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On Internal Presentation of Regular Graphs

Article · July 1999

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SLIDE 2 On In ternal Presen tation

f

Regular Graphs Didier Caucal 1 and T eo dor Knapik 2 1 IRISACNRS, Campus de Beaulieu 35042 Rennes, F rance email: caucal@irisa.fr 2 IREMIA, Univ ersit de la Runion, BP 7151, 97715 Saint Denis Messageries Cedex 9, Runion email: knapik@univreunion.fr Abstract. The study

f

innite graphs has p

ten

tial applications in the sp ec- ication and v erication

f

innite systems and in the transformation

f

suc h systems. Prexrecognizable graphs and regular graphs are

f

particular in terest in this area since their monadic secondorder theories are decidable. Although the latter form a prop er sub class

f

the former, no c haracterization

f

regular graphs within the class

f

prexrecognizable

nes

has b een kno wn, except for a graphtheoretic

ne
f

[2]. W e pro vide here three suc h new c haracterizations. In particular, a decidable, languagetheoretic, necessary and sucien t condition for the regularit y

f

an y prexrecognizable graph is established. Our pro

fs

yield a construction

f

a deterministic h yp eredgereplacemen t grammar for an y prexrecognizable graph that is regular. In tro duction Graphs are

ne
f

the most general structures in computer science. While the study

f

nite graphs has a long history , innite graphs ha v e

nly

recen tly at- tracted the atten tion

f

computer scien tists. As a matter

f

fact, innite graphs are indisp ensable for mo deling large

r

innite systems. In

rder

to put in practice v erication

f

suc h systems, in v estigation

f

the logical and algorithmic prop erties

f

innite graphs needs to b e pursued. Another direction where innite graphs ha v e promising applications is the elimination

f

redundancy in syn tactic

b

jects, suc h as e.g. recursiv e program sc hemes (see [12] for a surv ey). T

b

e studied b y a computer scien tist, an eectiv e presen tation is required for innite graphs. Let us men tion v arious presen tations in tro duced in the lit- erature esp ecially in connection with logic. One

f

the rst suc h presen tations uses pushdo wn automata [22]. Presen ted graphs are transition graphs

f

pushdo wn automata where the in ternal conguration form the set

f

v ertices. A quite dieren t approac h ma y b e found in [11] where presen tations consist

f

systems

f

equations

v

er h yp eredge replacemen t (HR)

p

erations. The graph presen ted b y a system is the least solution

f

the system. This ma y also b e understo

d

as an !

iteration
f

a deterministic graph grammar (see e.g. [7]). The class

f

graphs ha ving this presen tation is called equational in [11] and regular in [7]. A more general class

f

graphs is

btained

[1] when, instead

f

HR

p

erations, v ertex replacemen t (VR)

p

erations [14, 15] are used in systems

f

equations. This leads to an alternativ e presen tation

f

the class

f

graphs rst dened [9] and called recognizable. (T

b

e more precise, w e call these graphs prexrecognizable.) Since b

th

classes

f

graphs ma y b e presen ted b y systems

SLIDE 3 2

f

equations, w e prefer to sp eak

f

regular (resp. prexrecognizable) graphs instead

f

equational. 1 F

r

the class

f

prexrecognizable graphs, t w

kinds
f

presen tations are in tro duced in [9]. The rst

ne

is based

n

the comp

sition
f

t w

languagetheoretic
p

erations that act

n

paths

f

the innite binary tree: in v erse rational substitutions, and rational restrictions. The second

ne

consists

f

a kind

f

extension

f

recognizable relations whic h is further used for sp ecifying the edge relation b et w een v ertices. In the presen t pap er, w e call the latter prexrecognizable relations. An extension

f

this approac h to (not necessarily simple) h yp ergraphs is in tro duced in [25]. Finally , let us men tion t w

additional

graph presen tations. In [6] a graph presen tation consists

f

a nite stringrewriting system together with a rational subset

f

the set

f

irreducible w

rds.

The graph is dened similarly to the Ca yley graph

f

a group presen tation. Last but not least, an in teresting idea is dev elop ed in [17] where graphs are presen ted b y systems

f

equations with higherorder recursion. Most

f

the presen tations men tioned ab

v

e are related to the class

f

prex recognizable graphs

r

to its prop er sub class

f

regular graphs. Th us the transition graphs

f

pushdo wn automata are precisely the ro

ted

regular graphs

f

nite degree [7], and an alternativ e presen tation

f

this class is the

ne
f

[6] although more general classes

f

graphs ma y b e

btained

within the latter approac h b y appropriate restrictions

n

stringrewriting systems used in the presen tations [19, 20]. On the

ther

hand, the VRsystems

f

equations considered in [1] presen t prexrecognizable graphs

r

their extensions to h yp ergraphs. W e conjecture that the approac h

f

[25] leads to the same class

f

h yp ergraphs. On the con trary , the presen tations

f

[17] lead to a class

f

graphs that seems to b e more general than the prexrecognizable graphs, but this h yp

thesis

needs to b e in v estigated. The classes

f

graphs men tioned ab

v

e ha v e b een disco v ered mainly in logical in v estigations. Th us the decidabilit y

f

the monadic secondorder theory has b een established in [22] 2 for regular ro

ted

graphs

f

nite degree, in [11] for regular graphs, and in [9] for prexrecognizable graphs. As sho wn in [18], the class

f

graphs describ ed there has decidable CTL [10] and S1S [4], and the decidabilit y

f

the mo dal calculus [21] and ev en

f

the monadic secondorder logic is conjectured. V ery dieren t motiv ations app ear in [16]. The author

f

that pap er is in- terested in connections b et w een problems

n

nite graphs and their innite equiv alen ts. Since an y recursiv e presen tation mak es sense with resp ect to this goal, the author stipulates the use

f

either terminating T uring mac hines that recognize the edge relation

r

some

ther

equiv alen t means. Among the presen tations review ed ab

v

e t w

kinds

ma y b e distinguished: those where the graph is dened up to isomorphism and those where the v ertices are explicitly named. W e shall call the former external and the latter in ternal. According to this classication, presen tations

f

[6, 7, 16, 22, 25] are in ternal and those

f

[1, 11, 17] are external. Concerning [9],

ne
f

the t w

presen

tations

f

1 In

rder

to a v

id

a confusion, the terms

f

HRequational and VRequational are used in [2]. 2 In fact, the primary motiv ation comes here from the com binatorial semigroup theory .

SLIDE 4 3 the pap er is external while the

ther

is in ternal. As men tioned b efore, the latter is based

n

prexrecognizable relations (on the free monoid). More precisely , the v ertices

f

the dened graph are w

rds

and the edge relation is sp ecied as a nite union

f

elemen tary prexrecognizable relations. An elemen tary prex recognizable relation is a catenation

f

a cartesian pro duct

f

t w

rational

sets with an iden tit y relation

n

a rational set. (The usual catenation

p

eration extends pairwise to relations.) T aking in to accoun t the div ersit y

f

presen tations, w e b eliev e that it is im- p

rtan

t to establish connections b et w een them and to pro vide v arious c haracterizations

f

eac h in teresting class

f

innite graphs. Insofar as the pro

fs

are constructiv e, this kind

f

in v estigations leads to algorithms that allo w to mo v e b et w een dieren t presen tations

f

a same graph. It is also useful to disp

se
f

the mem b ership criteria for eac h kind

f

presen tations. W e b eliev e that the presen t pap er is a step to w ards these directions. Its main purp

se

is to pro vide an in ternal c haracterization

f

the class

f

regular graphs, viz the graphs presen ted b y deterministic graph grammars (or HRsystems

f

equations). This class

f

graphs is imp

rtan

t in so far as a generalization

f

regular trees. On the

ther

hand, in ternal presen tations are particularly adv an tageous for study- ing the bisim ulation

r

graph decomp

sitions

b ecause

f

the explicit naming

f

v ertices. The in ternal c haracterizations

f

regular graphs kno wn so far concern

nly

the sub classes

f

regular graphs

f

nite degree [7] and

f

ro

ted

regular graphs

f

nite degree [22]. Here, w e giv e the c haracterization

f

the whole class within the class

f

graphs presen ted b y the prexrecognizable relations [9] (briey called prexrecognizable graphs). The latter form another imp

rtan

t class since their monadic secondorder theory is decidable and their traces are con textfree languages. In

rder

to c haracterize the regular graphs within the class

f

prexrecognizable

nes,

w e in tro duce t w

tec

hnical notions: the n um b er

f

decomp

sitions
f

a binary relation (on the free monoid) and the dep endence lev el. Tw

more

familiar notions are also used, namely the n um b er

f

degrees

f

a graph and the treewidth. In sum, w e ha v e four notions, eac h

f

whic h yields an indep enden t c haracterization

f

regular graphs. One

f

these c haracterizations is turned in to a criterion

n

regular languages in v

lv

ed in a presen tation b y prexrecognizable relations and leads to simple decision metho d

f

the prop ert y

f

b eing regular for a prexrecognizable graph. Moreo v er, an algorithm for the construction

f

a deterministic graph grammar from a presen tation b y prexrecognizable relations ma y readily b e deriv ed from

ur

pro

fs,

as demonstrated b y an example. The criterion for innite graphs based

n

the treewidth app ears already in e.g. [2] and an algorithm that constructs an HRsystem

f

equations equiv alen t to a giv en VRsystem

f

equations

f

b

unded

treewidth is in tro duced in the extended v ersion

f

that pap er [3]. Moreo v er in [1] an equiv alence is established b et w een the VRsystems

f

equations and the external presen tations

f

[9]. No w, the results

f

the latter pap er allo w to mo v e b et w een these external presen tations and the presen tations b y the prexrecognizable relations. Hence, b y putting together these results, it is p

ssible

to construct a deterministic graph grammar equiv alen t to a giv en presen tation b y prexrecognizable relations. In view

f

these fact,

ur

rst con tribution consists in pro viding a direct construc-

SLIDE 5 4 tion instead

f

a threestep

ne.

Our second con tribution is the criterion for the regularit y applicable

n

presen tations b y prexrecognizable relations as w ell as additional c haracterizations

f

regular graphs. 1 Preliminaries F

r

an y set E , w e denote b y jE j its cardinal and b y } (E ) its p

w

erset. Let I N b e the set

f

nonnegativ e in tegers and for an y n 2 I N, w e let [n] = f1; : : : ; ng with [0] = ?. A disjoin t union

f

t w

sets

E and F is written E + F . Monoids A monoid is a set M equipp ed with an in ternal asso ciativ e

p

eration, written m ultiplicativ ely , and the neutral elemen t 1 l 2 M. A (monoid ) morphism

f

M in to M is a map

:

M ! M that asso ciates the neutral elemen t

f

M to the neutral elemen t

f

M and that is compatible with m ultiplication, viz (x 1 x 2 ) = (x 1 )(x 2 ) for all x 1 ; x 2 2 M. The p

w

erset } (M) is naturally equipp ed with the monoid structure: f1 lg is the neutral elemen t and M M = fxx jx 2 M ; x 2 M g for all M ; M 2 } (M) : The Kle ene star M

f

M is dened as S n0 M n . The set S n i=1 M i is written M n . Giv en M ; M

M,

the left (right ) r esidual

f

M b y M , written M 1 M (resp. M M 1 ), is the set fx 2 M j 9 x 2 M ; xx 2 M g (resp. fx 2 M j 9 x 2 M ; x x 2 M g). In the particular case

f

a singleton M = fy g w e write y 1 M (resp. M y 1 ). A subset M

f

M is r e c

gnizable

if there exists a nite monoid M and a monoid morphism

:

M ! M that satur ates M , viz

1

((M )) = M . The family

f

all r e c

gnizable

subsets

f

M is written Rec(M). The smallest subset

f

} (M) that con tains F in (M), viz the family

f

all nite subsets

f

M and that is closed under union, catenation and Kleene star forms the family

f

all r ational subsets

f

M, written Rat (M). By the McKnigh t Theorem, Rec(M)

Rat

(M) when M is nitely generated [5]. W e tak e a nite set

f

sym b

ls

N , called an alphab et. A w

rd

w

v

er N

f

length jw j 2 I N is a mapping from

jw

j

in

to N and is denoted b y the catenation w (1): : : w (jw j)

f

its letters (where the catenation

p

erator is

mitted).

