Computability by monadic second-order logic

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www.elsevier.com/locate/ipl

Computability

by

monadic

second-order

logic

Joost Engelfriet

LIACS,LeidenUniversity,P.O.Box9512,2300RALeiden,theNetherlands

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Articlehistory:

Received27August2020

Receivedinrevisedform20November2020 Accepted21November2020

Availableonline23November2020 CommunicatedbyLeahEpstein Keywords:

Theoryofcomputation Recursivelyenumerable Graphrelation

Monadicsecond-orderlogic

Abinaryrelationongraphsisrecursivelyenumerableifandonlyifitcanbecomputedby aformulaofmonadicsecond-orderlogic.Thelattermeansthattheformuladefinesaset ofgraphs,intheusualway,suchthateach“computationgraph”inthatsetdeterminesa pairconsistingofaninputgraphandanoutputgraph.

©2020TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

There aremanycharacterizations ofcomputability,but theonepresentedheredoesnotseemtoappearexplicitly in theliterature.1 _{Nevertheless,itisa naturalandsimple}

characterization, based on the intuitive idea that a com-putation ofa machine,ora derivation ofagrammar, can berepresentedbyagraphsatisfyingaformulaofmonadic second-order (MSO) logic. Assuming the reader to be fa-miliarwithMSOlogicongraphs(see,e.g.,[3,Chapter 5]), theMSO-computabilityofabinary relationongraphscan be givenin halfa page,see below.One advantage ofthe definition is that there isno need to code thegraphs as stringsornumbers.

For an alphabet



,we considerdirected edge-labeled graphsg

= (

V

,

E

)

over



whereV isanonemptyfiniteset ofnodesandE

⊆

V

× 

×

V isasetoflabelededges.We alsodenoteV byVg,andE byEg.Anedge

(

u

,

ψ,

v

)

∈

Eg iscalleda

ψ

-edge.Isomorphicgraphsareconsideredtobe equal. The set of all (abstract) graphs over



is denoted by

G

.

To modelcomputationswe use aspecial edge label

ν

thatisnotin



.Wedefineacomputationgraph over



to

E-mailaddress:j.engelfriet@liacs.leidenuniv.nl.

1 _{Thisfirstsentenceandthefirstpartofthenextsentencearetaken}

overfrom [8].

be agraphh over



∪ {

ν

}

withatleastone

ν

-edgesuch thatforeveryu

,

v

,

u

,

v

∈

Vh,

(1)

(

u

, ν,

u

) /

∈

Eh,and

(2) if

(

u

, ν,

v

),

(

u

, ν,

v

)

∈

Eh,then

(

u

, ν,

v

)

∈

Eh. The inputgraph in

(

h

)

is defined to be the subgraph ofh

induced by all nodes that havean outgoing

ν

-edge, and the outputgraph out

(

h

)

is the subgraph of h induced by allnodesthat haveanincoming

ν

-edge.By (2) above,the

ν

-edges of h connect every node of in

(

h

)

to every node ofout

(

h

)

, andso by (1) above, Vin(h) andVout(h) are

dis-joint.Infact,the roleof the

ν

-edges isjust tospecifyan ordered pair of disjoint subsets of Vh, in a simple way. Notethatthere maybe arbitrarily manynodesandedges inh thatbelongneithertoin

(

h

)

nortoout

(

h

)

.Also,there may be edges between in

(

h

)

and out

(

h

)

other than the

ν

-edges.Thisnotionofcomputation graphgeneralizesthe “pairgraph”of [9],whichonitsturngeneralizesthe “ori-gingraph”of [1].

