Graph Connectivity and Monadic NP
Thomas Schwentick
Institut fur Informatik
Universitat Maim
Abstract
from Monadic coNP (MonCoNP). Whereas Graph
Connectivity is expressible in MonCoNP,
Ehrenfeucht games are a useful tool an proving that
certain properties of finite structures are not expressible by formulas of a certain type.
I n this paper a new method is introduced that allows
the extension of a local winning strategy for Duplicator, one of the two players in Ehrenfeucht games, to
a global winning strategy.
As an application it i s shown that Graph Connectivity cannot be expressed b y existential second-order
formulas, where the second-order quantification is restricted to unary relations (Monadic N P ) , even in the
presence of a built-in h e a r order. This settles a n open
problem from [I] and [Ill.
As a second application it i s stated, that, o n the
other hand, the presence of a linear order increases
the power of Monadic N P more than the presence of a
successor relation.
1
0
0
de Rougemont (51 showed that Graph Connectivity cannot be expressed in MonNP even in the
presence of a built-in successor relation;
0
Fagin, Stockmeyer and Vardi [ll] showed that
Graph Connectivity cannot be expressed in
MonNP even in the presence of arbitrary builtin relations of low degree (i.e. all elements are
only related to few other elements).
For a definition of expressibility in tlie context of
built-in relations see Section 2.
For a discussion of the importance of built-in relations see, for example, [l]and [ll].It should be noted
that it was already shown by Ajtai [3], that, in a very
strong sense, MonCoNP is not captured by MonNP
plus arbitrary built-in relations.
It was stated as an open problem in [I] and [ll],
whether Graph Connectivity is expressible in MonNP
in the presence of a built-in linear order. In this paper
we show that, as was widely conjectured, this is not
the case.
For the proof of this result, we introduce a new
method that is interesting in its own right. It allows
to deal with relations of large degree, as long as they
are, in some sense, homogenous. It is based on the
extension, under certain circumstances, of a winning
strategy for Duplicator in an Ehrenfeucht game on
structures H and H', to a winning strategy on structures G and GI that contain H and HI as substructures.
In Section 2 we give some basic definitions and notations. In Section 3 we recall the basic facts about
Ehrenfeucht games. In Section 4 we prove the Extension Theorem. In Section 5 we show that Graph
Connectivity cannot be expressed in MonNP even in
the presence of a built-in linear order. A short discussion of our results will be given in Section 6 . There
Introduction
Fagin [8] showed that the complexity class NP coincides with the class of all sets of finite structures
that can be characterized by existential second-order
formulas (E:). This means that, at least in principle,
NP # coNP could be proved by showing that existential second-order formulas and universal second-order
formulas have not the same expressive power over finite structures.
Unfortunately there has been very little progress
in showing that the power of existential and universal
second-order formulas differ in general. But there have
been some results concerning a restricted version of
the problem, where the second-order quantification is
only allowed over unary relations. The class of all sets
of finite structures that can be characterized by existential unary second-order formulas is called Monadic
N P (MonNP).
Graph Connectivity has played the role of a key
natural example for the separation of Monadic NP
6 14
0272-5428/94 $04.00 0 1994 IEEE
Fagin [9] showed that Graph Connectivity cannot
be expressed in MonNP;
we also state, as another application of our technique,
that MonNP is more powerful in the presence of a
built-in linear order than in the presence of a built-in
successor relation.
2.1
Finite structures
A signature S is a finite set of relation symbols
R 1 , ... ,R,, each with a fixed arity ai, and constant
symbols c1,. .. ,ct. We do not make use of function
symbols.
A finite S-structure
G = (Uc,
Rf,... ,R t , cf, . . . ,cr) consists of
0
a finite universe
0
elements cf of
uG.
< G,x1,...,xm>.
If 6 is a function from UG to (0,... ,IC}, we write
(G, 6) for the (SU {Uo,... ,Uk})-structure
< G,U,",... ,UE >, where x E U? if and only if
S(x) = i (Here we assume that the Vi are unary relation sy,bols that do not already occur in S).
For a subset W UG,that contains all the cf, the
induced substructure G J- W has universe W, relations
Ri n W"i and constants cf.
