On Local and Non-Local Properties

On Local and Non-Local Properties

PROCEEDINGS OF THEHERBRAND SYMPOSILIM LOGIC COLLOQUIm ’81, J. Stem (editor) @ North-Holland F‘ublishing Company, 1982 ON L O C A L AND NON-LOCAL 105...
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PROCEEDINGS OF THEHERBRAND SYMPOSILIM LOGIC COLLOQUIm ’81, J. Stem (editor) @ North-Holland F‘ublishing Company, 1982

ON L O C A L AND NON-LOCAL

105

PROPERTIES

Haim Gaifman The Hebrew University Jerusalem and University Paris VI

SO.

INTRODUCTION

The result to be presented here was motivated by questions of first-order definability within the class of finite relational structures. These questions arose in the research of suitable languages for databases (cf [AHU],

[AUI and [CHI). A stan-

dard example is the following : Can we express in first-order language the property that a graph (i.e. a binary symmetric relation) is connected : The negative answer is easily proved either by a compactness argument, or by forming the ultrapower of a sequence of connected graphe, Mi, i
almost

have diameter > n. If, however, we pose the question in the domain of

finite graphs : is there a first-order sentence cp

M bcp iff M is connected

?

such that for all

finite M,

then these easy arguments do not carry over. The

negative answer neecis another kind of proof (cf [ A U I , [ C H I )

which, though not dif-

ficult, involves a finer analysis of the situation. Since detabases are, essentially, finite relational structures, their investigation leads naturally to questions concerning finite models. Some questions which have easy solutions for infinite models become not so easy and sometimes quite difficult when transferred to the finite domain. The method to be presented is an analysis of first-order formulas in terms of local properties. We use a natural simple metric in the model and define the concept of a k-local formula where k

is any natural number. Roughly speaking, a k-local

formula is one which asserts something about some k-neighborhood around a point,

x, i.e. about the model (Vck)(x), x), where V(k)(x) is the set of all pointsof distance < k from x ; this means that all quantifiers arerelativized to V(k)(x) and

x

is the free variable of the formula. The main theorem asserts that every

first order sentence, cp, is logically equivalent to a Boolean combination of sentences that assert, each, something of the following form : There exist

s

disjoint

r-neighborhoods, each satisfying the r-local formula JI.

H. GAIFMAN

106

If cp

is a formula, one has to add to the combination r-local formulas in the

free variables of CD. The theorem is proved by quantifier elimination. The proof yields an effective translation of the formula cp

into the Boolean combination, as well as upper

bounds for the neighborhood radius, r, and the number, s , of neighborhoods, in terms of the quantifier depth of cp. If the quantifier depth speaking, the upper bound for r

is

is n then, roughly,

7n-1. By considering particular examples,

one can establish a lower bound which is

Zn-'

.

(For the exact details see § I ) .

This poses the problem of narrowing the gap. The reader not interested in these details o r in the technicalities of the proof can go on directly to 82, once the theorem is understood. The point is

to

establish a precise sense in which first

order sentencesare local and to use it in order to show that such and such properties are not characterized by first-order sentences because they are not local, Thus, to cite one of the examples given in 8 2 , within the class of finite graphs one cannot define by a first-order sentence those that are planar. In general, let

Co

and

C1

be two classes of models. Assume that for every n

one can find M ECo, MIECl such that for all

r,s

< n,

every combination of

s

disjoint r-neighboods in one model is isomorphic to a similar combination in the other. Then Co

and

C1

cannot be seperated by means of a first-order sentence.

Moreover, as proved in §2, such classes cannot be seperated in any richer logic obtained by introducing predicates which denote local properties ties of k-neighborhoods,where k

-

i.e., proper-

ranges over the natural numbers. There is no

restriction on the properties allowed,provided only that they are preserved under isomorphisms and that they are local. The examples given in 62 are all from graph theory,which seems to be the most natural domain for applying the method. The proofs are very easy in drawing prototypes of the Mo

and

shows that the same combinations of a

M,

-

for they consist

mentioned above. A glance suffices to

"small" number of

"small" neighborhoods

are realized in both. In $ 3 a different kind of application is given

-

a much shorter proof of a previous

result of the author, answering a set-theoretical question of Levy, [ G I . As stated at the end, this leads to the existence of a certain curious transitive set. A s it is,the method does not apply to models possessing a more regular mathemati-

cal structure - s u c h as a linear ordering, for then the whole model is equal to some small neighborhood. Any extension of the method to such modelswill have to use a much more sophisticated distance function.