When w = uv then u (resp. v ) is called a pr ex (resp. sux )

f

w . The set

f

suxes

f

w is written su (w ). The r eversal w (jw j)w (jw j

1)

: : : w (1)

f

w is written e w and this notation is extended to sets in the usual w a y: f W = f e w j w 2 W g. The set N

f

the w

rds
v

er N is a monoid for the catenation extended to w

rds

with the unit w

rd

" b eing the w

rd
f

length 0: N

is

the free monoid generated b y N . The subsets

f

N

are

languages

v

er N and Rat (N

)

= Rec(N

)

according to the Kleene Theorem. W e do no recall basic denitions related to nite automata and semiau- tomata. These ma y b e found in e.g. [5].

SLIDE 6 5 Relations and Graphs Giv en a relation R 2 } (M

M

), Dom (R) (resp. Ran(R)) stands for the domain (resp. range)

f

R. When b

th

M and M are monoids, } (M

M

) is pro vided with monoid structure: the singleton f(1 l; 1 l )g is the neutral elemen t and RR = f(m 1 m 1 ; m 2 m 2 ) j (m 1 ; m 2 ) 2 R; (m 1 ; m 2 ) 2 R g for all R; R 2 } (M

M

). The family

f

recognizable relations is c haracterized b y the Mezei Theorem: R 2 Rec(M

M

) if and

nly

if R is a nite union

f

relations

f

the form M

M

with M 2 Rec (M) and M 2 Rec (M ). Putting together the Mezei and the Kleene Theorems, w e ha v e the follo wing c haracterization

f

the recognizable binary relations

n

the free monoid N

:

Rec (N

N
)

= n [ i2I U i V i

I

nite ; U i ; V i 2 Rat(N

)
:

An extension

f

Rec (N

N
)

is the class

f

pr exr e c

gnizable

r elations

n

N

.

A relation is prex recognizable if it is a nite union

f

the form (U

V

)W = f(uw ; v w ) j u 2 U; v 2 V ; w 2 W g with U; V ; W 2 Rat(N

)

: A single relation (U

V

)W is called elementary pr exr e c

gnizable.

Lik ewise recognizable relations, prexrecognizable relations form a b

lean

algebra [9]. The family

f

binary relations

v

er an y set D forms a monoid under comp

si-

tion

f

relations with the iden tit y relation Id D as the unit. Th us for R

D
D

, w e write R

(resp.

R + ) to denote the reexiv etransitiv e (resp. transitiv e) closure

f

R. Nev ertheless, eac h time w e refer to Rec (D

D

)

r

Rat(D

D

) the former monoid structure

n

} (D

D

) is understo

d

and this requires D to b e also a monoid. The restriction

f

a relation R 2 } (D

D

) to a subset C

f

D is written R jC , viz R jC = R \ (C

C

). Giv en an alphab et T , an e dgelab el le d simple

riente

d gr aph G

ver

T is a set

f

e dges, viz a subset

f

D

T
D

where D is an arbitrary set. Giv en d; d 2 D , an edge from d to d lab elled b y a 2 T , is written d a

!

G d , meaning that d a

!

d 2 G. Ob viously a

!

is a binary relation

n

D for eac h a 2 A and w e write a ! (resp. a ) for its symmetric closure (resp. in v erse). Con v ersely , an y binary relation

n

D is an unlab elled graph. The graphs w e consider ha v e no isolated v ertices and V G stands for the set

f

v ertices

f

G: V G = [ a2T Dom ( a

!

) [ Ran ( a

!)

: Similarly , giv en a binary relation R 2 D

D

w e let V R = Dom (R) [ Ran(R). A (nite directed) p ath in G

D
T
D

from some d 2 D to some d 2 D is a sequence

f

edges

f

the follo wing form: d a 1

!

d 1 ; : : : ; d n1 a n

!

d n , suc h that d = d and d n = d . A (nite) undir e cte d p ath in G b et w een some d 2 D and some d 2 D is a nite sequence

f

undir e cte d e dges d a 1 ! d 1 ; : : : ; d n1 a n ! d n suc h that d = d and d n = d . A path (resp. undirected path)

f

ab

v

e form is elementary when d i = d j ) i = j for all i; j 2 f0; 1; : : : ; ng.

SLIDE 7 6 A graph G

N
T
N
is

prexrecognizable, when it is

f

the form n [ i=1 (U i a i

!

V i )W i with n

and

8 i 2 [n]; U i ; V i ; W i 2 R at(N

)

: (1.1) where (U i a i

!

V i )W i = fuw a i

!

v w j u 2 U; v 2 V ; w 2 W g for eac h i 2 [n]. 2 Num b er

f

Degrees, Decomp

sition

Num b er and Dep endence Lev el As w e shall establish later, the regularit y

f

a prexrecognizable graph is related to the niteness

f

the set

f

the degrees

f

its v ertices. W e shall call the cardinal

f

this set the numb er

f

de gr e es and discuss it in the presen t section. W e consider a tec hnical notion

f

the n um b er

f

decomp

sitions
f

a relation and w e sho w that the n um b er

f

degrees and the latter notion are dep enden t. Indeed, as established in this section, w e are able to decide the niteness

f

the n um b er

f

degrees

f

an y prexrecognizable graph using the decomp

sition

n um b er. In this w a y , w e can decide the regularit y . T

mak

e precise (in Sect. 4) the connection b et w een the regularit y and the n um b er

f

degrees, w e need another tec hnical notion, namely the dep endence lev el

f

a relation. It is in tro duced at the end

f

this section. 2.1 Num b er

f

Degrees F

r

an y w

rd

x 2 N

and

an y binary relation R

N
N
,

w e dene d + R (x) := jR(x)j =

fy

j (x; y ) 2 Rg

the
utde

gr e e

f

x in R d

R

(x) := jR 1 (x)j =

fy

j (y ; x) 2 Rg

the

inde gr e e

f

x in R d

R

(x) := d

R

(x) + d + R (x) the de gr e e

f

x in R F

r

instance the follo wing ascendan t relation

n

a

:

f(a m+n ; a n ) j m

1

^ n

0g

= (a + f"g)a

satises

d +

(a

n ) = n and d

(a

n ) = 1 = d

(a

n ), for ev ery n 2 I N. Let us giv e a basic prop ert y

f

the v ertex degree

f

the union

f

t w

relations.

Lemma 2.1. F

r

every R; S

N
N
,

x 2 N

,
2

f+; ; g, we have maxfd

R

(x); d

S

(x)g

d
R[S

(x)

d
R

(x) + d

S

(x) : Pr

f.

(i) W e ha v e R(x)

(R

[ S )(x) hence d + R (x) = jR(x)j

j(R

[ S )(x)j = d + R[S (x). By replacing R b y S , d + S (x)

d

+ R[S (x) so maxfd + R (x); d + S (x)g

d

+ R[S (x). F urthermore (R [ S )(x) = R(x) [ S (x), hence d + R[S (x) = j(R [ S )(x)j

jR(x)j

+ jS (x)j = d + R (x) + d + S (x) :

SLIDE 8 7 (ii) As d

R

= d + R 1 and (R [ S ) 1 = R 1 [ S 1 , w e get b y (i) the inequalities for d

.

F

r

d

,

w e ha v e maxfd

R

(x); d

S

(x)g = maxfd

R

(x) + d + R (x); d

S

(x) + d + S (x)g

maxfd
R

(x); d

S

(x)g + maxfd + R (x); d + S (x)g

d
R[S

(x) + d + R[S (x) b y (i) and (ii) = d

R[S

(x)

[d
R

(x) + d

S

(x)] + [d + R (x) + d + S (x)] b y (i) and (ii) = d

R

(x) + d

S

(x) 2 F

r

resp ectiv ely

2

f; ; +g, w e denote b y d

(R)

:= maxfd

R

(x) j x 2 N

g

the inde gr e e, de gr e e,

utde

gr e e

f

R, and D

(R)

:=

fd
R

(x) j x 2 N

g
is

the n um b er

f

indegrees, degrees and

f
utdegrees
f

R. The follo wing lemma pro vides a basic prop ert y

f

the n um b er

f

degrees

f

the union

f

t w

relations.

Lemma 2.2. F

r

every R; S

N
N
and
2

f+; ; g, we have D

(R)

< 1 ^ D

(S

) < 1 = ) D

(R

[ S ) < 1 Pr

f.

By Lemma 2.1, d

R[S

(x) = 1 ( )

d
R

(x) = 1 _ d

S

(x) = 1

:

Assume that D

(R)

< 1 ^ D

(S

) < 1. Then the follo wing in tegers exist: D := maxfd

R

(x) < 1 j x 2 N

g

and D := max fd

S

(x) < 1 j x 2 N

g

with max(?) = 0. By Lemma 2.1, fd

R[S

(x) j x 2 N

g
f1g

[ f0; 1; : : : ; D + D g hence D

(R

[ S )

D

+ D + 2 < 1. 2 It is imp

rtan

t to notice that the con v erse

f

Lemma 2.2 is false. Indeed, for the in v erse

f

the previous ascendan t relation, namely f(a n ; a m+n ) j m

1

^ n

0g

= (f"g a + )a

w

e ha v e D + () = 1 and D + () = 1 = D + ( [

).

According to Lemma 2.2, in

rder

to establish the niteness

f

the n um b er

f

degrees

f

a nite union

f

relations, it is enough to establish it for eac h relation comp

sing

the union. In particular, for graphs presen ted b y prexrecognizable relations, w e need to able to decide the niteness

f

the n um b er

f

degrees

f

elemen tary prexrecognizable relations, viz the relations

f

the form: (U V )W = f(uw ; v w ) j u 2 U ^ v 2 V ^ w 2 W g where U; V ; W 2 Rat(N

).

W e address this problem via the notion

f

the decomp

sition

n um b er

f

a relation.

SLIDE 9 8 2.2 Decomp

sition

Num b er

f

a Relation W e consider the n um b er decomp

sitions
f

a w

rd

b y a binary relation and w e establish that for an y prexrecognizable relation, w e can decide the niteness

f

the maxim um decomp

sition

n um b er

f

the w

rds

(Prop

sition

2.6). F

r

an y w

rd

w 2 N

and

an y binary relation R

N
N
,

w e dene D ec(w ; R) :=

f(u;

v ) 2 R j uv = w g

the

de c

mp
sition

numb er

f

w according to R. Note that

D

ec(w ; R)

jw

j + 1 and b

th

b

unds

are p

ssible:

D ec(w ; ?) = ^ D ec(w ; N

N
)

= jw j + 1. F urthermore D ec(w ; R + S ) = D ec(w ; R) + D ec(w ; S ). By maximalit y , w e extend the decomp

sition

n um b er

f

a w

rd

to all w

rds:

D ec(R) := maxfD ec(w ; R) j w 2 N

g

is the maximum de c

mp
sition

numb er according to R

N
N
with

the con v en tion that max(E ) = 1 for an y innite set E

I

N. Let us giv e a basic prop ert y

f

Dec for the pro duct

f

languages. Lemma 2.3. F

r

every U; V

N
and

u 2 N

,

we have D ec(u 1 U V )

D

ec(U V )

min

fjU j; jV jg Pr

f.