As an abbreviation we write G 4 [ X I , .. . ,x,] for
the structure < G 1{XI,. ..,zm},21,.. . ,x, >.
3
variables x,y, xl,... and
0
quantifiers 3,V
Ehrenfeucht games
Ehrenfeucht games were introduced in [7] and
proved to be a useful tool in obtaining inexpressibility
results. The rules of a k-round first-order (FO) Ehrenfeucht game are as follows.
There are two players, Duplicator and Spoiler.
They play a fixed number, k, of rounds on two finite structures G,G' over some signature S. In every
round, Spoiler chooses one element of one of the two
structures. Then Duplicator chooses an elcirient of' the
other structure. At the end of the game there are elements X I ,... ,zk of G,and xi,.. . ,z/k of G', chosen,
where xi and xi are those chosen in round a . Duplicator wins, if G -1 [XI,... , x m ] 2 G' 1[xi,... ,xA]. The
Sometimes we are interested in relations with a specific meaning. Especially the interpretation of the binary relation symbol < will always have to be a linear
order.
Formulas and expressibility
First-order formulas over a signature S are built of
0
0
1,
+
Whenever possible we will omit the superscript G.
If RI;. . . ,RI are relations over U G we write
< G, RI,. . . ,R1 > for the structure that results by
expanding G with RI, .. . ,RI.
Sometimes we want to distinguish certain elements in UG. Then we view G with the distinguished elements 21,... , x m as a (SU {cl,.. . ,c,})structure, and we denote the resulting structure by
2.2
the logical connectives A, V,
We.say that a formula 4 expresses a property P
(characterizes a class C) of finite structures over some
signature S, if for every S-structure G it holds that G
has property P ( G is in C), iff G 4.
We write MonNP for the class of sets of finite structures that can be characterized by Monadic C: formulas.
We are especially interested in a modified form of
expressibility, where the structures have adtlitioiial relations, so-called built-in relations.
Let S be a signature, P a property of S-structures.
Let S' be a signature, disjoint from S. P is expressible
b y a formula 4 in the presence of 5"-bualt-in relations,
if for every SI-structure A it holds that 4 holds exactly
in those (SUS')-expansions of A whose S-reduct have
property P.
If P is expressible in MonNP in the presence of
a linear order we also write that P is expressible in
MonNP+ <.
set of the natural numbers, (mostly of the form
{ 1,... ,n}, for some natural number n),
relations RA over U G of arity ai, and
0
4", G I= 4.
UG,which will always be a sub-
0
the = relation,
in the usual way.
In second-order formulas over S quantification over
relational variables X, Y, .. . is allowed. C: formulas are second-order formulas in prenex normal form,
where relational variables are quantified only existentially. Monadic C: formulas are C: formulas where
only quantification of unary relation symbols occurs.
We make use of the usual Tarskian truth semantic
to define the meaning of "structure G fulfils formula
Definitions and notations
2
0
relation and constant symbols from S,
615
3.3 Lemma. Let S be a fixed sign.atwe. For every k
there is a constant h/ such that in every S-structure
G there are at most N diflerent k-types r , " ( x ) .
importance of Ehrenfeucht games results from the following
3.1 Theorem. [7, 101 Let C be a set of finite structures over some signature S. C is first-order definable,
iff there is a f i e d k, such that, for every G E C and
G'
C, then Spoiler has a winning strategy in the
k-round FO Ehrenfeucht game on G and G'.
3.4 Lemma. Duplicator has a winning strategy in
the k-round FO Ehrenfeucht game on the S-structures
G and GI if and only if {T,"_,(x) I x E G } =
{ T , " ~ ( X ) I x E GI}.
e
We write G M I . G' if duplicator has a winning strategy in the k-round FO Ehrenfeucht game on G and G'.
Accordingly we write
< G , x 1 , ..., x , > X k - m < G',x: ,...,x & > if Duplicator still has a winning strategy after rn rounds
have already been played, in which X I , . .. ,x m and
x i , . .. ,x& were chosen.
For showing that a set C of graphs is not in MonNP,
Ajtai and Fagin [l]introduced a more advanced game,
which we will call Ajtai-Fagin (c, k)-game over C.
