstruction(cf.Section4.1),particularlythefollowing.
TheLocallyFiniteFixpointanditsProperties 9
(a) Deterministicandordinarycontext-freelanguagesobtainedasthebehaviours
ofdeterministicandnon-deterministicstack-machines,respectively.
(b) ConstructivelyS-algebraicformalpowerseries,i.e.the“context-free”subclass
ofweightedlanguageswithweightsfromasemiringS,yieldedfromweighted
context-freegrammars.
2. ThemonadofCourcelle’salgebraictrees.
4.1 GeneralizedPowersetConstruction
Thedeterminizationofanon-deterministicautomatonusingthepowersetconstruction
is an instance of a more general framework, described by Silva, Bonchi, Bonsangue,
and Rutten [41] based on an observation by Bartels [10] (see also Jacobs [26]). In
that generalized powerset construction, an automaton with side-effects is turned into
anordinaryautomatonbyinternalizingtheside-effectsinthestates.TheLFFinteracts
wellwith thisconstruction,becauseitpreciselycapturesthebehavioursoffinite-state
automatawithsideeffects.Thenotionofside-effectisformalizedbyamonad,which
inducesthecategory,inwhichtheLFFisconsidered.
In the following we assume that readers are familiar with monads and Eilenberg-
Moorealgebras(seee.g.[29]foran introduction).Fora monadT onC wedenoteby
CT thecategoryofEilenberg-Moorealgebras.Recallfrom[8,Corollary2.75]thatifC
is lfp(in most of ourexamplesC is Set) andT is finitary thenCT is lfp, too,and for
everyfpobjectX thefreeEilenberg-MoorealgebraTXisfpinCT.Inalltheexamples
we consider below, the classes of fp and fg objects either provablydiffer or it is still
unknownwhethertheseclassescoincide.
Example4.1. InSections4.4and4.5wearegoingtomakeuseofMoggi’sexception
monad transformer(see e.g. [15]). Let us recall that for a fixed object E, the finitary
functor (−)+E togetherwith the unit η = inl : X → X +E and multiplication
X
µ =id +[id ,id ]: X +E+E →X +E formafinitarymonad,theexception
X X E E
monad. Its algebras are E-pointed objects, i.e. objects X, together with a morphism
E → X,andhomomorphismsaremorphismspreservingthepointing.Sotheinduced
Eilenberg-MoorecategoryisjusttheslicecategoryC(−)+E ∼=E/C.
Now,givenanymonadT weobtainanewmonadT(−+E)withobviousunitand
multiplication. An Eilenberg-Moore algebra for T(−+ E) consists of an Eilenberg-
MoorealgebraforT andanE-pointing,andhomomorphismsareT-algebrahomomor-
phismspreservingthepointing[25].
Now an automaton with side-effects is modelled as an HT-coalgebra, where T is a
finitarymonadonCprovidingthetypeofside-effect.Forexample,forHX =2×XΣ,
where Σ is an input alphabet, 2 = {0,1} and T the finite powerset monad on Set,
HT-coalgebras are non-deterministic automata. However, the coalgebraic semantics
using the final HT-coalgebra does not yield the usual language semantics of non-
deterministicautomata.To obtainthisoneconsidersthefinalcoalgebraofa liftingof
H toCT.DenotebyU :CT →C thecanonicalforgetfulfunctor.
Definition4.2. ForafunctorH : C → C andamonadT : C → C,aliftingofH isa
functorHT :CT →CT suchthatH ·U =U ·HT.
If such a (not necessarily unique) lifting exists, the generalized powerset construc-
tion transforms an HT-coalgebra into a HT-coalgebra on CT: For a coalgebra x :
10 S.Milius,D.PattinsonandT.Wißmann
X → HTX, HTX carries an Eilenberg-Moore algebra, and one uses freeness of
theEilenberg-MoorealgebraTX toobtainacanonicalT-algebrahomomorphismx♯ :
(TX,µT) → HT(TX,µT). The coalgebraic language semantics of (X,x) is then
givenbyX −η−X→ TX−x−♯→† νHT, i.e.bycomposingtheuniquecoalgebramorphismin-
ducedbyx♯ withη .ThisconstructionyieldsafunctorT′ :Coalg(HT)→CoalgHT
X
mapping coalgebras X −→x HTX to x♯ and homomorphisms f to Tf (see e.g. [12,
ProofofLemma3.27]foraproof).
NowouraimistoshowthattheLFFofHT characterizespreciselythecoalgebraic
languagesemanticsofallfp-carriedHT-coalgebras.As therightadjointU preserves
monosandisfaithful,weknowthatHT preservesmonos,andasT isfinitary,filtered
colimitsinCT arecreatedbytheforgetfulfunctortoC,andwethereforeseethatHT is
finitary.Thus,byTheorem3.8,ϑHT existsandisasubcoalgebraofνHT.By[37]and
[10,Corollary3.4.19],weknowthatνHT iscarriedbyνH equippedwithacanonical
algebrastructure.
Nowlet usturnto the desiredcharacterizationofϑHT. Formally,the coalgebraic
languagesemanticsofallfp-carriedHT-coalgebrasiscollectedbyformingthecolimit
k : K → HK ofthediagramCoalg HT −T→′ CoalgHT −U→ CoalgH.Thiscoalgebra
fg
K is not yet a subcoalgebra of νH (for C = Set that means, not all behaviourally
equivalentstatesareidentifiedinK),buttakingitsimageinνH weobtaintheLFF:
Proposition4.3. Theimage(I,i)oftheuniquecoalgebramorphismk† : K → νHT
ispreciselythelocallyfinitefixpointoftheliftingHT.
Onecanalsodirectlytaketheunionofalldesiredbehaviours,forC =Set:
Theorem4.4. The locally finite fixpointof the lifting HT comprises precisely the im-
agesofdeterminizedHT-coalgebras:
ϑHT = x♯†[TX]= x♯†·ηT[X]⊆νHT. (1)
X
x:X→HTX x:X→HTX
X[finite X[finite
Thisresultsuggeststhatthelocallyfinitefixpointistherightobjecttoconsiderinorder
torepresentfinitebehaviour.Wenowinstantiatethegeneraltheorywithexamplesfrom
theliteraturetocharacterizeseveralwell-knownnotionsasLFF.
4.2 TheLanguagesofNon-deterministicAutomata
Let us start with a simple standard example.We alreadymentioned that non-determi-
nisticautomataarecoalgebrasforthefunctorX 7→2×P(X)Σ.HencetheyareHT-
f
coalgebrasforH =2×(−)Σ andT =P thefinitepowersetmonadonSet.Theabove
f
generalizedpowersetconstructiontheninstantiatesastheusualpowersetconstruction
thatassignstoagivennon-deterministicautomatonitsdeterminization.
Now note that the final coalgebra for H is carried by the set L = P(Σ∗) of all
formallanguagesoverΣwiththecoalgebrastructuregivenbyo:L→2witho(L)=1
iffLcontainstheemptywordandt : L → LΣ witht(L)(s) = {w | sw ∈ L}theleft
languagederivative.ThefunctorH hasacanonicalliftingHT ontheEilenberg-Moore
categoryofP,viz.thecategoryofjoinsemi-lattices.ThefinalcoalgebraνHT iscarried
f
by all formallanguageswith the join semi-lattice structure given by union and ∅ and