W e ha v e w = xy ^ x 2 u 1 U ^ y 2 V = ) uw = (ux)y ^ ux 2 U ^ y 2 V : So D ec(w ; u 1 U V )

D

ec(uw ; U

V

) hence D ec(u 1 U V )

D

ec(U V ). By symmetry

f

U and V , it remains to v erify that D ec(U

V

)

jU

j. This in- equalit y is trivial for jU j = 1. F

r

jU j < 1, let U = fu 1 ; : : : ; u jU j g. F

r

an y w 2 N

;

D ec(w ; U

V

) =

f(u

i ; u 1 i w ) j 1

i
jU

j ^ u 1 i w 2 V g

jU

j. 2 W e are ready no w to establish a connection b et w een the niteness

f

the n um b er

f

degrees and the niteness

f

the n um b er

f

decomp

sitions.

Lemma 2.4. F

r

every U; V ; W

N
,

we have D + ((U

V

):W ) < 1 ( ) D ec(U

W

) < 1 _ jV j 2 f0; 1g; D

((U
V

):W ) < 1 ( ) D ec(V W ) < 1 _ jU j 2 f0; 1g: Pr

f.

Let U; V ; W

N
.

F

r

ev ery x 2 N

,

d + (U V ):W (x) = jfy j (x; y ) 2 (U V ):W gj = jfv w j v 2 V ^ w 2 W ^ 9u 2 U; uw = xgj = jV :(U 1 x \ W )j and jU 1 x \ W j = D ec(x; U

W

).

SLIDE 10 9 Th us d + (U V ):W (x)

jV

j:D ec(x; U

W

); if x 2 U:W ^ V 6= ?; d + (U V ):W (x) = 0; if x 62 U:W _ V = ?: (i) W e establish the rst equiv alence. ) Assume that D + ((U V ):W ) < 1 ^ < jV j < 1. The follo wing in teger D = maxfd + (U

V

):W (x) < 1 j x 2 N

g

exists. Consequen tly D ec(x; U

W

)

D

for ev ery x 2 N

.

Hence D ec(U

W

)

D

< 1 : ( Assume that D ec(U

W

) < 1 _ jV j 2 f0; 1g. If V = ? then D + ((U

V

):W ) = D + (?) = : If jV j = 1 then 8 x 2 N

d

+ (U V ):W (x) 2 f0; 1g : Hence D + ((U V ):W )

2.

If D ec(U W ) < 1 ^ < jV j < 1 then, b y (i), 8x 2 N

d

+ (U V ):W (x)

D

where D = jV j:D ec(U

W

) : Hence D + ((U V ):W )

D

+ 1 < 1. The second equiv alence follo ws from the rst

ne

and the fact that d

(U

V ):W (x) = d + (V U ):W (x) for ev ery x 2 N

:

2 Since the niteness

f

a rational language is decidable and according to ab

v

e lemma, w e ma y decide the niteness

f

the n um b er

f

degrees

f

an elementary prexrecognizable relation (U V ):W , pro vided that w e are able to decide the niteness

f

the decomp

sition
f

U W (and V

W

). Consequen tly , w e shall fo cus no w

n

deciding the niteness

f

the n um b er

f

decomp

sitions
f

a Carte- sian pro duct U W

f

rational languages U and W . Lemma 2.5. F

r

every U; W 2 R at(N

),

we have D ec(U W ) = 1 ( ) 9 x; y ; z 2 N

(

y 6= " ^ xy

U

^ y

z
W

) : Pr

f.

( F

r

ev ery n

0,

w e ha v e D ec(U

W

)

D

ec(xy n z ; U

W

)

f(x;

y n z ); (xy ; y n1 z ) : : : ; (xy n ; z )g

n

+ 1 :

SLIDE 11 10 ) As U and W are rational languages, w e tak e resp ectiv ely p =

fs

1 U j s 2 N

g
and

q =

fW

s 1 j s 2 N

g
the

n um b er

f

the left residuals

f

U and the n um b er

f

the righ t residuals

f

W . Let n

pq

. Since D ec(U

W

) = 1, there exists t 2 N

suc

h that D ec(t; U

W

) > n. So t = u w = : : : = u n w n with u ; : : : ; u n 2 U , w ; : : : ; w n 2 W and ju j < : : : < ju n j. As n

pq

, there exist

i

< j

n

suc h that (u i ) 1 U = (u j ) 1 U ^ W (w i ) 1 = W (w j ) 1 : W e set x = u i , z = w j and let y 2 N

suc

h that u i y = u j . In particular w i = y w j . Let m; n 2 I N. Observ e that (u i ) 1 U = (u j ) 1 U = y 1 ((u i ) 1 U ) : Th us u 1 i U = y 1 ((u i ) 1 U ) = (y 1 ) 2 ((u i ) 1 U ) =

=

(y 1 ) m ((u i ) 1 U ) = (y m ) 1 ((u i ) 1 U ) : Since W (w j ) 1 = W (w i ) 1 = (W (w j ) 1 )y 1 , w e establish similarly that W (w j ) 1 = (W (w j ) 1 )(y n ) 1 : Since u i 2 U , " 2 (u i ) 1 U = (u i y m ) 1 U i.e. xy m 2 U and " 2 W (w j ) 1 = W (y n w j ) 1 i.e. y n z 2 W 2 According the follo wing prop

sition,

it turns

ut

that the c haracterization giv en in Lemma 2.5 is decidable. Prop

sition

2.6. W e c an de cide the niteness

f

D ec(U

W

) for al l r ational U; W

N
.

Pr

f.

W e establish that the follo wing c haracterization

f

Lemma 2.5 is decidable: 9 x; y ; z 2 N

(y

6= " ^ xy

U

^ y

z
W

) ( ) 9 x; y ; z 2 N

(y

6= " ^ y

x

1 U \ W z 1 ) As U and W ha v e resp ectiv ely a nite n um b er

f

left residuals and righ t residuals, it is equiv alen t to b e able to decide for an y rational language Y whether 9 y 2 N

r

f"g; y

Y

The rationalit y

f

Y means that there is a nite set Q

f

states, a graph A

QN

Q, an initial state f 2 Q and a subset F

Q
f

nal states suc h that Y is accepted b y (A; f ; F ) viz

SLIDE 12 11 Y = L (A; f ; F ) := fy 2 N

j

9 (f ; y (1); q 1 ); : : : ; (q jy j1 ; y (jy j); q jy j ) 2 A; q jy j 2 F g i.e. Y is the lab el language

f

paths from f to a state in F . Th us w e ha v e the follo wing equiv alence 9 y 2 N

r

f"g; y

Y

( ) f 2 F ^ 9 f 1 ; : : : ; f n 2 F 9

j
n

L (A; f ; f 1 ) \ : : : \ L (A; f n1 ; f n ) \ L (A; f n ; f j ) r f"g 6= ? whic h is decidable. 2 When instead

f

prexrecognizable relations, w e fo cus

n

regonizable

nes,

then w e can compute the decomp

sition

n um b er. Prop

sition

2.7. The mapping D ec : R ec(N

N
)

! I N [ f1g is r e cursive. As this prop

sition

is not necessary for the sequel, its pro

f

is giv en in app endix. 2.3 Dep endence Lev el As demonstrated in Sect. 4, the tec hnical notion

f

dep endence in tro duced b elo w is useful for the construction

f

a deterministic graph grammar for a regular graph presen ted b y prexrecognizable relations. F

r

an y w

rd

z 2 N

and

an y binary relation R

N
N
,

w e dene D ep(z ; R) :=

ft

j 9 y 2 N

;

(t; y z ) 2 R [ R 1 ^ jtj < jz jg

the

(prex) dep endenc e level

f

z according to R. Note that

D

ep(z ; R)

jN

j + : : : + jN j jz j1 D ep(z ; R)

D

ep(z ; R [ S )

D

ep(z ; R) + D ep(z ; S ) By maximalit y , w e extend the dep endence lev el

f

a w

rd

to all w

rds:

D ep(R) := maxfD ep(z ; R) j z 2 N

g

is the dep endenc e level

f

R (with max(E ) = 1 for an y innite set E

I

N). F

r

instance D ep((f"g fag)a

)

= 1 and D ep(((aa)

af"g)(aa)
)

= 1. The follo wing lemma establishes a relation b et w een the niteness

f

the decomp

sition

n um b er and the niteness

f

the dep endence lev el

f

an elemen tary prexrecognizable relation. Lemma 2.8. F

r

every U; V ; W

N
jU

j < 1 ^ D ec(V W ) < 1

=

) D ep((U

V

)W ) < 1 : Pr

f.

Let u 2 N

and

D ec(V ; W ) < 1. Since D ep(R [ S ) < 1 ( ) D ep(R) < 1 ^ D ep(S ) < 1; it is sucien t to sho w that D ep((fug V )W ) < 1. Let then z 2 N

.

W e ha v e to sho w that D ep(z ; (fug V )W ) has a nite upp er b

und

indep enden t

f

z . W e ha v e to consider the t w

cases

b elo w.

SLIDE 13 12 1. Let (t; y z ) 2 (fug V )W with jtj < jz j. Then t = uw and y z = v w for some v 2 V and w 2 W . In particular jw j < juw j = jtj < jz j. Hence sw = z and y s = v for some s 2 N + . In particular s = y 1 v 2 y 1 V . Using Lemma 2.3, the n um b er

f

suc h w is b

unded

b y D ec(z ; y 1 V W )

D

ec(y 1 V W )

D

ec(V W ) : Th us the n um b er

f

suc h t is b

unded

b y D ec(V W ). 2. Let (y z ; t) 2 (fug V )W with jtj < jz j. So y z = uw and t = v w for some v 2 V and w 2 W . In particular jv w j = jtj < jz j < jy z j = juw j. Hence jv j < juj and jw j < jz j. So u = y s and sw = z for some s 2 N + . Th us 1

jsj
juj

implying that there are at most juj c hoices for w . The n um b er

f

suc h t is b

unded

b y (jN j + : : : + jN j juj1 )juj. 2 3 T reewidth W e sa y that a binary relation

r

a graph R is a tr e e if R has a v ertex

f

n ull indegree, d

R

(r ) = 0, that is the r

t
f

R, viz R

(r

) = V R , and ev ery another v ertex is

f

indegree 1, viz d

R

(s) = 1 for an y s 6= r . The v ertices

f

a tree are also called no des. The subtr e e T =s

f

a tree T from a no de s is T =s := T jT

(s)

. A standard notion, in tro duced in [23], is the treedecomp

sition
f

a graph. Denition 3.1. A tr e ede c

mp
sition
f

a binary r elation R

N
N
is

a tr e e T

N
N
with

a (vertex lab el ling) mapping ` : V T ! } (V R ) such that (1) al l vertic es

f

R ar e c

ver

e d by `, viz for al l (u; v ) 2 R ther e exits a no de s

f

T such that u; v 2 `(s), (2) `(s) \ `(t)

`(r

) for al l s; t 2 V T and any no de r

n

an elementary undir e cte d p ath

f

T b etwe en s and t. W e denote by T w (T ; `) := maxfj`(s)j

1

j s 2 V T g the tr e ewidth

f

(T ; `), T w (R) := min fT w (T ; `) j (T ; `) tr e ede c.

f

Rg the tr e ewidth

f

R. " B 3 B 2 B A A 2 A 3 A 2 B AB 2 AB Fig. 3.1.

SLIDE 14 13 F

r

instance the relation R = f(A n ; A n+1 ) j n

0g

[ f(A m+1 B n ; A m B n+1 ) j m; n

0g

[ f(B n+1 ; B n ) j n

0g

depicted

n

Fig. 3.1 has the tree decomp

sition
f

treewidth 2 (see Fig. 3.2) whic h is minimal, th us T w (R) = 2. A 4 B 4 A 3 B " A B A B A 2 A 2 B B 2 A 2 B 2 A 3 A 3 B 2 B 3 A 3 B 3 A 4 A 4 B 3 B 4 A 3 B B 4 A 2 B 2 A 2 B 2 AB 3 B 4 A 3 B 3 A 2 B A 2 B B 3 AB 2 A 2 B 2 AB Fig. 3.2. A treedecomp

sition

Note that R

S

= ) T w (R)

T

w (S ) T w (R) < jV R j Let us consider the treewidth

f

the cartesian pro duct. Lemma 3.2. min (jE j; jF j)

1
T

w (E

F

)

min

(jE j; jF j) Pr

f.