For a set H of computation graphs over



we

de-fine the graph relation computed by H to be rel

(

H

)

=

{(

in

(

h

),

out

(

h

))

|

h

∈

H

}

⊆

G

×

G

. Finally, for an al-phabet



, we say that a graph relation R

⊆

G

×

G

is

MSO-computable if thereare an alphabet



andan MSO-definable set H of computation graphs over



∪ 

such

https://doi.org/10.1016/j.ipl.2020.106074

0020-0190/©2020TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

γ γ β β α β β α d d d ν

Fig. 1. Acomputationgraphh foraninducedsubgraph,with



= {α, β, γ}.Theinputgraphin(h)andoutputgraphout(h)aresurroundedbyovals,and theν-labelededgefromtheleftovaltotherightovalrepresentsthe12ν-labelededgesfromeachnodeofin(h)toeachnodeofout(h).

thatrel

(

H

)

=

R.Asobservedbefore,weassumethereader tobefamiliarwithMSOlogicongraphs.2 TheclosedMSO formula

ϕ

thatdefinesthesetH canbeviewedasa “ma-chine” ofwhichthe computationsare representedby the graphsinH .Wewillalsosaythatrel

(

H

)

isthegraph rela-tioncomputedby

ϕ

.Foreachh

∈

H ,theinputgraphin

(

h

)

andtheoutput graphout

(

h

)

mustbegraphs overthe in-put/outputalphabet



.Theauxiliaryalphabet



isneeded to allow the edges of a computation graph that are not partofits input oroutput graph,tocarry arbitrary infor-mation intheir label;it issimilar tothe “working alpha-bet”ofamachine.ThisnotionofMSO-computability gen-eralizesthe“MSO-expressibility” ofgraphrelationsof [9],3 whichonitsturngeneralizestheMSOgraphtransductions of [3,Chapter 7] (asshownin [9,Section 7.1]).

Examples.(1) Let R

⊆

G

×

G

be the set of all

(

g

,

g

)

such that g is an induced subgraph of g. The graph re-lation R isMSO-computable because itcan be computed

by an MSO-definable set H of computation graphs over



∪ 

,with



= {

d

}

.We note that,by definition,the set ofall computationgraphs h over



∪ 

isMSO-definable, andthesets ofnodesVin(h) and Vout(h) canbe expressed

inMSOlogic.Theset H consists ofcomputation graphsh

suchthatVh

=

Vin(h)

∪

Vout(h),in

(

h

)

andout

(

h

)

aregraphs

over



,andthed-edgesformanisomorphismfromout

(

h

)

toaninducedsubgraphofin

(

h

)

.Thelastconditionmeans, indetail,thatforeveryu

,

v

,

u

,

v

∈

Vh,

•

if

(

u

,

d

,

v

)

is an edge of h, then u

∈

Vout(h) and v

∈

Vin(h),

•

ifu

∈

Vout(h),thenu hasanoutgoingd-edge,

•

if

(

u

,

d

,

v

)

and

(

u

,

d

,

v

)

areedgesofh,then

– u

=

uifandonlyifv

=

v,and

– for every

γ

∈ 

,

(

u

, γ

,

u

)

∈

Eh if and only if

(

v

, γ

,

v

)

∈

Eh.

There may be

γ

-edges in h between in

(

h

)

and out

(

h

)

, with

γ

∈ 

;thoughtheyareharmless,wecould addition-ally forbid them. For an example of such a computation

2 _{TheatomicformulasofMSOlogicarex=y,x∈X ,andedge}

ψ(x, y),

wherex and y arenodes,X isasetofnodes,andedgeψ(x, y)expresses

thatthereisa

ψ

-edgefromx to y.

3 _{TherelationR is“MSO-expressible”,inthesenseof [9,Section 3.1],if}

itisMSO-computablebyasetH ofpairgraphs,whereapairgraphisa computationgraphh suchthatVh=Vin(h)∪Vout(h).

graphseeFig.1.Obviouslytheaboveconditionscanbe ex-pressedbyanMSOformula

ϕ

,whichdefines H .Moreover rel

(

H

)

=

R,andhence R isMSO-computable.Notethat R

iseven“MSO-expressible”,inthesenseof [9].