Local and non-localproperties

107

We should mention that a result of L. Marcus [M 1 which is an immediate corollary of the theorem, can be regarded as a forerunner of it. There seems-to be no direct proof of the theorem, or the applications, from the corollary.

$ 1 . THE THEOREM

Let L be a first-order language with finitely many predicates including equality and no function symbols (and no individual constants). The restriction on function symbols is not essential, since they can be eliminated in the usual way by using predicates. Our results can be formulated in general, but it is more convenient to formulate and prove them for a language that has only predicates. "x", "y", "z"

We shall use

both for individual variables in L

as well as for

members of models. We shall also use the members of models as names of themselves

--

__ ...,_x,y,z

and occasionally substitute them for free variables in formulas. a,b, are used for tuples. If

_x,y_

=

xo,.

..

are in X. Let M

x = xo,. ..

yo,. . .

By

By

and '%€ X"

-

y = yo,.

..,yn-l

then

-

we mean that all the elements of x

"Fcj" we mean that every element in X

is also in 7 .

be a model for L, say M = (M, Ro, ...). Define a metric

d = dM

over M

as follows : d(x,x) =

Dfo

for some predicate d(x,y) Q 1 eDF

-

tuple x E M

containing both

d(x,y)

If L

= m

x

R (including equality) and some

and y. M bR(x).

wDFfor all n ,

d(x,y) $ n.

has one binary predicate besides equality then M dM

path connecting x

and

If all predicates of

y, disregarding the direction of the edges.

L are monadic or binary then d(x,y)

the formula Wi(Ri(x,y)

can be regarded as a

is the usual distance : dM(x,y) = length of the smallest

directed graph and

v Ri(y,x))

=

1

is definable by

H. GAIFMAN

108

where the Ri's

are all the binary predicates in L.

In general, d(x,y)

= 1

is definable by a formula that involves quantifiers. Here the finiteness of the number of predicates is used. Since the forthcoming result concerns single formulas and each single formula involves finitely many predicates, the result holdsfor any language provided that we define the distance using only the predicates that occur in the formula in question.

d(x,y)

Q

is definable by

n

3v0,.

..,v

n

If M'cM

> dM(x,y)

then evidently dM,(x,y)

(x= v hy = v A M d(vi,vi+l) (1). O i
for all

x,yEM'. If L

has only

monadic o r binary predicates then d (x,y) = 1 Q dM,(x,y) = 1. But in general M the additional members of M which cause dM(x,y) to be 1 need not be in M'. V(k)(x)

=

V(k)(x)

is the k-neighborhood of x.

Iyi d(x,y)

Q

is definable in L

Since V(k)(x)

fiers ~vEV(~)(X)

kl

we can express inside L

the bounded quanti-

and VVEV(~)(X).

Definition : A local k-formula around x

is a formula with a single free varia-

ble x, in which all quantifiers are of the form QuEV(~)(X). The satisfaction of a k-local formula around ("(k) (x) , x)

.

x

depends only on the model

-

These notions are generalized in the obvious way for tuples. If k = ko,..., km- 1 and

-

-

x = X~,...,X~-~ then define the r-neighborhood around x

-

-

and define a k-local

formula around x

which all quantifiers are bounded by-()'(V We use

V(~)(Z)'' for

speak accordingly of We put

A

) Z ( ) ' ( V

a formula with free variables

-

for all

x

icm

and of k-local formulas around kand then V (x) = {y: d(?,y) Q kl.

k-local formula is a formula which is k-local around some

local formula is one which is k

-

x

in

4.

for the case where ki = k

k-neighborhoods of

d6,y) = DfMinicm d(xi.y)

as

as :

local around some

X.

x

and X.

and a simple k-

Similarly k-neighborhoods

109

Local and non-localproperties around a single x ,p(k)

are referred to as simple k-neighborhoods. We use

(T) , $ ( k ) (T) ,...,,p(k) (2),

formulas around

$ J ( ~ )

-

(x), . ..

to range over k-local (k-local)

x. Before the main theorem we bring some helpful trivial obser-

vations. and

If xl, X,€V(~)(X)

d(xl, x2) = 1 <’=> V ( k )

-

occur in x

k

(x)

(d(xl, x2)

then Min(d(x,xl), (Evidently if

in V(k)(x). k xl, X ~ E V(x)

=

< k

d(x,x2))

M kR(T)

and

x1,x2

implies that all members of

x

are

L has only monadic or binary predicates then

is sufficient.)

possibly, the last are i n

then

For if

1).