Without loss

f

generalit y , w e ma y assume that jE j

jF

j. W e establish rst that T w (E F )

jE

j. Let (f n ) n0 b e a sequence with ff n j n 2 I Ng = F and suc h that if f i = f j for some i < j then f k = f i for ev ery i

k
j

. W e ma y

bserv

e that the follo wing liform tree T = f(a n ; a n+1 ) j n 2 I Ng with a 2 N and the lab elling `(a n ) = E [ ff n g dene a tree decomp

sition
f

E

F
f

treewidth jE j. W e establish no w that jE j1

T

w (E

F

). Let (T ; `) b e a tree decomp

sition
f

E

F

and let us sho w that T w (T ; `)

jE

j

1.

This is the case if there exists s 2 V T suc h that j`(s)j

jE

j. Assume therefore that j`(s)j < jE j for ev ery s 2 V T . By strong induction

n

n 2 I N, w e sho w that there is a path s ! s 1 ! : : : ! s n in T suc h that D n =

f(s

; s 1 ); : : : ; (s n1 ; s n )g [ T =s n ; `

is

a tree decomp

sition
f

E F with `(s )

`(s

1 )

:

: :

`(s

n ).

SLIDE 15 14

Induction

basis W e ha v e T =s = T .

Induction

step By h yp

thesis

j`(s n )j < jE j and let e 2 E r `(s n ). Since jF j

jE

j, F r `(s n ) 6= ?. Let then f 2 F r `(s n ). Since e = 2 `(s n ) (resp. f = 2 `(s n )), also e = 2 `(s i ) (resp. f = 2 `(s i )) for eac h i 2 [n] b ecause, b y induction h yp

thesis,

`(s )

`(s

1 )

:

: :

`(s

n ) : (i) F urthermore, D n is a tree decomp

sition
f

E F . Th us, there is a successor s n+1

f

s n suc h that e 2 S `

T
(s

n+1 )

.

Moreo v er, suc h a successor is unique. If not, w e w

uld

ha v e t w

no

des s and s connected b y an elemen tary path passing through s n and suc h that e 2 `(s) and e 2 `(s ). Then `(s) \ `(s )

`(s

n ) and this w

uld

con tradict e = 2 `(s n ). Similarly , w e conclude that there is a unique successor s n+1

f

s n suc h that f 2 S `

T
(s

n+1 )

.

But the edge (e; f )

f

E F has to b e co v ered b y the decomp

sition

D n , th us either e; f 2 `

T
(s

n+1 )

r

e; f 2 `

T
(s

n+1 )

.

Consequen tly s n+1 = s n+1 . In fact, all edges

f

feg F (resp. E ff g) ha v e to b e co v ered b y D n , hence feg [ E

S

`

T
(s

n+1 )

(resp.

E [ ff g

S

`

T
(s

n+1 )

.

In sum, E [ F

[

`

T
(s

n+1 )

:

(ii) But then, for D n to satisfy (2)

f

Denition 3.1,

ne

m ust ha v e [ `

T
(t)
`(s

n ); for ev ery t 2 T (s n ) r fs n+1 g. F rom this fact together with (i) , w e ma y conclude that D n+1 satises (2)

f

Denition 3.1 and that D n+1 co v ers all v ertices

f

E F . Hence D n+1 is a tree decomp

sition
f

E F . F urthermore, according to (ii) , for ev ery x 2 `(s n ), there is a no de s 2 T =s n+1 suc h that x 2 `(s). Again, due to (2)

f

Denition 3.1, w e ha v e x 2 `(s n+1 ). Hence `(s )

`(s

1 )

:

: :

`(s

n )

`(s

n+1 ). Finally w e ha v e

btained

a liform tree T = f(s n ; s n+1 ) j n 2 I Ng suc h that (T ; `) is a tree decomp

sition
f

E F with `(s )

:

: :

`(s

n )

:

: : and S n0 `(s n ) = E [ F . If jE j < 1 there exists n 2 I N suc h that j`(s n )j

jE

j whic h is imp

ssible.

Consequen tly jE j = 1 and then jE j

1

= 1 = T w (T ; `)

T

w (T ; `) : 2 According to ab

v

e lemma, when b

th

U and V are innite, T w (U w

V

w ) = 1, for eac h w 2 W . Hence T w

(U
V

)W

=

1. On the

ther

hand, as sho wn in [8], when b

th

U and V are nite, the graph (of the relation) (U

V

)W is regular (and

f

nite degree). Since the treewidth

f

eac h regular graph is nite [11], w e shall address the case when either U

r

V is p

ssibly

innite. Due to the symmetry , it is enough to consider

nly

the case when V is nite and, for

ur

purp

se,

it is ev en enough to tak e V = fv g with v 2 N

.

Th us a necessary condition for a nite treewidth in this case is giv en b elo w in terms

f

decomp

sition

n um b er.

SLIDE 16 15 Lemma 3.3. F

r

al l U 2 R at(N

);

v 2 N

and

W

N
T

w ((U fv g)W ) < 1 = ) D ec(U

W

) < 1 : Pr

f.

By con trap

sition.

Assume that D ec(U

W

) = 1. Let p = jfx 1 U j x 2 N

gj

b e the n um b er

f

the left residuals

f

the rational language U and let n

p.

Since D ec(U

W

) = 1, there exist u 2 U and w 2 W suc h that D ec(uw ; U

W

) > n. Consequen tly uw = u w = : : : = u n w n with u ; : : : ; u n 2 U , w ; : : : ; w n 2 W and ju j < : : : < ju n j. Since ju p j

p,

u p = xy v p with y 6= " and (xy ) 1 U = x 1 U : By induction

n

m 2 I N, x 1 U = (y m ) 1 (x 1 U ). F

r

ev ery p

i
n,

w e let u i = xy v i viz v i = (xy ) 1 u i 2 (xy ) 1 U and v i 2 (xy m ) 1 U , for eac h m 2 I N. Th us xy

fv

p ; : : : ; v n g

U

. No w (xy

fv

p w p ; : : : ; v n w n gfv w p ; : : : ; v w n g)

(U
fv

g)W b ecause for ev ery m

and

i; j 2 fp; : : : ; ng, w e ha v e (xy m v i w i ; v w j ) = (xy m v j w j ; v w j ) = (xy m v j ; v )w j 2 (U fv g)W : By Lemma 3.2, w e

btain

T w ((U

fv

g)W )

T

w (xy

fv

p w p ; : : : ; v n w n gfv w p ; : : : ; v w n g)

n
p

: Hence T w ((U

fv

g)W ) = 1. 2 Note that the assumption

f

rationalit y

f

U in Lemma 3.3 is necessary . Indeed, it is sucien t to consider U = fabab 2 : : : ab n j n

1g

and W = fab m : : : ab n j 2

m
ng

: F

r

ev ery n

1,

D ec(ab: : : ab n ; U W ) = n

1

hence D ec(U

W

) = 1. But (U f"g)W is acyclic, hence T w ((U

f"g)W

) = 1. 4 An in ternal represen tation

f

regular graphs W e

p

en this section with reminders

n

regular graphs. W e giv e a simple pro

f
f

the fact, rst established in [13], that the treewidth

f

a regular graph is nite. W e also recall some kno wn c haracterizations

f

in teresting sub classes

f

regular graphs. Then w e pro v e a series

f

lemmas that yield a construction

f

a deterministic graph grammar from a graph presen tation b y prexrecognizable relations, pro vided that they satisfy some additional niteness assumptions. As established in the main theorem dev

ted

to sev eral c haracterizations

f

regular graphs, these assumptions turn

ut

to b e equiv alen t to the niteness

f

decom- p

sition

n um b er (com bined with

ther

simple and decidable criteria) that is decidable (see Prop

sition

2.6).

SLIDE 17 16 4.1 Graph Grammars and Regular Graphs W e tak e a nite set T

f

lab els with non n ull arities. A hyp er ar c is a w

rd

as 1 : : : s p lab elled b y a

f

arit y p > and joining in

rder

the v ertices s 1 ; : : : ; s p . In particular an arc s a

!

t is the w

rd

ast with a

f

arit y 2. A hyp er gr aph H is a set

f

h yp erarcs. A gr aph gr ammar R is a nite set

f

rules

f

the form ax 1 : : : x p

!

H where x 1 ; : : : ; x p are distinct v ertices (and a is a lab el

f

arit y p) and H is a nite h yp ergraph. The lab els

f

the left hand sides

f

R are the nonterminals

f

R . The

ther

lab els in R are the terminals and are

f

arit y 2 (they

nly

lab el arcs). W e sa y that R is deterministic if there is

nly
ne

rule p er nonterminal. An example

f

a deterministic graph grammar is depicted

n

Fig. 4.1. A rewriting M

!

R N consists in c ho

sing

a nonterminal h yp erarc

1 2 3 1 2 1 2 3 1 2

et A B c a A A a c B b d a Fig. 4.1. A deterministic graph grammar R X = as 1 : : : s p in M and a rule ax 1 : : : x p

!

H in R to b e applied. The v ertices x i in H indicate ho w to replace X b y H : N = (M

X

) [ fbg (t 1 ): : : g (t p ) j bt 1 : : : t p 2 H g for some matc hing function g mapping x i to s i , and the

ther

v ertices

f

H injectiv ely to v ertices

utside
f

M . This rewriting is denoted b y M

!

R; X N . Note that

!

R is not in general a functional relation, ev en when R is deterministic. Nev ertheless M

!

R; X 1

:

: :

!

R; X n N i M

!

R; X

(1)
:

: :

!

R; X

(n)

N for an y X i 2 M and for an y p erm utation

n

[n]. Th us, it mak es sense to dene a c

mplete

p ar al lel r ewriting = ) R as follo ws: M = ) R N if M

!

R; X 1

:

: :

!