Asanother(similar)example,ifR consistsofall

(

g

,

g

)

such that g has atleast two, disjoint,induced subgraphs isomorphic to g, then we take



= {

d1

,

d2

}

, we re-quirethatthedi-edgessatisfy thesameconditionsasthe

d-edges above(foreachi

∈ {

1

,

2

}

),andwerequirethatno node of in

(

h

)

has both an incoming d1-edge and an in-comingd2-edge.

(2)Letg0 beafixedgraphover



,andlet R

⊆

G

×

G

be the set of all

(

g

,

g0

)

such that the number of nodes of g withanoutgoing

α

-edgeequalsitsnumberofnodes with an outgoing

β

-edge, with

α,

β

∈ 

. There is an

MSO-definable set H of computation graphs over



∪ 

such that rel

(

H

)

=

R, where



= {

d

,

e

}

.It consistsof all graphs h thatareobtainedbyadding

ν

-,d- ande-edgesto thedisjointunionofg,g,andg0,whereg isanarbitrary graphover



andgisisomorphicto g.The

ν

-edges deter-minethatin

(

h

)

=

g andout

(

h

)

=

g0.Thed-edgesestablish anisomorphismbetweeng andg,andthee-edges estab-lishabijection betweenthe nodesof g withan outgoing

α

-edgeandthenodesofgwithanoutgoing

β

-edge.Since theserequirementscan easily beexpressed inMSO logic,

R isMSO-computable.Itisnot difficultto showthat R is

not“MSO-expressible”,cf.theConclusionof [9].



Ouraimisnowtoprovethefollowingtheorem. Theorem.AgraphrelationisMSO-computableifandonlyifit isrecursivelyenumerable.

Recursive enumerability of a graph relation R means

that there is a (single tape) nondeterministic Turing ma-chine M such that

(

g

,

g

)

∈

R if andonly if, on input g, M hasacomputationthatoutputsg.Inonedirectionthis theoremisobvious:everyMSO-computablegraphrelation isrecursivelyenumerable.Infact,oninput g

∈

G

(coded as a string in an appropriate way) M guesses a compu-tation graph h over



∪ 

such that in

(

h

)

=

g, checks whetherh satisfiestheMSOformula

ϕ

(cf. [3,Chapter 6]), andifso, outputs the(coded) graphout

(

h

)

.To show the other direction we first consider the caseof string rela-tions.ForthenotionofMSO-computabilitywerepresenta stringw

=

γ

1

γ

2

· · ·

γ

kover



bythegraphgr

(

g

)

∈

G

such that Vgr(g)

= {

1

,

2

,

. . . ,

k

+

1

}

and Egr(g)

= {(

j

, γ

j

,

j

+

1

)

|

i α β B

β i β B

β α i B

β α f B

ν

Fig. 2. A computationgraph h over  ∪ ,with



= {α, β}and  = {i, f, B, ∗}.It representsthecomputationofa(verysimple)Turing ma-chineM thatchangeseveryαoftheinputstringinto

β

andviceversa. Heretheinputstringisαβ,andM usesspacem=3 andtimen=4.The initialstateofM isi,thefinalstateis f ,andtheinstructionsareiα βi, iβαi,andi Bf B,whereB istheblank.Thestringsw1, w2, w3, w4

correspondingtoM’scomputationareiαβB,

β

iβB,

β

αi B,and

β

αf B.The ∗-labelsoftheverticaledgesofh areomitted.