It consists of the following steps.
Proof. For the if-part: Duplicator can play in such
a way that after round j it holds that
For the only-if-part: Spoiler can play in such a way
that after round j it holds that
0
(1) Duplicator selects a graph G E C.
( 2 ) Spoiler colors the vertices of G with c colors.
(3) Duplicator selects a graph
with c colors.
G'
4
C and colors it
Before we formally state the Extension Theorem,
we are first going to discuss its main features in a
simplified context.
Let H and H' be two graphs' such that Duplicator has a winning strategy in the k-round Ehrenfeucht
game on H and HI. The question is:
(4) Spoiler and Duplicator play a k-round FO Ehrenfeucht game on the colored graphs G and GI.
3.2 Theorem. [ l ] C is an MonNP, zff there are c and
k such that Spoiler has a winning strategy in the AjtaiFugin ( c ,k)-game over C.
graphs G and GI respectively, can we also
extend Duplicator's winning strategy to the
In the following we define the k-type, T,"(x), of an
element x of G and show two useful properties of ktypes (cf. [SI).
0
Duplicator has a winning strategy in the k-round
FO game on G and GI,iff G and G' have the same
set of (k - 1)-types.
0
For a fixed signature the number of different ktypes depends only on k.
We define the k-type of a sequence
elements of G inductively.
0
.,"(xl,.
21,
...,X I
How to extend a local winning strategy of Duplicator
We will argue in the following that the answer is yes,
if the extensions of the graphs fulfil conditions (1) and
(2) below.
We write 5 and 6' for the distance functions on the
graphs G and GI, defined as usual. By N , ( H ) we
denote the e-neighbourhood of H, i.e., the subgraph
induced by all vertices z with S ( x , H) 5 e .
of
The two conditions are
..,x i ) is the S-isomorphism class of
(1) Duplicator's winning strategy can be extended to
N2k(H) and
(H') in such a way that in every round the two chosen vertices have the same
distance from H and HI respectively.
G 3. [ x i , . . . ,211.
T P + ~ ( x..~. ,, x i ) := { . , " ( X i , . .. , x i , X ) I x E G } .
We will make use of the following lemmas, the first
of which can easily be proved by induction.
'For the moment, our graphs consist only of vertices and
edges. There are neither colors nor other built-in relations.
616
(2) There is an isomorphism a from G - H to G'- HI
which on &k(H) respects the distance from H
and HI, i.e., 6(z, H) = S'(cx(z), HI) for every
z E N,k(H) - H .
Now we are going to transform the ideas of the previous discussion into a more general situation.
Instead of graphs we have arbitrary finite structures;
The idea for the winning strategy of Duplicator is
Instead of the usual distance function on graphs,
we will allow more general, "user-defined", distance functions;
as follows:
At the beginning of the game we view the vertices
in H and HI as inner vertices and the vertices outside
N2k (H) and N2&( H ' ) as outer vertices. The status of
the vertices in the buffer area between inner and outer
vertices remains open. By definition, at the beginning,
the distance from every outer vertex to every inner
vertex is more than 2 k .
During the game, Duplicator's strategy depends on
the distance from H of vertex
chosen by Spoiler
(where wlog we assume that z E G). If z is
Instead of one pair of subgraphs H , H', we have
to combine winning strategies of a lot of pairs of
substructures of G and G';
Before, the only relation, namely the edge relation, was homogenous in the sense that every pair
of vertices with distance more than one, was not
related. Now we have a more complicated kind of
homogenicity, namely condition (iii) below.
an inner vertex, Duplicator chooses the other vertex according to his winning strategy on the inner
vertices;
Now we are ready to formulate
an outer vertex, Duplicator plays according to the
isomorphism a;
4.1 Theorem. Let k > 0. Let S be a signature with
relational symbols 7-1,. .. ,r , of arities a ( l ) ,.. . ,a ( s )
and constant symbols c1,. . . ,ct .
Let G,G' be finite struct.ures over S ruitli in.. ,R,,Cl,.
. . ,Cl and
, . ,R:,
terpretations RI,.
Ci,... ,Cl respectively. Let 6,s' be distance fumctions"
on G and GI.