R; X n N where X 1 ; : : : ; X n are all nonterminal h yp erarcs

f

M . F

r

instance, t w

rst

steps

f

the parallel deriv ation from the h yp erarc A according to the grammar R

f

Fig. 4.1 are depicted as follo ws:

1 2 1 2 1 2

A a c B a c c a a A A b d

SLIDE 18 17 W e denote b y [M ] = fast 2 M j a a terminal

f

R g the set

f

terminal arcs

f

M . A graph G is gener ate d by a deterministic gr aph gr ammar R from a h yp ergraph H if G is isomorphic to S n0 [H n ] where H = H and H n = ) R H n+1 for ev ery n

0.

F

r

instance, the graph G generated b y R is depicted

n

Figure 4.2.

1 2

c a a c b d a Fig. 4.2. Graph G generated b y the grammar R (of Fig. 4.1) Denition 4.1. A r e gular gr aph is a gr aph gener ate d by a deterministic gr aph gr ammar fr

m

a nite hyp er gr aph. Let us recall that these graphs are the equational graphs

f

[11] and, as stipulated in [2], should rather b e called HRequational since eac h grammar rule represen ts a h yp eredge replacemen t

p

eration. Whenev er a regular graph is a tree, socalled r e gular tr e e, then it has a nite n um b er

f

nonisomorphic subtrees. The innite parallel deriv ation

f

a regular graph G b y a grammar induces a regular deriv ation tree. The v ertices

f

this tree ma y b e lab elled b y the pro duced v ertices

f

G. In this w a y , w e get a treedecomp

sition
f

G and the follo wing lemma ma y b e established. Lemma 4.2. A ny r e gular gr aph is

f

nite tr e ewidth. Pr

f.

Let G b e a regular graph: there is a deterministic graph grammar R , a nonterminal h yp erarc X 2 Dom(R ), and an innite parallel deriv ation fX g = H = ) R H 1 = ) R : : : = ) R : : :

SLIDE 19 18 suc h that S n0 [H n ] = G. By denition

f

the parallel rewriting, 8 n 2 I N 8 X 2 H n r [H n ] 9 H X ;n X

!

G H X ;n ^ H n+1 = [H n ] [ [ fH X ;n j X 2 H n r [H n ]g : W e dene the follo wing (unlab elled) tree: T = f(X ; n)

!

(Y ; n + 1) j n

^

X 2 H n r [H n ] ^ Y 2 H X ;n r [H X ;n ]g : As the grammar R has a nite n um b er

f

nonterminals, note that this tree is regular. No w w e dene the lab elling

f

T as follo ws: `(X ; n) = V H X ;n for ev ery n

and

X 2 H n

[H

n ] : In particular T w (T ; `) < max fjV H j j H 2 Ran (R )g < 1 : F urthermore (T ; `) is a tree decomp

sition
f

G hence T w (G)

T

w (T ; `) < 1. 2 W e n um b er the v ertices

f

a graph G b y a mapping g from V G in to I N, called a gr aduation

f

G. W e will dene the regularit y

f

a graph b y v ertices

f

increasing graduation. Precisely , for ev ery n

0,

w e denote b y G g ;n := G

fs

j g (s)ng = fs a

!

G t j g (s)

n

^ g (t)

ng

the n rst lev els

f

G according to g , and w e denote b y @ g ;n G := fs j g (s)

n

^ 9 t(s ! G t ^ g (t) > n)g the nth fr

ntier
f

G b y g . Note that, for G connected, w e ha v e @ g ;n G = V G g ;n \ V GG g ;n : W e sa y that a graph G is r e gular by g if there exists a deterministic graph grammar R , a nite h yp ergraph X and an in teger m suc h that for ev ery n

0,

R generates from X b y n parallel rewritings the graph G g ;m+n

f

terminal arcs, plus a set

f

nonterminal h yp erarcs

f

v ertex set @ g ;m+n G i.e. 8 n 2 I N 9 H ( X = ) R n H ^ [H ] = G g ;m+n ^ V H [H ] = @ g ;m+n G ) : In particular [X ] = G g ;m and the v ertex set V X [X ]

f

nonterminal h yp erarcs

f

X is the set @ g ;m G. Note that for G connected, w e ha v e @ g ;m+n G = ? ( ) G g ;m+n = G : When V G is a language, then the length ma y b e used as a graduation. F

r

SLIDE 20 19

1 2 3 3 1 2 1 2 3 3 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 3 3 1 2

c a E a C c a a D F A D A a c B B C d c a c a E G d c a D F c a a G A E b b d b Fig. 4.3. A deterministic graph grammar R 1 generating G b y distance instance, the graph (" a

!

A)A

B
is

regular b y length. Another usual graduation is the distanc e from a giv en v ertex set C : d G (s; C ) := min fd G (s; t) j t 2 C g where d G (s; t) := min (fn j s ! G n tg [ f1g) F

r

instance the graph G (see Fig. 4.2) remains regular b y distance from the set

f

v ertices f1; 2g using the deterministic graph grammar R 1 (see Fig. 4.3). Suc h a transformation is general and eectiv e. Prop

sition

4.3. [22],[7] A ny c

nne

cte d r e gular gr aph

f

nite de gr e e is ef- fe ctively r e gular by distanc e fr

m

any nonempty nite vertex set. An in ternal represen tation

f

the regular graphs

f

nite degree is giv en b y the rational restrictions

f

the transition graphs

f

pushdo wn automata [22], [7]. Another in ternal represen tation is b y prex rewriting

f

string rewrite systems rationally righ tcon trolled. Prop

sition

4.4. [8] The r e gular gr aphs

f

nite de gr e e ar e, up to isomorphism and in an ee ctive way, the gr aphs

f

the form: n [ i=1 (u i a i

!

v i )W i with n

and

8 i 2 [n]

u

i ; v i 2 N

^

W i 2 R at(N

)
and

these gr aphs ar e ee ctively r e gular by length.

SLIDE 21 20 F

r

instance (A a

!

E G)L + (B a

!

E D )M + (E a

!

A)M + (E b

!

H )M + (F d

!

H )M + (F G c

!

B )L + (F D c

!

C )M + (H a

!

B )M where M = (D + G)

G

and L = " + M , represen ts the graph G

f

Fig. 4.2 as a subset

f

N

T
N
where

N = fA; B ; C ; D ; E ; F ; G; H g and T = fa; b; c; dg. Prop

sition

4.4 ma y b e extended to all regular graphs (including those

f

innite degree) pro vided that lefthand and righ thand sides

f

rewrite rules are rational languages. Prop

sition

4.5. [9] A ny r e gular gr aph is, up to isomorphism and in an ef- fe ctive way, pr exr e c

gnizable.

W e should p

in

t

ut

that the con v erse

f

the ab

v

e prop

sition

is false. F

r

instance (A + a

!

")A

is

not a regular graph. Ho w ev er Barthelmann has established that, among VRequtional graphs (hence equiv alen tly prexrecognizable [1]), the regular

nes

are precisely those

f

nite treewidth [2]. 4.2 Innite Graph Grammars for Prexrecognizable Graphs W e shall use the length as graduation for the construction

f

a deterministic graph grammar R G generating a graph G presen ted b y prexrecognizable relations. More precisely , the grammar will generate the v ertices

f

G in the

rder

increasing their length, at eac h step

f

= ) R G . F

r

this aim, the follo wing

bserv

a- tion is useful. Consider t w

v

ertices s; t 2 V G and let z b e the longest common sux

f

s and t, viz there exists x; y 2 N

suc

h that s = xz , t = y z and su (x) \ su (y ) = f"g. W e are going to establish that under some uniform condition based

n

lengths

f

x and y , there is no edge b et w een s and t, pro vided that G satises the requiremen ts

f

the next lemma. Henceforth G is the graph

f

the form (1.1) satisfying 8 i 2 [n]

jU

i j < 1 _ jV i j < 1

:

(4.1) Then the follo wing in teger p G = max

min

fjuj; jv jg

9i

2 [n]; u 2 U i ^ v 2 V i

(4.2)

exists b ecause p G = max

min

fmax u2U i (juj); max v 2V i (jv j)g

i

2 [n]

:

Lemma 4.6. xz a

!

y z = 2 G for e ach a 2 T , al l z ; z 2 N

and

al l x; y 2 N

satisfying

jxj

p

G , jy j

p

G and neither z is a sux

f

z nor vic eversa. Pr

f.

Let x; y ; z ; z 2 N

suc

h that jxj

p

G , jy j

p

G and neither z is a sux

f

z nor vicev ersa. Assume b y con tradiction that xz = uw and y z = v w where u 2 U i and v 2 V i for some i 2 [n]. Then jz j > jw j and jz j > jw j b ecause neither z is a sux

f

z nor vicev ersa. Hence jxj < juj and jy j < jv j. No w min (juj; jv j) > p G , that con tradicts (4.2) . 2

SLIDE 22 21 This lemma suggests considering subgraphs

f

G, the v ertices

f

whic h ha v e a common sux with remaining big prex together with small adjacen t v ertices. Th us giv en z 2 N

,

w e let H z =

xz

a

!

y z 2 G

jxj
p

G ^ jy j

p

G

K

z =

t

a

!

y z 2 G [ G 1

jtj

< jz j + p G ^ jy j

p

G

G

z = H z [ K z Fig. 4.4 depicts G z with v ertices

rdered

from left to righ t according to their length. K z t H z y z jz j + p G Fig. 4.4. Let us construct no w a deterministic graph grammar R G that generates G b y length from a nite h yp ergraph. A t rst glance, w e construct a graph grammar R 1 G that is p

ssibly

innite. Then w e sho w that b y iden tifying appropriate rules, pro vided that G fullls some conditions, w e get a nite graph grammar still generating G. Finally , in Subsection 4.4, w e pro vide a shortcut for a direct construction

f

a nite graph grammar. W e consider an injection

from

N

in

to a new set

f

non terminal sym b

ls

and, giv en z 2 N

,

w e let V z = V K z r V H z = ft j t a

!

y z 2 G [ G 1 ^ jtj < jz j + p G ^ jy j

p

G g : The grammar R 1 G consists

f

the follo wing rules, for eac h z 2 N

:

the lefthand side

z

= (z )s 1 : : : s n with fs 1 ; : : : ; s n g = V z and n = jV z j the terminal rhs

z

=

t

a

!

y z 2 G [ G 1

jtj
jz

j + p G ^ jy j = p G

the

nonterminal rhs

z

= f az j a 2 N ^ G az 6= ?g where rhs means righ thand side. In sum, R 1 G = f z

!
z

[

z

j z 2 N

g

: By construction, w e get the follo wing lemma.

SLIDE 23 22 Lemma 4.7. R 1 G gener ates G (by length) form the fol lowing nite hyp er gr aph H =

"

[ fx a

!

y 2 G j jxj < p G ^ jy j < p G g and, for e ach z 2 N

,

gener ates G z (by length) fr

m

the hyp er e dge

z

. Pr

f.

Since G = G " [

x

a

!

y 2 G

jxj

< p G ^ jy j < p G

,

it is enough to establish that, for eac h z 2 N

,

G z is generated from

z

b y R 1 G . Let then z 2 N

.

Let r a

!

s 2 G z . W e claim that r a

!

s deriv es from

z

b y R 1 G . Assume that jr j

jsj.