Now, if d(x,xl)

then all the members of the path of length n also holds with V(k)(x)

< k

Min (d(x,xl) d(x,x2))

=

d(xl,X2) = n

m,

and m+nGk

that connects x1 and

V(k-l) (x). Hence V(k)(x)

1 (d(xI,x2)

replaced by any submodel M ’

=

n)

x2 except, and this

such that M‘3V(k)(x).

From this we get : (1)

For m+n

Q

3u€V(”)(y)

k,

d(T,y) S m

cp <->

implies

~ u E V ( ~ ) ( ~ )[(d(u,y)

is obtained from $

where $v(k)G)

4 n)

by bounding all quantifiers in it by

Vck)(F). This enables one to use, instead of quantifiers bounded by V(n) (Y), quantifiers bounded by V(k)(G). In particular, if k G k ’ then every cp(k)(x) can be rewritten as a

I)(~’)(:).

...,v are such that d(vi, v ~ + ~S )1 , x, = vo and x2 = vn, and vk ,...,v n E V (n-k) (x,). From this we get vo ,...,V,EV(~)(X~) If vo,

(2)

If

k,+k2

>n

then

then (kl,k2) <-->

d(x1,x2) G n

(d(xl,x2) S n) V

Main Theorem. Every first-order formula a(;)

-

u = u,,

(XI .x2).

with the free variables

is logically equivalent to a Boolean combination of

(I) Sentences of the form

3v0 where

,...,vs-l

$(r)(vi)

[Mi
A

Mi<.
is obtained from $(r)(vj)

by substituting vi

for v..

H. GAIFMAN

110

q~(~)(z)where w c u.

(11) Local formulas

If

has no free variables then only the sentences of

a

(I) occur in the

Boolean combination.

If n

=

quantifier depth of a r

s 4 m+n,

7n-1, (2)

Explanation. From

then the following inequalities can be guaranteed t4

it follows that

1

(7n-1).

d(v.,v.) 1

J

>

2r

can be written in the

form o(~~)(v;,v~). Hence, by ( l ) , sentences of form (I) are also of form

and the bound on

r

becomes : k 4 2 . 7 “ - ’ .

(I*) asserts the existence of a k-

neighborhood satisfying a certain first-order property (where a “k-neighborhood” is to be understood here as a submodel of the form (Vk (vo V ~ - ~ ) , V ~ v~-~)).

There are at least

s

$.

Hence

(I)

,...,

(I ) . A s is easily seen d(x1,x2)>kl+k2

(I) is of course a very special case of (kl) iff V (x1)”fl2)(x2) =

,...,

*

asserts :

pairwise disjoint simple r-neighborhood, each satisfying

the first-order property expressed by I$. In the case where

d(x1,x2)

< k,

for all

x1,x2€M, the whole model coincides

with some simple k-neighborhood and for models for which this k

is small the

theorem is of no interest. For example, in models for set theory the distance between any two points i s 4 2 (take have a ternary relation R

z

such that

such that x E z Vx,y3z R(x,y,z)

and

yEz). Or if we

holds, the distance is

always G 1. On the other hand many families of graphs considered by graphtheorists give rise to models with non-trivial distance functions. That is, roughly speaking, t o say, that for

siderably simpler

k which is not very large the simple k-neighborhoods are conthan the whole model. In such cases the theorem is very help-

ful in describing what can and what cannot be expressed in first-order logic. A l s o models with trivial distances may sometimes be described in terms of a different collection of basic relations which give rise to non-trivial distances. For example,if for some binary predicate R, our models satisfy Vx,yR(x,y)

then the dis-

tance is alway 4 1, but then one can omit this R

without reducing the expressive

power of the language (i.e., formulas involving R

are translatable into formulas

not involving it). In general, we can consider any change of basic relations provided that we do not reduce the expressive power of our language. We shall later

Local and non-localproperties

111

use this procedure in one of our applications. Note that if all our predicates are monadic then x # y implies d(x,y) = m . # vj) and we get the Consequently the sentence in (I) becomes 37(Mi
cp(T) there exist k and a formula

$ );(

such that M

M

and any formula

(x)I=

cp(G)++-V(k)