Let H I , . .. ,HI and H i , . . , ,HI be sequences of su.bsets of U G and UG' respectively, such th.aP N2k ( H i )n
N 2 c ( H j ) = 8 = N Z L ( H ; ) ~ N ~ ~f (OHF ;i #) j .
a vertex of the buffer area, then there are two
cases:
e
n:,,
if z is closer to the inner vertices, the area of
the inner vertices is extended to N ~ ( = , (HH) )
and Na(=,H)( H I ) ,and Duplicator chooses the
other vertex according to his winning strategy on the inner vertices,
otherwise, the area of the outer vertices
will be extended to all vertices outside
N6(z,H)-l(HI and N6(z,H)-l ( H I ) , and Duplicator plays according to the isomorphism
Duplicator has a winning strategy in the k-round
FO Ehrenfeucht game on G and G', i f the following
conditions are fulfilled.
CY.
(i) For every j 5 1, Duplicator has a winn.ing
strategy in the k-round Ehrenfeucht game on
(GJ. N2k(Hj),6j) and (GI 1N2k(H(i),6(i),where
S j ( z ) := man {S(z,y)ly E H j } .
It is easily shown, by induction on i, that after
round i the distance from every outer vertex to every
inner vertex is more than 2"-'. In particular, after k
rounds, no inner vertex is adjacent to any outer vertex.
The selected vertices induce subgraphs I and I' of G
and GI, consisting only of inner vertices and 0 and 0',
consisting only of outer vertices. I and I' are isomorphic because on the inner vertices Duplicator played
according to a winning strategy; 0 and 0' are isomorphic, because on the outer vertices Duplicator played
according to a. Because there is neither an edge between I and 0 nor between I' and 0', we get that
I U 0 is isomorphic to I' U 0', hence the induced subgraphs of all selected vertices are isomorphic. Thus,
Duplicator has indeed a winning strategy on G and
(ii) There is a n S-isomorphism a from
G J. ( U G - ( H I U - - - U H ~to
) ) G' 1(UG'-(HiU...U
H I ) ) , such that f o r every 2 E UG - ( H IU . . U H I )
and every j 5 1
21.e. 6 : G x G + Bv, 6' : G' x G' + lV.
3As before, w e write N , ( H ) for the set of all
G'.
6(z,H )
617
5 e.
vertices
with
(iii) For all p 5 s, all sequences 21,. ..
and xi,.. .
E UG' and every e
f o r every i
E UG
that
< 2k
5 u ( p ) and j 5 1:
Sj(Zi)
# e + 1 # Sj(z:),
for every i 5 u ( p ) :
i f Sj(xi) 5 e f o r some j
5 1,
We show first how the theorem follows from Lemma
4.3.
We set q := IC. Let x 1 , . . . , x k and x i , . .. ,x)k be the
selected vertices.
Let p 5 s and let i l , . . . ,i a ( p )be a secpcnce of indices from (1,. . . ,k}.
If we set e := d,, the sequences zil,.. . , x i a ( p , and
xi,,... ,z!' a b ) fulfil the prerequisites (a), (b), (c) from
condition (iii). Hence
then Sj(z;) =
qx:),
otherwise a ( z i )= xi,
f o r every j 5 1: i f xj,,. . . ,xi, are exactly
the elements of 21,. .. ,z,,(~) in N , ( H j ) then
G J [Zj,, . ,z'
l r]
G' J [xj, ,. ,x i q ] ,
it holds that
Rp(z1,.
.
*
,"a(p))
-
RL(4,. . . ,"h(,)).
Rp(zil,
4.2 Remark. (iii) is the homegeneity condition men-
e R;(x:,,. . . > x:~(,,,
)*
,xi,(,))
As this holds for every relation R,,, we gct
tioned above. It roughly says: If
G J- [ X I , . . . , X k ]
(a) there is a gap between inner and outer vertices in
both tuples,
Z
G' J. [x:,. . . ,.CL.].
0
(b) corresponding xi, xi are either both inner vertices,
in which c u e they have the same short distance to
the inner subgraph with the same number H j , Hi
or both outer vertices and then a maps xi to xi,
and
Proof of Lemma 4.3.
duction on q.