(The

ther

case is symmetric.) Then jsj

jz

j + p G and there exist x 2 N p G and y 2 N

suc

h that s = xy z . Hence r a

!

xy z 2

y

z , viz r a

!

xy z is a terminal arc

f

the righ thand side

f

the rule with the lefthand side

y

z . No w, b y induction, it follo ws that the non terminal h yp erarc

y

z deriv es from

z

b y R 1 G . Con v ersely assume that r a

!

s is a terminal arc deriv ed from

z

b y R 1 G . Then r a

!

s 2

y

z for some y 2 N

.

By denition,

y

z

G

z . 2 4.3 F rom Innite Graph Grammars to Finite

nes

In

rder

to iden tify rules

f

R 1 G w e need few lemmas. The rst

ne

attests that the lengths

f

lefthand sides

f

R 1 G are b

unded.

Lemma 4.8. L et G b e a gr aph

f

the form (1.1) satisfying 4.1. Then max z 2N

jV

z j

D

ep(G) . Pr

f.

Assume that (4.1) holds and let z 2 N

.

Observ e rst that V z = ft a

!

y z 2 G [ G 1 j jtj < jz j + p G ^ jy j

p

G g = ft a

!

r sz 2 G [ G 1 j jtj < jz j + p G ^ jr j

^

jsj = p G g = ft a

!

r sz 2 G [ G 1 j jtj < jsj + jz j ^ jsj = p G g = ft a

!

r sz 2 G [ G 1 j jtj < jsz j ^ jsj = p G g

ft

a

!

r sz 2 G [ G 1 j jtj < jsz jg : It follo ws that jV z j

ft

j t a

!

r sz 2 G [ G 1 ^ t < jsz jg

=

D ep(sz ; G) : Hence max z 2N

jV

z j

max

z 2N

(D

ep(sz ; G))

D

ep(G) < 1 . 2 W e shall in tro duce no w an equiv alence relation b et w een the rules

f

R 1 G . W e iden tify a rule

f

a lefthand side

z

with a rule

f

a lefthand side

z

so as to ha v e the graphs generated from

z

and

z

, namely G z and G z , isomorphic. In fact these rules are considered as equiv alen t when z

G

z according to an equiv alence relation

n

N

.

In

rder

to dene this equiv alence, w e pro ceed as follo ws. Let T b e a new alphab et in bijection with T . Giv en z 2 N

,

w e dene the mapping g z : V z ! } ((T [ T )N

)

b y: t 7 ! g z (t) = fay j t a

!

K z y z g [ fa y j y z a

!

K z tg :

SLIDE 24 23 This leads to the equiv alence

G
n

N

dened

b y z

G

z if

W

i z 1 = W i z 1 for ev ery i 2 [n]; 9 h z ;z : V z ! V z bijectiv e and g z = g z

h

z ;z : Lemma 4.9. L et G b e a gr aph

f

the form (1.1) satisfying (4.1) . Then, for e ach z ; z 2 N

,

G z and G z ar e isomorphic, whenever z

G

z . Pr

f.

Let z ; z 2 N

suc

h that z

G

z . Then W i z 1 = W i z 1 for ev ery i 2 [n]. In

rder

to pro v e that H z and H z are isomorphic, it is enough to sho w that H z z 1 = H z z 1 where H z z 1 = fx a

!

y j xz a

!

G y z ^ jxj

p

G ^ jy j

p

G g : By symmetry , it is enough to establish that H z z 1

H

z z 1 . Let x a

!

y 2 H z z 1 . Then xz a

!

y z 2 G with jxj

p

G and jy j

p

G . There exists i 2 [n], u 2 U i , v 2 V i , w 2 W i suc h that xz = uw , y z = v w and a = a i . By denition

f

p G , w e ha v e min fjuj; jv jg

p

G

min

fjxj; jy jg. Th us juj

jxj

_ jv j

jy

j hence jw j

jz

j. It follo ws that there exists t 2 N

suc

h that tz = w . Consequen tly x = ut and y = v t. F urthermore tz 2 W i so t 2 W i z 1 = W i z 1 i.e. tz 2 W i . Hence utz a

!

G v tz i.e. xz a

!

G y z with jxj; jy j

p

G . Th us x a

!

y 2 H z z 1 meaning that H z z 1

H

z z 1 and b y symmetry w e ha v e H z z 1 = H z z 1 : (4.3) Consequen tly H z and H z are isomorphic. T

close

the pro

f,

it is enough to establish that the bijection h z ;z extends to an isomorphism b et w een K z and K z . Indeed h z ;z is compatible with edges adjacen t to V z and V z b ecause g z = g z

h

z ;z . In sum, the follo wing mapping h : V G z

!

V G z s 7 ! h(s) :=

h

z ;z (s) if s 2 V z (i.e. jsj < jz j + p G ) (sz 1 )z

therwise

is an isomorphism from G z to G z . 2 In

rder

to apply

G

to the construction

f

a nite graph grammar generating G, w e need to establish that

G

is

f

nite index. Lemma 4.10. L et G b e a gr aph

f

the form (1.1) satisfying ( 8i 2 [n]; jU i j < 1 _ jV i j < 1 ) ^ D ep(G) < 1 : (4.4) Then

G

is

f

nite index. Pr

f.

Let z ; z 2 N

.

W e write ^ g z to denote the follo wing (m ulti) set n g z (t);

fs

j g z (s) = g z (t)g

t

2 V z

:

F

r

z

G

z , w e need a bijection h z ;z . The latter exists if and

nly

if ^ g z = ^ g z . Consequen tly [z ]

G

is c haracterized b y b

th

(W i z 1 j i 2 [n]) and ^ g z . In

rder

to establish that f[z ]

G

j z 2 N

g

is nite, w e ha v e to pro v e the niteness

f

SLIDE 25 24

f(W

i z 1 j i 2 [n]) j z 2 N

g,

that is

b

vious, and

f

^ g z j z 2 N

g.

F

r

the latter,

bserv

e that, for an y t 2 V z , g z (t) = [ a2T a

N

p G \ [ i2[n] a i =a t2U i W i V i W i z 1

[

a

N

p G \ [ i2[n] a i =a t2V i W i U i W i z 1

!

: Th us g z (t) 2 M z , where M z stands for the follo wing nite family

f

rational sets: M z = (

[

i2I a i (N p G \ V i W i z 1 )

[
[

i2J a i (N p G \ U i W i z 1 )

I
[n]

^ J

[n]

) : Consequen tly g z (V z ) = fg z (t) j t 2 V z g is a subset

f

M z . But jg 1 z (g z (t))j

jV

z j for eac h t 2 V z , hence ^ g z

g

z (V z )

jV

z j

M

z

jV

z j

M

z

max

z 2N

jV

z j

:

Then, b y Lemma 4.8, ^ g z

M

z

[D

ep(G)] viz ^ g z is a subset

f

a nite set b ecause D ep(G) < 1. Since ^ g z 2 } (M z

[D

ep(G)] ), w e ha v e f ^ g z j z 2 N

g
[

z 2N

}

(M z

[D

ep(G)]) : T

close

the pro

f,

it is enough to

bserv

e that the righ thand side

f

ab

v

e inclusion is a nite set. Indeed, b y examining the denition

f

M z , w e notice that

[

z 2N

}

(M z

[D

ep(G)])

dep

ends

n

constan ts jT j, jN j, n, p G and

n

the n um b er

f

deriv ativ es

f

eac h rational set U i W i and V i W i , for i 2 [n]. 2 In

rder

to compute the equiv alence classes

f
G

, the follo wing lemma is useful. Lemma 4.11. L et G b e a gr aph

f

the form (1.1) satisfying (4.4) . Then

G

is stable

n

the left. Pr

f.

Let z 2 N

and

b 2 N . W e establish rst that V bz

V

z [ (V H z \ N p G z ). Let t 2 V bz . By denition

f

V bz w e ha v e t < p G + jbz j = p G + jz j + 1 viz jtj

p

G + jz j. The follo wing cases ma y therefore b e distinguished.

Case
f

jtj < p G + jz j. Then

b

viously t 2 V z .

SLIDE 26 25

Case
f

jtj = p G + jz j. Here, w e claim that t 2 V H z \ N p G z . Since t 2 V bz , there exists y 2 N p G suc h that t a

!

y bz 2 K bz (resp.

r

y bz a

!

t 2 K bz ) for some a 2 T . Let x; z 2 N

with

jxj = p G (and jz j = jz j) suc h that t = xz . By Lemma 4.6, w e can conclude that z = z . Hence t 2 V H z \ N p G z . W e ha v e th us established that V bz

V

z [ (V H z \ N p G z ). Let z 2 N

suc

h that z

G

z and let us sho w that bz

G

bz . Ob viously , W i (bz ) 1 = (W i z 1 )b 1 = (W i z 1 )b 1 = W i (bz ) 1 , for all i 2 [n], b ecause W i z 1 = W i z 1 for all i 2 [n]. It remains to establish the existence

f

a bijection h bz ;bz : V bz ! V bz suc h that g bz = h bz ;bz

g

bz . Since V z \ (V H z \ N p G z ) = ?, this bijection ma y b e dened as follo ws: h bz ;bz (t) = ( h z ;z (t); if t 2 V z ; (tz 1 )z ; if t 2 V H z \ N p G z ; for all t 2 V bz . Let t 2 V bz and let us sho w that g bz (t) = g bz (h bz ;bz (t)).

Case
f

t 2 V z . g bz (t) = fay j t a

!

y bz 2 K bz g [ fay j y bz a

!

t 2 K bz g = fay b j t a

!

y bz 2 K bz gb 1 [ fay b j y bz a

!

t 2 K bz gb 1 =

fay

b j t a

!

y bz 2 K z ^ jy j

p

G g [ fa y b j y bz a

!

t 2 K z ^ jy j

p

G g

b

1 = g z (t)b 1 \ (T [ T )N p G = g z (h z ;z (t))b 1 \ (T [ T )N p G = g z (h bz ;bz (t))b 1 \ (T [ T )N p G = g bz (h bz ;bz (t)):

case
f

t 2 V H z \ N p G z W e set t = xz for some x 2 N p G . Then g bz (t) = fay j xz a

!

y bz 2 K bz g [ fa y j y bz a

!

xz 2 K bz g = fay j xz a

!

y bz 2 H z ^ jy j

p

G g [ fa y j y bz a

!

xz 2 H z ^ jy j

p

G g = fay j x a

!

y b 2 H z z 1 ^ jy j

p

G g [ fa y j y b a

!

x 2 H z z 1 ^ jy j

p

G g. No w, using (4.3) established in the pro

f
f

Lemma 4.9, w e ha v e g bz (t) = fay j x a

!

y b 2 H z z 1 ^ jy j

p

G g [ fa y j y b a

!

x 2 H z z 1 ^ jy j

p

G g = fay j xz a

!

y bz 2 H z ^ jy j

p

G g [ fay j y bz a

!

xz 2 H z ^ jy j

p

G g = fay j xz a

!

y bz 2 H z ^ jy j

p

G g [ fay j y bz a

!

xz 2 H z ^ jy j

p

G g = fay j xz a

!

y bz 2 K bz g [ fa y j y bz a

!

xz 2 K bz g = g bz (xz ) = g bz ((tz 1 )z ) = g bz (h z ;z (t)): 2

SLIDE 27 26 Using the ab

v

e lemma, w e ma y readily compute the equiv alence classes

f
G

. As the represen tativ e

f

a class, w e tak e the minim um elemen t w.r.t. the follo wing (hierarc hic) w ellordering: x 4 y ( ) jxj

jy

j _ (jxj = jy j ^ x 6 lex y ) where 6 lex stands for the lexicographic extension

f

an

rdering

6

f

N . W e note M

G

the set

f

suc h represen tativ es

f

the equiv alence classes

f
G

: M

G

= fmin 4

[z

]

G
j

z 2 N

g

: W e construct a nite deterministic semiautomaton A G = (M

G

; ";

)
v

er N , where M

G

is the set

f

states

f

A G , " is the initial state and

(z

; a) = min 4

[az

]

G
.