1

≤

j

≤

k

}

. The proof is similar to the one of [3, The-orem 5.6]. Let M be a nondeterministic Turing machine that computes the recursively enumerable string relation

R

⊆ 

∗

× 

∗.Consideracomputationof M that,foran in-put string w,outputs the string w. Supposethat it uses

space m and time n. Thus, it can be viewed as a

se-quence ofstrings w1

,

. . . ,

wn,each of lengthm

+

1, such that wi is the content of M’s tape at time i (including thestate ofM), w1 contains w (plusthe initialstateand blanks),andwncontains w(andafinalstateandblanks). Clearly, thissequencecan berepresented bya gridof di-mensionn

× (

m

+

2

)

.Therowsofthegridarethegraphs gr

(

w1

),

. . . ,

gr

(

wn

)

,whichareconnectedby

∗

-labeled col-umn edges fromthe j-thnode of wi tothe j-thnodeof

w_i+1 forevery 1

≤

i

≤

n

−

1 and1

≤

j

≤

m

+

2.It iseasy to turn that grid into a computation graph h by adding

ν

-edges fromthenodesofgr

(

w

)

inthefirstrowtothose ofgr

(

w

)

inthe lastrow. Thus,h isacomputation graph over



∪ 

such that in

(

h

)

=

gr

(

w

)

andout

(

h

)

=

gr

(

w

)

, where the alphabet



consists of the columnsymbol

∗

, the workingsymbolsof M (includingthe blank),andthe states of M. For an example see Fig. 2. Since the set of grids isMSO-definable(asshownin [3,Section 5.2]),itis a straightforward exercise inMSO logic to show that the computationgraphs h,obtainedfromthe(successful) com-putations of M, can be defined by an MSO formula

ϕ

M. Inparticular,

ϕ

M shouldexpressthattheconsecutiverows ofthegrid(correspondingtostrings wi and wi+1) satisfy the (local) changes determined by the instructions of M. This shows that the graph relation computed by

ϕ

M is gr

(

R

)

= {(

gr

(

w

),

gr

(

w

))

| (

w

,

w

)

∈

R

}

, and so, gr

(

R

)

is MSO-computable.

Foran alphabet



, letthe graphencodingrelation enc

consistofallpairs

(

g

,

gr

(

w

))

suchthatg

∈

G

andw isan appropriateencodingofg asastring(whichwewill spec-ify later).4 _{By definition,if a graphrelation R}

_⊆

_G

_×

_G

4 _{Appropriatenessmeansthattheencodingandthecorresponding}

de-codingarecomputableinastraightforwardintuitivesense.Inparticular,it

isrecursivelyenumerablethen thereisa recursively enu-merablestringrelation R such that R is thecomposition ofenc,gr

(

R

)

,andenc−1. Hence,to obtainour theorem

forgraph relationsit now sufficesto prove the following

twolemmas.

Lemma 1.The class of MSO-computable graph relations is closedunderinverseandcomposition.

Lemma2.Forevery



,thegraph encodingrelationenc is MSO-computable.

Proof of Lemma1. Closure under inverseis obvious: just reversethedirectionofall

ν

-edges.Toproveclosureunder composition, let R1 and R2 be graph relations computed byMSOformulas

ϕ

1 and

ϕ

2.Wemayassumethat

ϕ

1 and

ϕ

2 usethesameauxiliary alphabet



.Moreover,we may assumethat every computationgraphh defined by

ϕ

1 or

ϕ

2 isconnected:ifnot,thenaddaspecialsymbol

μ

to



and require that every node u of h that is not in in

(

h

)

orout

(

h

)

,hasa

μ

-edgetoin

(

h

)

orout

(

h

)

.Finally,we as-sumethat

ϕ

1 usesthe label

ν

1 insteadof

ν

,and

ϕ

2 uses

ν

2 instead of

ν

, with

ν

1

=

ν

2. The MSO formula

ϕ

that computesthecompositionofR1andR2,usestheauxiliary alphabet



∪ {

ν

1

, ν

2

,

d

}

anddefinescomputation graphsh that are obtained asthe disjointunion of a computation graph h1 of

ϕ

1 and a computation graph h2 of

ϕ

2, en-richedbyd-edgesthat establishanisomorphismbetween out

(

h1

)

andin

(

h2

)

,andby

ν

-edgesfromin

(

h1

)

toout

(

h2

)

. It should be clearthat this can be realizedby

ϕ

; for in-stance, it expresses that the connected components ofh

minus itsenriching edges satisfy

ϕ

1 or

ϕ

2, depending on whethertheycontaina

ν

1-edgeora

ν

2-edge.