M. This was first proved syntactically by Leo Marcus [ MI

for all €;

[ GI

of one monadic function. In

JIG),

for models

it was generalized and used by the author who

gave it a shorter semantic but non-constructive proof. Some years later the author found an easy semantic non-constructive proof for a weaker version of the theorem

*

where (I) is replaced by (I ) . Then, using this as a guideline he arrived at the present proof which amounts to a quantifier elimination). rile shall use the following fact : (3)

For any formula

.

u ( . .yi(F)

yi

and

-

-

M

M = M

U M2, M ll M

1

(3)

i E I, and 6 . (V), j E J , such that J

which is the disjoint sum of the two models

x E M 1 , y € M 2 . Here

by replacing every Qu by 1

bers of

there is a Boolean combination

of formulas yi(c),

6. have no common free variables and

holds in any model for any

5(L,v)

... 6.J (T) . . .)

2

= $

hMi

h

is the relativization of

QuE Mi). By a

M1,M2 and

to Mi

(obtained

"disjoint sum" we mean that

and no relation holds between members of

MI and mem-

M2.

is a special case covered by the results of Fefferman-Vaught, [FV]. It is 5 : If

also provable directly by induction on

5

is quantifier-free, replace

each atomic formula which contains variables both from L and from (This is the only place where the disjointness of induction step use the fact that

3vy

MI and

is equivalent in M

M2 to

v

by

uo# uo

is used). For the (3v€Ml)yv(3v€M2)y.

Proof of the Main theorem : By induction on a. The main step uses the following lemma. Lemma: Let

-

6 = 5(k)

(v,z) be a k-local formula around v,z, where

v = V~,...,V~-~.Then

328 is logically equivalent to a Boolean combination of

sentences of the form

3Zt.m

1
IJJ(~)(X;) A

d(x x.) > 2r] M i
H. GAIFMAN

112

-

x = xo t

Q

,...,xs-l, r =

<

2k+l, s

P,+ 1 ,

and of local formulas cp(t)(T),

with

7k+ 3 .

Proof : -

is equivalent to

37.8

B1

Call the first disjunct

E 2 . We shall transform each into the

and the second

required form ; first - B 1 . Assuming d(T,z) 4 2k+l it follows that

Wick(V(k)(Z)

(l),instead of using quantifiers bounded by v ~ Y E V ‘ ~ )(z)cp

3y€V(k)(;)cp

and ~Y€V(~)(;) V(3k+1) (v). Hence

quantifiers bounded by

where all quantifiers in (3k+l) rewritten as cp (v). Now consider YEV(~)(Z),

B2. d(x,y)

If

8;

d(;,z)

and, by

we can use quantifiers

can be eliminated in favour of

B 1 can be rewritten as

are bounded by

> 2k+l

V(3k+1)

(v). This, again can be

then for every x E V ( ~ )

(7) and every

> 1 . This is easily seen t o imply that the submodel V(k)(T,~)

is the disjoint sum of to

(2)

v (3k+l)(vi))

(G). ALSO ~ Y E V ‘ ~(V,z)cp ) is equivalent to

bounded by

lent

V(k)

c

Vck)(T)

and V ( k ) ( z ) .

By (3),

6 is in this case equiva-

a Boolean combination of-formulas of the forms Y(~)(;)

and

6(k)(~).

Putting this combination in disjunctive normal form and using the fact that the k-local formulas around the same string of variables are closed under Boolean combinations, E2

can be rewritten as

It remains to transform the formula of the form 3z[(d(v,z) Since z

> 2k+l)

A

~(~’(v)

A6(k)(Z)].

does not occur in Y ‘ ~ ) , this becomes :

Y(~)(;)

A

3z[(d(J,z)

> 2k+l)

A

6(k)(z)1.

Local and non-localproperties

113

Thus, we have reduced o u r problem to

It asserts the existence of

q(y).

Call this formula

se distance from each vi,

i = 0,...,Q-1 z's

serting the existence of at least m tances are >2(2k+l).

=

Bm

(For m

=

1

B

who-

be the sentence as-

satisfying 6(k)(z)

whose mutual disAm(zo

,...,zm- 1)

is

J

the second conjunct disappears.) Since q(v)

satisfying 6(k)(z),

Mi 2(2k+l)).