We show thc lemma by in-
For q = 0 statements (1) and (3) follow immediately, and ( 2 ) follows from (i).
(c) restricted to every inner subgraph the tuples are
isomorphic
Let Duplicator have played according to (1)-(3) in
the first q rounds. Let wlog x,+l E UG be the element
chosen by spoiler in round q + 1. We show how duplicator can select
such that (1)-(3) hold again.
then the tuples relate the same with respect to R p .
Note that we do not demand that inner and outer vertices are not related.
We distinguish two cases.
Proof. Duplicator will follow the strategy described
above, maintaining a buffer zone of distance 2 k - q , after q rounds. We set A(z) := min{6j(z) l j 5 l } , (accordingly A') and
d, := max{A(zi) I i
1. A(z,+~)5 d ,
This means that x,+l is closer to the inner vertices, hence the area of the inner vertices will be
extended to contain x ~ + ~ .
5 q,x; inner vertex},
A(Z,+i) 5 dq+2'"-(,+') means that for soniej we
have S j ( x , + l ) 5 d, 2'"-(9+'). By the iiiduction
hypothesis we know that
do := 0. Which vertices are considered inner vertices
+
will be defined during the game. We show
4.3 Lemma. Duplicator can play in such a way that,
for every q 5 IC, after round q, it holds that
(1) for every i
< (G1Ndq+2k-r(Hj),Sj),
2 1 , . . . , ~ >xa-q
g
< (G'J- Nd,+Zk--s ( H ( ) Si)
, 1,xi,.. . ,X: > .
5q
(a) eithhr f o r some j,Sj(Zi) 5 d,, then Sj(xi) =
6;<x:),
(b) or, otherwise, A(xi) > d , 2 k - 9 , A'(z:) >
d , 2 k - 9 and a ( z i )= xi,
+
+ 2"-(9+')
Therefore there is an
that
+
E Nd,+2k.--s(Hi)
such
< ( G J N d q + Z k - - p ( H i ) , 6 j ) , x 1 , . . . ,xq+1 > = k - ( q + l )
< (G' N d q + 2 k - 9 (Hi),
Si),x : , . .. ,x;+l > .
618
Let k, the number of rounds of the Ehrenfeucht
game, and c, the number of colors be fixed.
We will use graphs that are closely related to sequences of permutations.
In Particular S j ( z q + l ) = S~(Z>+~).
Hence (1) follows.
On the other hand, because dq+'
d, 2k-q, it follows
+
+ 2k-(q+') <
-
For n > 0 and a sequence, P = T I , . .. , x i , of permutations over { 1,. .. ,n}, we define Gp, the graph of
P, in the following way:
The vertices of Gp are numbered v i j ,
i = 1,. , I
1 , j = 1,. ..,n and are ordered lexicographically. There is an edge between vij and viljl
iff
.. +
So (2) is also fulfilled. Finally, (3) holds because
dq+'
5
5
dq+2k-(q+')
2k
+ 1 and j' = n i ( j ) , or
(b) i = 1, i' = 1 + 1, and j = j ' .
- p--9+ 2"(q+')
(a) i' = i
= 2k - 2"(Q+').
As an example let n = 4,n1 = ( 1 2 4 ) , ~ 2 =
(243),n3 = (24). Then G, looks as follows:
In this case zq+l will be viewed as outer vertex.
Duplicator chooses zb+' := a ( z q + l ) .
(1)
5
- (3) follow immediately.
Graph Connectivity
MonNP+ <
is
not
The reader may easily verify that, Gp is connected
if and only if
xi is an n-cycle. We note that in
this case Gp is simply a cycle, in the sense of graph
theory.
We define a distance function 6 on Gp by setting
6 ( v i j , u p j , ) := min(1i' - iI,I - ' ;1 - il).
Thus, the distance of two vertices is given by the
distance of their respective columns.
The idea of the proof is as follows:
in
nf=,
In this section we are going to prove our main result.
In the proof we will make use of the following grouptheoretical result of Coppersmith. With S, we denote
the group of all permutations over { 1,. ..,n}. We say
a subgroup U of S, fulfils condition (o), if for every
g E U the product of the n-cycle ( 1 2 - - - n )and g is
again an n-cycle.