This semiautomaton has the follo wing prop ert y: L A G ("; z ) = ] [z ]

G

: Of course, in practice, w e should dev elop A G b y length, starting from ". 4.4 Construction

f

a Finite Graph Grammar In view

f

the lemmas

f

this section w e are able to giv e a direct construction

f

a nite deterministic graph grammar R G that generates (a graph isomorphic to) G. The grammar R G consists

f

the follo wing rules, for eac h z 2 M

G

the lefthand side

z

= (z )s 1 : : : s n with fs 1 ; : : : ; s n g = V z and n = jV z j the terminal rhs

z

=

t

a

!

y z 2 G [ G 1

jtj
jz

j + p G ^ jy j = p G

the

nonterminal rhs

z

= fh z ;az ( z ) j a 2 N ^ z = min 4

[az

]

G
^

G az 6= ?g where h z ; az ( z ) = h z ; az ((z )s 1 : : : s jV z j ) = (z )h z ; az (s 1 ) : : : h z ; az (s jV z j ). On the whole R G = f z

!
z

[

z

j z 2 M

G

g : Example 4.12. Let N = fAg b e a singleletter alphab et and T = fa; b; cg. Consider the follo wing graph G

N
T
N
G

= (AAA a

!

")A

[

(" b

!(AA)
)"

[ (" c

!

A(AA)

)"

represen ted

n

Fig. 4.5. W e sk etc h no w the construction

f

R G . First w e compute p G = 0. Then w e dev elop A G starting from its initial state ". In

rder

to c hec k whether a transition

f

A G lab elled b y A from some state z leads to a new state Az

r

to some state already constructed, w e need to test the equiv alence

f

Az with formerly constructed states. F

r

this aim, w e need to compute the mapping g z for eac h p

ten

tial state z

f

A G . This is summarized in T able 4.6. F

r

k 2 [5],

SLIDE 28 27 A 6 A 5 A 4 A 3 " A A 2 a a a a a b b b b c c c c Fig. 4.5. w e c hec k that A k = 2 S k 1 i=0 [A i ]

G

. F

r

A 6 w e notice that there is a mapping h A 4 ;A 6 : V A 4 ! V A 6 suc h that g A 4 = g A 6

h

A 4 ;A 6 , namely h A 4 ;A 6 (") = "; h A 4 ;A 6 (A) = A 3 ; h A 4 ;A 6 (A 2 ) = A 4 ; h A 4 ;A 6 (A 3 ) = A 5 : Moreo v er W i (A 4 ) 1 = W i (A 6 ) 1 , for i 2 f1; 2; 3g. (Recall that W 1 = A

and

W 2 = W 3 = ".) Hence A 4

G

A 6 . This leads to the semiautomaton A G depicted " A A 2 A 2 A 3 A 4 A 5 A A A A A A A Fig. 4.7. A G

n

Fig. 4.7 meaning that

G

has the follo wing classes: ["] = f"g; [A] = fAg; [A 2 ] = fA 2 g; [A 3 ] = fA 3 g; [A 4 ] = A 4 (A 2 )

;

[A 5 ] = A 5 (A 2 )

:

In the next step, w e compute the rules

f

R G , as summarized in T able 4.8. These rules are depicted

n

Fig. 4.9. No w H =

"

[ fx a

!

y 2 G j jxj < p G ; jy j < p G g =

"

b ecause fx a

!

y 2 G j jxj < p G ; jy j < p G g = ?. Hence G is generated b y R G from ["]. 4.5 Sev eral Characterizations

f

Regular Graphs In the sequel, w e summarize the results

f

this pap er. In addition to the c haracterization based

n

treewidth [1], w e giv e three

ther

c haracterizations

f

the regular graphs, within the graphs presen ted b y prexrecognizable relations. Theorem 4.13. Given a pr exr e c

gnizable

gr aph G = S n i=1 (U i a i

!

V i )W i with n

0,a

1 ; : : : ; a n 2 T and U 1 ; V 1 ; W 1 ; : : : ; U n ; V n ; W n 2 R at(N

)
f?g,

the fol lowing c

nditions

ar e e quivalent:

SLIDE 29 28 V " = ? V A = f"g g A (") = faA 2 ; b(A 2 )

A;

c(A 2 )

g

V A 2 = f"; Ag V A 3 = f"; A; A 2 g g A 2 (") = faA; b(A 2 )

;

c(A 2 )

Ag

g A 3 (") = fa; b(A 2 )

A;

c(A 2 )

g

g A 2 (A) = faA 2 g g A 3 (A) = faAg g A 3 (A 2 ) = faA 2 g V A 4 = f"; A; A 2 ; A 3 g V A 5 = f"; A 2 ; A 3 ; A 4 g g A 4 (") = fb(A 2 )

;

c(A 2 )

Ag

g A 5 (") = fb(A 2 )

A;

c(A 2 )

g

g A 4 (A) = fag g A 5 (A 2 ) = fag g A 4 (A 2 ) = faAg g A 5 (A 3 ) = faAg g A 4 (A 3 ) = faA 2 g g A 5 (A 4 ) = faA 2 g V A 6 = f"; A 3 ; A 4 ; A 5 g g A 6 (") = fb(A 2 )

;

c(A 2 )

Ag

g A 6 (A 3 ) = fag g A 6 (A 4 ) = faAg g A 6 (A 5 ) = faA 2 g T able 4.6. (1) G is r e gular (2) T w (G) < 1 (3) 8i 2 [n]; (jU i j < 1 ^ D ec(V i W i ) < 1) _ (jV i j < 1 ^ D ec(U i W i ) < 1) (4) 8i 2 [n]; (jU i j < 1 ^ D

((U

i

V

i )W i ) < 1)_ (jV i j < 1 ^ D + ((U i V i )W i ) < 1) (5) (8i 2 [n]; jU i j < 1 _ jV i j < 1) ^ D ep(G) < 1 (6) G is r e gular by length. Pr

f.

(6) = )(1) Immediate. (1) = )(2) By Lemma 4.2. (2) = )(3) Supp

se

that T w (G) < 1 and let i 2 [n]. As (U i a i

!

V i )W i

G;

T w ((U i a i

!

V i )W i ) < 1 . Let u 2 U i ; v 2 V i ; w 2 W i . W e ha v e T w ((fug V i )W i ) < 1 hence D ec(V i W i ) < 1 b y Lemma 3.3 T w ((U i fv g)W i ) < 1 hence D ec(U i W i ) < 1 b y Lemma 3.3 T w (U i

V

i ) = T w ((U i V i )w ) < 1 hence jU i j < 1 _ jV i j < 1 b y Lemma 3.2 (3) ( ) (4) By Lemma 2.4. (3) = ) (5) By Lemma 2.8. (5) = ) (6) By construction, in view

f

Lemmas 4.7, 4.9, 4.10 and 4.11. 2 T

close

this section, w e com bine Prop

sition

2.6 with condition (3)

f

The-

rem

4.13.

SLIDE 30 29

"

= ["]

"

= f" b

!

"g

"

= fh A;A ( A )g = f A g

A

= [A] "

A

= f" c

!

Ag

A

= fh A 2 ;A 2 ( A 2 )g = f A 2 g

A

2 = [A] " A

A

2 = f" b

!

A 2 g

A

2 = fh A 3 ;A 3 ( A 3 )g = f A 3 g

A

3 = [A 3 ] " A A 2

A

3 = f" c

!

A 3 ; " a

A

3 g

A

3 = fh A 4 ;A 4 ( A 4 )g = f A 4 g

A

4 = [A 4 ] " A A 2 A 3

A

4 = f" b

!

A 4 ; A a

A

4 g

A

4 = fh A 5 ;A 5 ( A 5 )g = f A 5 g

A

5 = [A 5 ] " A 2 A 3 A 4

A

5 = f" c

!

A 5 ; A 2 a

A

5 g

A

5 = fh A 4 ;A 6 ( A 4 )g = f[A 4 ] " A 3 A 4 A 5 g T able 4.8.

; ; ;

A 4 A 2 A 2 A 5 " A 2 A 3 A 2 A " " A 4 A 2 A 4 A 3 " A A 2 A A 3 A " A " A " A 3 A 2 A A 3 " " " " a [A] b ["] [A 3 ] b [A 2 ] [A 4 ] [A 5 ] b a a c [A 2 ] [A] [A 3 ] [A4] c c [A 4 ] [A 5 ] Fig. 4.9. Corollary 4.14. The r e gularity pr

blem

for pr exr e c

gnizable

gr aphs is de cid- able. 5 Conclusion W e ha v e in v estigated the regularit y problem for prexrecognizable graphs, viz the graphs

f

the form: n [ i=1 (U i a i

!

V i )W i where U i ; V i ; W i 2 R at(N

),

for all i 2 [n]. W e ha v e established that among the graphs

f

this form the regular

nes

are precisely those that, for eac h i 2 [n], satisfy at least

ne
f

the follo wing:

jU

i j < 1 and there are no w

rds

x; y 6= "; z suc h that xy

V

i and y

z
W

i ,

jV

i j < 1 and there are no w

rds

x; y 6= "; z suc h that xy

U

i and y

z
W

i .

SLIDE 31 30 The disjunction

f

the ab

v

e conditions is equiv alen t to the niteness

f

U i

r

V i and

f

the dep endence lev el. This, in turn, enables the construction

f

a deterministic graph grammar that generates the graph b y length. The k ey ingredien t

f

this construction is an equiv alence relation

n

v ertices, the index

f

whic h has to b e nite. This relation is someho w compatible with adjacen t edges. The results

f

Sect. 4 can b e view ed as an analogue to

ne

direction

f

the MyhillNero de theorem (nite index implies the regularit y). W e b eliev e that the con v erse is w

rth

b eing studied. In addition to the c haracterizations based

n

the decomp

sition

n um b er and

n

the dep endence lev el, another

riginal

c haracterization

f

this pap er is based

n

the n um b er

f

degrees. This c haracterization seems to b e close to a graph theoretic

ne
f

[2]: a VRequational graph G is HRequational if and

nly

if the set

f

bipartite complete subgraphs (up to

rien

tation

f

edges)

f

G is nite up to isomorphism. The rst

ne
f

the aforemen tioned c haracterizations, namely the

ne

b y prexrecognizable relations, is adv an tageous for pro ving n umerous prop erties

f

regular graphs. F

r

instance,

ne

ma y readily sho w that, giv en a v ertex, the accessible subgraph

f

a regular graph is again regular. In this con text, the in ternal c haracterization seems promising for establishing the decidabilit y

f

the bisim ulation for the whole class

f

regular graphs. (Up to no w, the bisimilarit y is kno wn as b eing decidable for normgenerated regular graphs [8] and for regular graphs

f

nite

utdegree

[24].) The external presen tation

f

prexrecognizable graphs dened in [9] is also

f

in terest for c haracterizing regular graphs. W e recall that suc h a presen tation consists

f

t w

p

erations. The rst

ne,

called in v erse rational substitution, is applied

n

the innite binary tree. This

p

eration adds new edges. The second

ne,

called rational restriction, remo v es some v ertices from the graph resulting form the rst

ne.