A

$~(~~+')(z~), the sentences Bm Evidently

z

>2k+l. Let

,...,z m-1 Am ( z o ,...,z ~ - ~ ) where ,

3zo

Mi
is

,

for m

I,

=

6(k)(zi)

...,Q+l

q(v) A 3 ~ 6 ( ~ ) ( z and )

is equivalent to

is also of the form

are as required in the lemma. 3 ~ 6 ( ~ ) ( z is ) equivalent

to : A

(B 1

i B2)

V

i B3 )

A

(B2

... V

V

( B E A T BP.+l)

Hence it suffices to consider the formulas q(v) and

n (v)

A

A

B. A J

~

V

BP.+l. for ~ + I G~j G k + l ,

B

BQ+l.

can belong to a simple

...,zP.

zo,

First, assume BQ+* and let

satisfy AQ+l ( z o , .

.., z E ) .

No two

z.'s

i # j,

(2k+l)-neighborhood, because, for

d(zi, z . ) > 2(2k+l) ; hence some zi does not belong to any of the V(2k+l) (Vj), J j < P.. This implies n(T). Thus, q(v) A is equivalent to 1* 1 G j G P.

For (q():

A

where C.(y) A.(z,, J

J

Bj

E

1

Bj+l

Note that A.(zo, J

C.(v))

as :

B. J

v

(n(v)

A

B

j

j 2;'s

A - I B ~ A+ l ~c . ( Y ) ) J

in V(2k+1) (v) -

such that

:

v (2k+l) (7)

Z~,...,Z~-~

A

A

asserts the existence of

,...,z J-1 . ) 3Zo

A

rewrite q

... 3Zj-l

...,Z J-1 . )

V(2k+l) (v) - Aj ( z o , .

can be written as a

. ., z .

J-1

(2k+l)-local

)

. formula around

(because d(zi,z.) > 2(2k+l) J

..

is expressible by a (2k+l)-local forE V(2k+l) (T), then quantifiers bounded by

If z o , . ,z mula around zi, z j ) . j-1 V(2k+l) can be eliminated in favour of quantifiers bounded by (Z~,...,Z~-~) V(4k+2) (v). This implies that C

j

is equivalent to a

(4k+2)-local formula around

V.

Now

H. GAIFMAN

114

B. 3

AT

implies that, for some z

C.(T) 3

C.(G) implies q(T) B. ~1 3

3

B. ATBj+l 3

1 B .

3+1

every

A

x

i C ,

A

J

(7)

not in V(2k+1) (v) - , 6 (z)

which is of the required form.

C (G) implies the existence of j 'j satisfying 8(k)(x) has distance

case, all x ' s

holds. Hence

and the second disjunct can be replaced by

in V(2k+1) (v)

2;'s Q

2(2k+l)

such that

from some zi. In this

satisfying L S ( ~ ) ( X ) must be found in V3(2k+1)(v)

and

n(G) be-

comes equivalent to : 3xE "(6k+3) (v) - [6(k)(x) If X E V (6k+3) the quantifiers in

A

d(v,x)

> 2k+l].

IS'^) (x) , being

bounded by

V(k) (x) , are eli-

d(G,x) > 2k+l V(2k+l) (v) (7k+3) Consequently 7 B A C.(T) implies n ~ <-> ) q(7k+3)(~) for some cp (v) I J j+l (7k+3) and the first disjunct can be rewritten as B. h i B A C . 6) A cp (v)

minable in favour of quantifiers bounded by

V(6k+3+k)(G).

Also

.

is expressible by a formula in which all quantifiers are bounded by

3

q.e.d.

j+l

J

lemma.

End of the proof : Since every quantifier-free formula is of the form the claim holds for n

=

quantifier depth = 0. The claim carries over, trivially,

to Boolean combinations. It suffices to prove the claim for a = 3ua'(;,u) has quantifier depth n, n depth. Let

-

v = vo

cp(o)(z),

>

,...,vm-,.

that

1 , assuming it for all formulas of smaller quantifier

Apply the claim to

a ' ( v , u ) , write the resulting

Boolean combination of formulas of forms (I) and (11) in disjunctive normal form and distribute 3u

-

around v,u

over this disjunction. Using the fact that k-local formulas

are closed under Boolean combinations we get a disjunction of formu-

las, each of the form 3U[B(k)(G,u)

A

M Ail

iE1 where the

A.'s

are sentences of form

(I) or their negations. This is equiva-

lent to (3u B C k )

( 7 , ~ )A )

M Ai.

iEI By the induction hypothesis each

.., z s,-l[Mi
(zi)

sentence, where r' Q 7n-2

and

3z0,.

variables of a'

A

Xi

is either

d(zi, z . ) > 2r'I or a negation of such a M i
is more by 1, its quantifier depth-less by 1).