Proof of Lemma2. We firstspecify therelation enc.Let g

∈

G

. We may assume that Vg is the set of strings

{

a

,

a2

,

. . . ,

an

}

overthealphabet

{

a

}

,forsomen

≥

1,where

a

∈ 

/

. Let Eg

= {(

u1

, γ

1

,

v1

),

. . . ,

(

um

, γ

m

,

vm

)

}

for some

m

≥

0.Weencodeg,inastandardway,asthestring

w

=

#a#a2#

· · ·

#an$u1

γ

1v1$

· · ·

$um

γ

mvm$

over the alphabet



= 

∪ {

a

,

#

,

$

}

, and we define the graph encoding relation enc

⊆

G

×

G

to consist of

all pairs

(

g

,

gr

(

w

))

. Note that since w depends on lin-ear orderings of Vg and Eg, a graph g has in general

more than one encoding. On the other hand, the

rela-tion enc−_1 is a function. The set of strings over



that

encode graphs over



is not a regular language, and

hence the set enc(G) of graphs over



is not

MSO-definable [2,6,12].However,by enrichingeachgr

(

w

)

with

α

-edgesand

δ

-edges (where

α

and

δ

arespecialsymbols not in



), we can turn enc(G

)

into an MSO-definable set of graphs. Fora string w as displayed above we de-finegr+

(

w

)

tobethegraphgr

(

w

)

towhich

α

-edgesand

δ

-edges are added asfollows.For an example see Fig. 3. The

α

-edgesallowanMSOformulatoexpressthefactthat

isdecidablewhetherornotagivenstringistheencodingofsomegraph. Anystandardencodingofgraphssatisfiestheserequirements.

# a # a a # a a a

a γ a $ a γ a a $

α α α α α

δ δ δ δ δ δ δ δ δ

$

Fig. 3. The graph gr+(w) for the string w=#a#aa#aaa$aγa$aγaa$, which is an encoding of the graph g with Vg= {a, aa, aaa} and Eg= {(a, γ, a), (a, γ, aa)}.Thegraphgr(w)isobtainedfromgr+(w)byremovingallα- and

δ

-edges.

x y γ x y # ai _{# # a}j _# x y $ ai γ _aj _$ e e δ δ δ δ

Fig. 4. Parts of a computation graph h showing the MSO-computability of enc. The nodes x and y belong to in(h), all other nodes to mid(h).

thefirsthalfofw isoftheform#a#a2#

· · ·

#an$.Foreach substring#ai#ai ofw (1

≤

i

≤

n

−

1)thereare

α

-edgesin gr+

(

w

)

fromthenodesofthefirstoccurrenceofgr

(

ai

₎

_in gr

(

w

)

tothe nodesof thesecond occurrenceof gr

(

ai

₎

_in gr

(

w

)

,suchthattheyformanisomorphismbetweenthese two subgraphs. An MSO formula on gr+

(

w

)

can express that w isintheregularlanguage#a

(

#a∗

)

∗

(

$a∗



a∗

)

∗$,and, using the outgoing

α

-edges of gr

(

#ai#

)

, it can enforce that each substring #ai# is followed by ai+1# or ai+1$. The

δ

-edgesingr+

(

w

)

witnessthefactthatforeach sub-string $uj

γ

jvj$ of w (1

≤

j

≤

m) both uj and vj are in

{

a

,

a2

,

. . . ,

an

}

,i.e.,ujandvjare“declared”inthefirsthalf of w.Thus, thereare

δ

-edges fromthenodesofgr

(

uj

)

to thenodesofsomegr

(

#ai_#

₎

_orgr

₍

_#ai_$

₎

_{inthefirsthalfof} gr

(

w

)

that establishan isomorphismbetweengr

(

uj

)

and gr

(

ai

₎

_{,andsimilarlyforgr}

₍

_v

j

)