Within this approac h,

nly

regular graphs

f

nite degree ha v e a kno wn c haracterization. The c haracterization

f

the whole class

f

regular graphs remains to b e in v estigated. Ac kno wledgmen ts The authors are grateful to Damian Niwiski for n umerous suggestions that help ed impro ving the presen tation

f

this pap er. References 1. K. Barthelmann. On equational simple graphs. T ec hnical Rep

rt

9/97, Johannes Guten- b erg Univ ersitt, Mainz, 1998. 2. K. Barthelmann. When can an equational simple graph b e generated b y h yp eredge replacemen t. In L. Brim, J. Grusk a, and J. Zlatusk a, editors, Mathematic al F

undations
f

Computer Scienc e, LNCS 1450, pages 543552, Brno, Aug. 1998. 3. K. Barthelmann. When can an equational simple graph b e generated b y h yp eredge replacemen t. T ec hnical Rep

rt

2/98, Johannes Guten b erg Univ ersitt, Mainz, 1998. 4. J. R. Bc hi. W eak secondorder arithmetic and nite automata. Z. Math. L

gik

und Grund lag. Math., 6:6692, 1960. 5. J. Berstel. T r ansductions and ContextF r e e L anguages. B. G. T eubnner, Stuttgart, 1979. 6. H. Calbrix and T. Knapik. A stringrewriting c haracterization

f

con textfree graphs. In V. Arvind and R. Raman ujam, editors, 18 th International Confer enc e

n

F

undations
f

Softwar e T e chnolo gy and The

r

etic al Computer Scienc e, LNCS 1530, pages 331342, Chennai, Dec. 1998.

SLIDE 32 31 7. D. Caucal. On the regular structure

f

prex rewriting. The

r

etic al Comput. Sci., 106:61 86, 1992. 8. D. Caucal. Bisim ulation

f

con textfree grammars and pushdo wn automata. v

lume

53

f

L e ctur e Notes, pages 85106. CSLI, Stanford, 1995. 9. D. Caucal. On innite transition graphs ha ving a decidable monadic secondorder theory . In F. M. auf der Heide and B. Monien, editors, 23th International Col lo quium

n

A utomata L anguages and Pr

gr

amming, LNCS 1099, pages 194205, 1996. 10. E. M. Clark e and E. A. Emerson. Design and syn thesis

f

sync hronisation sk eletons using branc hing time temp

ral

logic. In D. Kozen, editor, Pr

c

e e dings

f

the W

rkshop
n

L

gic
f

Pr

gr

ams, LNCS 131, pages 5271, Y

rkto

wn Heigh ts, Ma y 1981. 11. B. Courcelle. The monadic secondorder logic

f

graphs, I I: Innite graphs

f

b

unded

width. Mathematic al System The

ry,

21:187221, 1989. 12. B. Courcelle. Recursiv e applicativ e program sc hemes. In J. v an Leeu w en, editor, F

rmal

Mo dels and Semantics, v

lume

B

f

Handb

k
f

The

r

etic al Computer Scienc e, pages 459492. Elsevier, 1990. 13. B. Courcelle. The monadic secondorder theory

f

graphs I I I: T reedecomp

sitions,

minors and complexit y issues. RAIR O Informatique Th

rique

et Applic ations, 26(3):257 286, 1992. 14. B. Courcelle. Structural prop erties

f

con textfree sets

f

graphs generated b y v ertex replacemen t. Information and Computation, 116(2):275293, 1995. 15. J. Engelfriet. Con texfree graph grammars. In G. Rozen b erg and A. Salomaa, editors, Beyond W

r

ds, v

lume

3

f

Handb

k
f

F

rmal

L anguages, pages 125213. Springer V erlag, 1997. 16. D. Harel. T

w

ards a theory

f

recursiv e structures. In L. Brim, J. Grusk a, and J. Zlatusk a, editors, Mathematic al F

undations
f

Computer Scienc e, LNCS 1450, pages 3653, Brno, Aug. 1998. 17. H. Hungar. Bey

nd

nitestate mo del c hec king: V erifying large and innite systems. Habilitation, Carl v

n

Ossietzky Univ ersitt Olden burg, 1998. 18. H. Hungar. Mo del c hec king and higherorder recursion. In M. Kut yo wski and L. P a- c holski, editors, Mathematic al F

undations
f

Computer Scienc e, Szk arsk a P

reba,

Sept. 1999. 19. T. Knapik and H. Calbrix. The graphs

f

nite monadic semiTh ue systems ha v e a decidable monadic secondorder theory . In C. S. Calude and M. J. Dinneen, editors, Combinatorics, Computation and L

gic

'99, pages 273285, Auc kland, Jan. 1999. Springer V erlag. 20. T. Knapik and H. Calbrix. Th ue sp ecications and their monadic secondorder prop erties. F undamenta Informatic, 38, 1999. 21. D. Kozen. Results

n

the prop

sitional

calculus. The

r

etic al Comput. Sci., 27:333354, 1983. 22. D. E. Muller and P . E. Sc h upp. The theory

f

ends, pushdo wn automata and secondorder logic. The

r

etic al Comput. Sci., 37:5175, 1985. 23. N. Rob ertson and P . Seymour. Graph minors iv, treewidth and w ellquasiordering. Journal

f

Combinatorial The

ry,

Series B, (48):227254, 1990. 24. G. Snizergues. Decidabilit y

f

the bisim ulation problem for equational graphs

f

nite

utdegree.

In R. Mot w ani, editor, Pr

c

e e dings F OCS'98, pages 120129. IEEE Computer So ciet y Press, 1998. F ull pro

fs

in tec hnical rep

rt

1183-97

f

LaBRI,

(A)
(B

)?, pages 1-113. 25. S. L. T

rre

and M. Nap

li.

Represen ting h yp ergraphs b y regular languages. In L. Brim, J. Grusk a, and J. Zlatusk a, editors, Mathematic al F

undations
f

Computer Scienc e, LNCS 1450, pages 571579, Brno, Aug. 1998.

SLIDE 33 i App endix In Sect. 2 the follo wing prop

sition

(2.7) has b een stated without pro

f.

Prop

sition.

The mapping D ec : R ec(N

N
)
!

I N [ f1g is r e cursive. W e need sev eral notations for the pro

f.

F

r

an y binary relation R

N
N
and

an y w

rd

u 2 N

,

w e denote b y R(u) = fv j (u; v ) 2 Rg, the image

f

u b y R, u 1 R = f(v ; w ) j (uv ; w ) 2 Rg, the left residual

f

R b y (u; "), R es " (R) = fRg [ fu 1 R r f"g N

j

u 2 N

g;

R es (R) = fL 1 \ : : : \ L n j n

^

L 1 ; : : : ; L n 2 S u2N

R

es(R(u))g . Note that for ev ery v 2 N + , (u 1 R r f"g N

)(v

) = (u 1 R)(v ) = R(uv ) : F

r

R recognizable, R es " (R); R es (R) are nite and can b e constructed. Pr

f.

Let R 2 R ec(N

N
).

W e can construct the follo wing nite graph: H :=

(S

; L)

!
u

1 S r f"g N

;

u 1 L \ S (u)

S

2 R es " (R); L 2 R es(R); u 1 L \ S (u) 6= ?

and

the restriction G

f

H to the v ertices accessible from (R; N

):

G:=f(S ; L)

!(S

; L ) j (R; N

)

! H

(S

; L)

!

H (S ; L )g : W e will sho w the follo wing

prop

ert y: D ec(R) = maxfn j (R; N

)

! G n g : This implies that D ec(R) < 1 ( ) G is acyclic whic h is a decidable prop ert y , and for a nite ro

ted

directed acyclic graph, w e can compute the maximal length

f

the paths from the ro

t.

Th us, this

prop

ert y means that D ec(R) is computable. T

pro

v e the

prop

ert y , w e lab el the graph G in to the follo wing graph: G :=

(S

; L) Z

!(S

; L ) j (S ; L)

!

G (S ; L ); Z = fu j u 1 S r f"g N

=

S ; u 1 L \ S (u) = L g

:

(i) Let 1

i
n

and (R ; L ) Z 1

!

G (R 1 ; L 1 ) : : : Z n

!

G (R n ; L n ). Let us sho w that Z 1 : : : Z i Z i+1 : : : Z n L n

R

. By denition

f

G, the existence

f

an edge (S ; L) Z

!

G (S ; L ) implies that 8u 2 Z 8v 2 L (uL

L

^ L

S

(u) ^ S

u

1 S ) :

SLIDE 34 ii Let u 1 2 Z 1 ; : : : ; u n 2 Z n . Then R 1

u

1 1 R ^ : : : ^ R i1

u

1 i1 R i2 ; L i

R

i1 (u i ) ^ u i+1 L i+1

L

i ^ : : : ^ u n L n

L

n1 : Th us u i+1 : : : u n L n

L

i

R

i1 (u i )

(u

1 : : : u i1 ) 1 R

(u

i ) : This means that for ev ery v 2 L n w e ha v e u 1 : : : u i R u i+1 : : : u n v . (ii) Let us v erify that R = S fZ L j 9S ; (R; N

)

Z

!

G (S ; L)g

Let

(u; v ) 2 R. Then (R; N

)
!

G (S ; L) where S = u 1 R r f"g N

and

L = R(u). In particular v 2 L. F urthermore for (R; N

)

Z

!

G (S ; L), w e ha v e u 2 Z .

Let

(R; N

)

Z

!

G (S ; L). Let u 2 Z and v 2 L. By denition

f

G and G (or b y i), v 2 R(u) i.e. (u; v ) 2 R. (iii) Let (R; N

)

= (R ; L ) Z 1

!

G (R 1 ; L 1 ) : : : Z n

!

G (R n ; L n ) and w 2 Z 1 : : : Z n L n with n

maximal.

Let us sho w that D ec(w ; R) = n. By h yp

thesis

w = z 1 : : : z n x n for some z 1 2 Z 1 ; : : : ; z n 2 Z n and x n 2 L n . By denition

f

G, z 2 ; : : : ; z n 6= ". W e get b y (i) (z 1 : : : z i ; z i+1 : : : z n x n ) 2 (Z 1 : : : Z i )(Z i+1 : : : Z n L n )

R

for ev ery i 2 [n]. Th us D ec(w ; R)

n.

It remains to pro v e that D ec(w ; R)

n

whic h is the case for D ec(w ; R) = 0. Let m = D ec(w ; R)

1.

Then w = u 1 v 1 = : : : = u m v m with ju 1 j < : : : < ju m j and u 1 ; : : : ; u m 2 U; v 1 ; : : : ; v m 2 V . Let (R ; L ) = (R; N

)

and u = ". By induction

n

i 2 [m], there exists R i ; L i ; Z i suc h that (R i1 ; L i1 ) Z i

!

G (R i ; L i ) ^ u 1 i1 u i 2 Z i ^ v i 2 L i : Hence (R; N

)

Z 1

!

G (R 1 ; L 1 ) : : : Z m

!

G (R m ; L m ) and w = u m v m = u 1 (u 1 1 u 2 ): : : (u 1 m1 u m )v m 2 Z 1 : : : Z m L m : By maximalit y

f

n, D ec(w ; R) = m

n.

Finally D ec(w ; R) = n. 2

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