Also

Local and non-localproperties

115

kG (7n-'-l). Applying the lemma we can rewrite 3u f3(k)(v,u) as a Boolean com2 bination of the desired form, such that in the sentences of form (I) we have we r = 2k+l S 2 1 (7n-1-1) + 1 = 7"-', s Q m+l 4 m+n and in the formulas Qt(;) 2 1 1 (7"-1). have t 4 7k+3 4 7. - (7n-1-1) + 3 = 2 q.e.d.

:

The question that naturally arises is

Can the bounds on

r,s,t, be improved 7

Let us introduce the following classification of sentences, where

"LS"

stands

for "Local Sentence". Definition : LS(r,s) 3vo where k 4 r

and

is the set of all sentences of the form

,...,vL-l (i!L$(k)(vi)

A

L 4 s . This is defined for

The theorem asserts that if

d(vi,v.)

M

J

i
> 0,

s

> 2k)

> 1.

is a sentence of quantifier depth n

equivalent to a Boolean combination of sentences from LS(7"-',n) formulawith m members of

then it is

and if it is a

free variables then it is equivalent to a Boolean combination of

LS(7n-1, n+m)

and of

1

formulas. A careful checking of

(7"-l)-local

the end of the proof shows, however, that we actually get, in the case of sentences an

equivalent Boolean combination of members of

u.i
In the case of formulas this is to be replaced by

...,n-I.

i = 0,

Now the following fact holds :

ELS(r,s), such that for every sequence "r,s 1 4 so < sl<. > r >0 ,(rk-l,skMl) in which r > r >

There exist sentences

..

(ro,so), . the conjunction

0

A...h

U

U

o
i

Tlr

.. .

k-1

..< s k- 1

i s not equivalent to a Boolean combina-

k-1 "k-1

qro,So tion of members of

1

LS(i,j)

unless the combination contains members

14j =

...,k-1.

0,

In particular

any Boolean combination of members of

qr,s

is not equivalent to

H. GAIFMAN

116 This means that the LS(r,s)

form a doubly indexed proper hierarchy in a strong

.

n Having them we can get r,s The lower the quantifier depth that is needed for the

sense. There are many ways of constructing such

r

lower bounds on

s.

and

n the higher (and therefore the better) the lower bound. Our n will be r,s r.s in a language consisting of equality, one bindary predicate R and one monadic (which is used only for no,s). For r > 0, n asserts the exisr,s disjoint neighborhoods V(r) (xi), i
predicate P tence of

s

where

6 (x)

=

= r-I) A T

3u(d(x,u)

n

As

in the firm cp(r)(x).

...,Z L - I ,

points : 0,

take (;3 0,s

R

with

P interpreted as 0. Let Co preted as empty and let

R

tacking on it a

CL, for

L>l,

be the model consisting of { : i>L}U{<2k-1,0>)

interpreted as

consist of one point, 0 , with R

*

consist of, 0, with P

C-l

L 2 0

as empty. Finally, for

r) ; evidently Sr(x) can be put M P(xi) A M d(x. ,x,) > 0). i
3u(d(x,u)

To prove the desired properties let 22

from xi :

from its centre, xi, but no point of distance r

tance r-l

and

interpreted as

L+1 ; i.e.,

"tail" of length

ao, ...,aL and ,..., . Now consi-

I

also its

in the copy of

* CL L

*

...,

Ck

Note also that the

in the copy of

*

Ck

and let M

exactly

s

copies of

Cr-]

from x, this is

L+l

CA

but that for all x

from x

(because of the

Cr-l, where O < s < m , and arbitrary num-

M'

. . .,V(k) ( x ~ - ~ )of

that the corresponding neighborhoods ) ' ( V This is also true for copies of

L+l

L-neighborhood of the point that

be obtained from M

L>r, provided that

..,V(L)(xJ-l)

(xi)

j<

and s

by dele-

O
and

j disjoint simple L-neighbor-

there is a similar collection V(')(x,!J,.

Cral in MI).