.Thiscanalsoeasilybe ex-pressedbyanMSOformula.Moreover,the

δ

-edgescanbe used to expressthat an edgeis not encodedtwice in w,

i.e., if j

=

k then$uj

γ

jvj$

=

$uk

γ

kvk$; infact,uj

=

uk if andonlyifthetwo

δ

-edgesthatstartfromthefirstnodes ofgr

(

uj

)

andgr

(

uk

)

ingr+

(

w

)

,leadtothesamenode(and similarlyforvj

=

vk).Wenowdefineenc+ toconsistofall

pairs

(

g

,

gr+

(

w

))

where w encodes g.It followsthat the setenc+_

(

G

)

isMSO-definable.5

Finally,weshowthat enc

⊆

G

×

G

is

MSO-comput-able by describing thecomputation graphs h over



∪ 

in an MSO-definable set H such that rel

(

H

)

=

enc. The

auxiliary alphabet is



= {

α

,

δ,

d

,

e

}

. Let mid

(

h

)

be the

subgraph of h induced by the nodes of h that are not

5 _{Werecallthatthesetofgraphsgr}

(w),wherew isanarbitrarystring over



,isMSO-definable,seeforinstance [3,Corollary 5.12] or [9, Exam-ple 2.1].

incident with a

ν

-edge, i.e., that are not in Vin(h) or Vout(h). First, we requirethat mid

(

h

)

is in enc+

(G

)

,i.e.,

mid

(

h

)

=

gr+

(

w

)

where w encodes some graph g in

G

. Second, werequirethat thereare d-edges fromout

(

h

)

to mid

(

h

)

thatestablishanisomorphismbetweenout

(

h

)

and the graph obtainedfrom mid

(

h

)

by removing all

α

- and

δ

-edges.Thismeansthatout

(

h

)

=

gr

(

w

)

.Third,itremains to require that in

(

h

)

is isomorphic to g. To realize this, we require that in

(

h

)

∈

G

and that there are e-edges

from in

(

h

)

to mid

(

h

)

that establish a bijection between

Vin(h) and the nodes of mid

(

h

)

that have an incoming

#-edge (thus representinga bijection between Vin(h) and Vg

= {

a

,

a2

,

. . . ,

an

}

). Since we wish this bijection to rep-resent an isomorphism betweenin

(

h

)

and g, we require forevery

(

x

, γ

,

y

)

∈

Vin(h)

× 

×

Vin(h) that

(

x

, γ

,

y

)

isan

edgeof in

(

h

)

ifandonly ifthereexistnodes x

,

x

,

y

,

y

ofmid

(

h

)

suchthat

(1)

(

x

,

e

,

x

)

and

(

y

,

e

,

y

)

areedgesofh,

(2)

(

x

,

δ,

x

)

and

(

y

,

δ,

y

)

areedgesofmid

(

h

)

, (3) x hasanincoming$-edgeinmid

(

h

)

,and

(4) there is a directed path from x to y in mid

(

h

)

, of which the consecutive edge labels form a string in a∗

γ

.

This situation is sketched in Fig. 4. Condition (1) means thatx and y correspondtosubstrings#ai

∗

and#aj

∗

of w (with

∗

∈ {

#

,

$

}

),i.e., to nodesai _andaj _{of g,and} condi-tions(2)-(4)meanthatw hasasubstring$ai

_γ

_aj_$,i.e.,that

(

ai

, γ

,

aj

)

isanedgeof g.Itshouldbeclearthatallthese requirementscanbeexpressedinMSOlogic,andthat the graphrelationcomputedbyH isenc.



Lemma 2 is trivial from the point of view of Turing computability:ifw encodesg,thenboth g andgr

(

w

)

can

be represented by w on the tape of a Turing machine.