5.1 Lemma.
group U of s,
0
141 For large
enough n and every subthat fulfils condition (o), it holds
For some large n, Duplicator selects a graph
G = Gp for some sequence P, consisting of many
identical subsequences, each containing all permutations over { 1,. .,n};
.
0
we show that, irrespective of how Spoiler colors
G,in every subgraph corresponding to one of the
subsequences, there is one permutation, that can
be replaced by a lot of other permutations, such
that every single replacement is not detectable in
a k-round game;
0
finally, we show that there exists a combination of
such undetectable replacements in different. subsequences of P that results in a sequence, P', of
permutations, such that
5.2 Theorem. Graph Connectivity cannot be expressed in MonNP even an the presence of a built-in
linear order.
Proof. We have to show that, for every k and c,
Duplicator has a winning strategy in the Ajtai-Fagin
(c, k)-game over the set of connectedgraphs on linearly
ordered vertex sets.
619
- P' does not multiply to an n-cycle, and
- the corresponding disconnected graph, G' =
Gpr cannot be distinguished from G in a k-
from the two inner columns of
( L i j Jij)
... ,T ~ - ' ; ' '' ( w ~ ( ~ be
) ) )the vector of
(IC-1)-types of the vertices of Lij, ordered according to
the vertex ordering. We know from Section 3 that the
number, N , of different (k - 1)-types of vertices does
not depend on n, but only on k, c and the size of the
range of S i j , which is 2&+ 1. At most Np(")= No(")
of the type vectors T(Lij) are different.
As there are n! subgraphs Lij of K i , this means,
that for every i, there is a type vector, Ti, that occurs
many times in I C ~ .We fix, for every i,
at least
such a type vector Ti. With Ai we denote the set of
those j with T ( L i j )= Ti.
Then any two subgraphs Lij, Lijt with indices j , j'
in Ai only differ in their central permutation. In particular, all Lij are colored identically. Moreover, from
Lemma 3.4 it follows immediately that
Let ~ 1 , ...
. gn! be an enumeration of all permutations over { 1,....n} and let
n!
p := ( r J U i ) - l
i= 1
--
We write Q; as an abbreviation for the sequence4
(),...,(),~i,O,---
, ( ) e
2k
Furthermore we write R for the sequence
P , Q 1 , * 9 Qn!.
Now let the sequence P consist of the n-cycle
[ 123. . n) followed by n!2"! copies of R.
(#) if j , j ' E Ai, then Duplicator has a winning strategy on
-
Lij
and L i y , that respects
(12.. . n), R I , .... Ri, ....Rn!2n!
Xi := {
7
6ij
and
Sijt.
Now fix some j i E A, for every i.
We write R : ( j ) for the sequence of per111utiitions
that arises, if in Ri we replace permutation uj, by a,.
We denote the product of the permutations of R : ( j )
with nij. Now let
P
0,
Let T ( L ; j ) :=
( T ~ - ~( w l ) ,
round Ehrenfeucht game.
2k
Lij.
(L..6 . - )
I
~ i jj
E Ai}.
That is, for every i, X i is the set of those permutation products, that are obtainable by undetectable
changes of Liji.' Because there are only 2"! subsets of
S n , there is a set of permutations Y C S, that occurs
at least n! times among the X i .
Let U 5 S, be the group of permutations that is
generated by Y.We have
0, U j , (1, ., (1
As the product of the permutations of every Ri is
the identity permutation, the product of all permutations of P equals (12.a.n). Hence G p is connected.
Duplicator will choose G := G p for some large
enough n. We will write ICi for the subgraph of G
corresponding to Ri and Lij for the subgraph of K i
corresponding to Q j . Hence Lij looks like this.
Because depends only on k and c, we can choose
n large enough, such that
;:::q
M....
M....
........
........
M....
-.f
..
..
.
.
.
....
"
I
:
---
''._
. : .
o-pL&+
\
2k
. -
.
Then it follows from Lemma 5.1, that there is a g E U,
such that (12.. .n)g is not an n-cycle.