L-neighborhood coincides

Crm1. It is not difficult to see that, for all

for each collection V(L)(xo), hoods in M

i>-l

contain infinitely many disjoint copies of each

bers of copies of the other basic models. Let ting one copy of

the

is isomorphic to

there are points of distance

tail). Now fix r > O CL, L = O,l,

in a copy of

CL and as there is no point of distance

(L+l)-neighborhood.

corresponds to

C.'s,

(where any number of isomorphic copies of each of these models

may be used). Note that for x with the copy of

{ O } and

add new members

,

der models which are disjoint unions of isomorphic copies of the

*

and

inter-

let CL be obtained from the cycle CL by

add to the binary relation the pairs and the C.'s, j > O

P

V(')

(xi)

in M'

such

are isomorphic.

(for in that case each of the

that occur in the collection in M

can be matched by a copy of

Consequently Boolean combinations of sentences from

Local and non-localproperties

u LS(i,j)

U

U

i < r 1Gj

O<

U

U

6 i

lGj
LS(i,j)

117

have the same truth-values in M

and

rl is seen to hold in M and to fail in M'. Our claim can be now der,s rived by considering a model which is the disjoint union of so copies of Cr -1,

M'. But

0

s1

copies of Cr -l,...,~k-l copies of Cr 1 k * C9., L = 0,l)

and of infinitely many copies of

... .

Now d(vI,v2) Where

log

=


(for n>O) can be expressed using quantifier depth 'log 'n

log2 and

-

'x'

=

smallest integer >

by induction, using the equivalences : d(u,v) G 1

d (u,v) G 2k

R(u,v),

3w[d(u,w) G k

d(u,v)GZk+l

A

-

X.

.

This is easily established,

3w[d (u,w) G k A d (v,w) k]

and

d(v,w) G k+l].

Consequently d(u,v) > n, which is equivalent to 1 (d(u,v)Gn), needs ?og2z quantifier depth and this is also true for d(u,v) = n (which is equivalent to (d(u,v) G n A d(u,v) > n-1)). Hence, for r > 0, cr(x) can be written with rlog rl + 1

quantifier depth. From this it follows easily that, for r > 0, rl bog rl + 1 +

can be expressed in quantifier depth

s.

s

quantifier depth. Evidently,

Theorem : For each n > 2

there exists a sentence, u , of quantifier depth n

such that any equivalent Boolean combination of sentences of U must contain members form each of LS(2n-2-i, =

n-1, LS(2-',

0,s

r,s needs

From this we get the following lower bound.

~~

i

rl

..

S i

. U LS(i.j) lGj

i+l) , i = 0,. ,n-1 where, for

is to be reread as LS(0,n).

n)

This simply means that we get the lower bound by replacing in the upper bound everywhere

7"-'

by

2"-'.

(and rereading 2-1 as 0).

Problem : Narrow down the gap between the upper bound 2n-2

7"-'

and the lower bound

In the case of formulas a similar construction yields the same kind of lower bound where, in addition,

1

(7"-1)

is to be replaced by

As

2"-'.

far as the applications of the next section are concerned there is no need for estimates on the values of r and s . Here is an outline of a shorter semantic

proof of the theorem as stated :

H. GAIFMAN

118

For €;

M

k = 0,I , .

-

let the local type of a be the set of all local formulas cp(k)

..

such that M ~ ~ ( k ) ( Now ~ ) assume . that for all

-a € LS(i,j), b

=

bo,

M1

/= a

...,bmdl €

e,

M2

i> 0 , j> 1

(G),

and all

1 u . Assume furthermore that a = ao,. ..,am-1 E M 1 , and a and b have, in their respective models, the same

M2

local type. Finally, assume that M1 and M2 are u-saturated. Then for every am E M1 there exists bm € M2 such that a,a and b,b, have the same local type. that

(This holds also for m = 0). Playing a FraissG-Ehrenfeucht game we deduce

-

(Ml,a) : (M2,%).

extensions MY

If M1 and

and Mi

M2

are not w-saturated we can get elementary

that are w-saturated and from

*-

(Ml,a)

(M2,%)

5

we dedu-

(Ml,Y) ? (M2,b). Now use the theorem that if every two models (of a theory T) that satisfy the same sentences out of some class S are elementary ce again

equivalent, then every sentence is equivalent (in T) to a Boolean combination of S.

members of

12. LOCAL INSEPERABILITY

Definition : Let Co and

C1 be two classes of models for L and

a sentence. Say that cp seperates Co

and

C1 if

is true in all members of

(0

one class and false in all members of the other. C and C

ble if

0

some sentence

if no sentence in L If

P

cp

in L

and

kq,

If

4

p

is first-order within the

{MEC : M does not satisfy F 1

first-order seperable. This means that for some M

are first-order sepera-

seperates them.