This is however based on the intuition that our encod-ingofgraphsasstringsiscomputable.Sincethenotionof MSO-computabilitydiscussedhereusesgraphsasdatatype ratherthanstrings,wewereabletogiveaformalproofof thatintuition.Thereadermayobjectthattheformal proof is basedonthe intuitionthat theencoding of astring w

asthe graphgr

(

w

)

is computable.One mightthen argue thatthelatterencodingissimplerthantheformer.

Traditionally, it hasbeen shownthat MSO logic is re-latedtoregularity,e.g.,toregularstringlanguages [2,6,12] and regular tree languages [4,13]. If one identifies regu-larity with computability by a finite-state machine, then this approach failsforMSO logic on graphs, because “no

notion of finite graph automaton has been defined that

would generalize conveniently finite automata on words and terms” ([3, Section 1.7]). For this reason, the MSO transducers of [3, Chapter 7] were proposed to play the roleoffinite-statetransducersofgraphs,andinthecaseof stringstheyindeedturnedouttobeequivalenttotwo-way finite-state transducers [5]. We have shown above how, droppingthefinite-statecondition,MSOlogicisrelatedto computabilitybyanymachine.

If, ontheother hand,oneidentifies regularitywith ra-tionality,i.e.,withasmallestclasscontainingallfinitesets of objects and closed under a number of natural opera-tions onsetsofobjects (union,concatenation,andKleene starin thecaseof stringlanguages),then the classof all MSO-definable sets of graphs hasa rational characteriza-tion [7]. Sincethe recursivelyenumerablestring relations also havearational characterization(asdiscussed in [8]), the question remains whether there is a naturalrational characterization of the MSO-computable graph relations. Such a characterization wouldatleast involvethe opera-tionsofunion,composition,andtransitiveclosureofgraph relations.

Theabovequotefrom [3,Section 1.7] referstothe non-existence of a finite-state graph automaton that accepts exactly theMSO-definablesetsofgraphs. In [11] a finite-state graphacceptorisintroducedofwhichthe computa-tions are “tilings” of the input graphs (which have tobe graphsofboundeddegree).All“tiling-recognizable”setsof graphsacceptedbythesemachinesareMSO-definable,and the reverse is true forstrings andtrees. Ifwe would al-lowthenodesofourgraphstohavelabels,thenwecould modelthe inputgraphin

(

h

)

andtheoutput graphout

(

h

)

ofacomputationgraph h bytwospecialnodelabelsrather than by

ν

-edges. Then, similar to MSO-computability, we coulddefine agraphrelationtobe“tiling-computable” by requiring the set H of computation graphs to be tiling-recognizablerather thanMSO-definable.This leadsto the followingquestionforgraphsofboundeddegree: isevery recursively enumerable graph relation tiling-computable? Notethat,asshownin [11,Example 3.2(b)],thesetofgrids istiling-recognizable.

Descriptive complexity theory investigates logics that characterize complexity classes. By Fagin’s theorem (see, e.g., [10, Theorem 5.1]), the complexity class NP equals

the set of problems that can be specified by existential second-order formulas. In terms of graphs, such a for-mula requires the existence of an extension of the in-putgraph by additionallabeled hyperedges (wherea hy-peredge is a sequence of nodes), such that the resulting (hyper)graph satisfies a first-orderformula. In our notion ofMSO-computability we require that the input graph is an induced subgraph ofa graphthat satisfies a monadic second-order formula, and we obtain all recursively enu-merableproblems.

We finally note that the notion of MSO-computability caneasily be generalizedtodeal witharbitraryrelational structures(cf. [3,Section 5.1]).

Declarationofcompetinginterest

The authordeclares that he hasno known competing

financialinterestsorpersonalrelationshipsthatcouldhave appearedtoinfluencetheworkreportedinthispaper. Acknowledgement

Ithankthereviewersfortheirhelpfulsuggestions. That’sallfolks! Thiswas my last paper. Thankyou, dear reader,andfarewell.

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