It follows from standard finite groiip thcory, that
there exist an m 5 n! and y l , ... ,ym E I', such that
9 = Y 1 * * * Ym.
As Y occurs a t least n! times among the X i , there
are indices i l , ... ,i, and j 1 , . .. ,j,, such that for all
9 l m
.
"
2k
Now let G be arbitrarily colored with c colors. Our
first goal is to show that, in every ICi, Duplicator will
find a large set Ai of indices, such that he has a winning strategy on every pair L i j , Lij, for j , j' E Ai.
Every Lij consists of p ( n ) := n(2k 2
2k)
vertices. Let Si,(.) be the distance of vertex 2
+ +
5That these changes are undetectable will be proved below,
L i j i and Lij are indistin-
For the moment, we only know that
guishable for every j E A ; .
4() denotes the identity permutation.
620
e
6
j q E Ai, and
Theorem 4.1 can also be used to prove the following
A i q j o = Yq-
Now let G' result from G by replacing Liqji, by
Li,j,, for every q 5 m. By the choice of the yq, the
product of the permutations of G' is not an n-cycle,
hence G' is not connected.
We have to show, finally, that Duplicator has a winning strategy in the k-round FO Ehrenfeucht game on
G and GI.
For every q, we write Hq for the subgraph of G,
induced by the middle two columns of vertices of the
subgraph Li, j i q Accordingly we define the subgraphs
Hi of G'.
We show now, that G, H I , .. ,H, and
of Theorem
GI,H{,. ,Hk fulfil conditions
4.1.
.
.
..
e)-(%)
e With distance functions 6 , 8 , defined as above, we
have that
N2k(Hi)nNp(Hj) = 0 = Nzk(H[)nN2L(Hj)for
ifj;
e
0
(i) follows from (#) above;
(ii) follows by setting a
(
.
)
v outside of the
Discussion
:= o for every vertex
Hq.
It remains to be shown, that the conclusion of condition (iii) is fulfilled for the edge relation and the
linear order.
For the edge relation this follows from the fact that
no two vertices x,y E G with 6 ( x , y) > 1 are connected
(the same holds in GI).
The linear order also fulfils (iii), because
e
for every k e d outer vertex x and every j , either
all vertices in N e ( H j ) are greater than x (and
if so, the same holds for Ne(H;) and ~ ( x ) )or
,
they are all smaller (and again the same holds for
Ne(H;) and a ( x ) ) , and
e
for different j and j' either every vertex from
Ne(Hj)is greater than every vertex from N,(Hj,)
(and if so, the same holds for N e ( H i ) and Q(z))
or every vertex from Ne(Hj) is smaller than every
vertex from Ne(Hj#)(and again the same holds
for Ne ( H i ) and CY(z)).
Hence we can apply Theorem 4.1.
U
6.1 Theorem. There is a graph property that can be
expressed in MonNP in the presence of a b d t - i n Linear
order, but not in the presence of a built-in successor
relation.
We only can give the idea of the proof here. As
a separating example we take the set of graphs that
consist of disjoint unions of even-sized cliqiies. In the
presence of a linear order one can express, in MonNP,
that every clique has even size by simply coloring the
vertices in every clique alternatingly with respect to
the order, and by verifying that in every clique the
smallest and the largest vertex get different colors. A
successor relation, on the other hand, if distributed
carefully over the whole graph, cannot help in testing
whether a clique has even size or not.
We note that Theorem 4.1 could also be used in
several other proofs, like [5] and [ll]. But it seems
to be, in general, incompatible with the technique of
Hanf [12], which is described in [ll] and the related
technique of [2].
We hope that our technique will be useful to obtain
other inexpressibility results. Maybe with its help or
with the help of some other techniques developed recently (cf. [ll,2]), it will be possible to attack the case
of existential second-order formulas that allow quantification over binary relations.
Acknowledgements
I am very grateful to Clemens Lautemann for uncountably many helpful discussions, valuable suggestions and improvements, and not least because of attracting me to this wonderful subject.
I am also indebted to Don Coppersmith for finding
Lemma 5.1, (which is actually a theorem) within very
short time, and the runners of dmanet, where I posted
the question, whether a result like Lemma 5.1 already
existed.
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