IMEC : M satisfies P I

if

1

seperates them. They are first-order inseperable

is some property of models then we say that

class C

let cp be

cp

are

we have : M satisfies P

<=>

for all M E C . is any set of formulas, then

(M,ao

,...,a,-l)

5

@(M',b0

,...,bk-l)

means

Definition : ( I ) Say that Co and C1 are locally inseperable if for every natural number n there exist a pair of models Mo E Co, M1 E C1 such that, for all r,s

< n, the same (up to isomorphism) collections, of

r-neighborhoods are realized in Mo lection V(r)(xo),

.. .,V(r)(xs-,)

corresponding collection )'(V (~(~'(x~),

J = (~(~'(y~), x.)

(11) Say that

s

disjoint simple

and M1. By this we means that for every col-

of disjoint neighborhoods in Mi (yo),

yj)

CO

,

...,V(r) (ys-l) in Midi

there is

a

(also disjoint) such that

for all j
and

C1 are locally first-order inseperable if

119

Local and non-localproperties there exist Ma E Co, M1 and every collection of disjoint neighborhoods

for every n and every finite set of formulas Q r,s G n

such that for all

. . .,V (r) ( x ~ - ~ )in (yo), . . . v(r) in

V(r)(xo),

Mi, (i = 0 , l )

"(r)

M1-i such that

there is a corresponding collection (V(r)(x.),x.) J

all j
Co

and

O(V(r)(yj),yj),

f

J

for

C1 are locally inseperable they are also locally first-

order inseperable, but the converse need not hold. As a corollary of the Main Theorem we have : Theorem 2.1.

: If

C,

and

C,

are locally first-order inseperable then they are

first-order inseperable. Proof :

Let cp be any first-order sentence. By the Main Theorem cp

is equiva-

lent to a Boolean combination of sentences of the form 3G(M $(r)(vi)

A.M.

i
d(vi,v.)>2r).

Let

be greaterthan all the r

R

and

s

that

l
consist of all the J l ' s such that

let 0

are involved and

this combination. Let M

E Co

M1 E C 1

0 rements of the definition with respect to n

hence cp does not seperate Co

and

Jl(r)(vi)

occurs in

be the two models satisfying the requiand

0. Then

Mo

I= cp

0

M1 (= cp,

C1.

In order to show that a certain property P

q.e.d. is not first-order within

C

it

{MEC : M satisfies P} and {MEC : M does not satisfy PI are

suffices to show that

locally first-order inseperable. In most examples that we have thought of these two classes satisfy the stronger requirement of local inseperability and it is this that one proves directly. Yet there are cases in which only the first property holds and for this reason we have introduced this weaker version. If two classes Co

and

C1

are locally inseperable then they are inseperable in

the somewhat stronger logic obtained as follows : Let

{P i}iEI

be any family of properties of models of the form

is a model for L

and

aEM. There is no restriction on the

be preserved under isomorphisms : If then

(M',a')

te P F )

to

satisfies Pi. For each L

and

let L*

for L, enlarge it to a model of xEM

such that

put : M

(VCk)(x),x)

I= cp(z) 0 Df M* I=

P.

(M,a)

where M

except that they

satisfies P. and (M',a')=(M,a) P i and each kEw add a monadic predica-

(M,a)

be the language thus obtained. If M

is a model

for L* by interpreting each P F ) as the set

M*

satisfies

cp(g).

P.. If cp(F)

is a formula of

L*

H. GAIFMAN

120

We shall call the logic thus obtained a monadic arbitrary local logic (of L), or for short, a MAL-logic. The word

"arbitrary"

is no restriction on the properties

is used to indicate that there

of the neighborhoods. We could define

Pi this logic, equivalently, by adding, instead of monadic'standardlyinterpreted predicates, new quantifiers 3!k), exists x

such that

Theorem 2.2.

If

Co

such that

(V(k);x),x) and

In order to prove it we first show

____ Lemma 2.3.

If

Co

are Mo E Co,

and

y = yo

is interpreted as

and

,...,ym-l E

"there

...'I.

L*

seperates them).

:

C1 are locally inseperable then for every m > 1

MI E C1 which realize the same m-neighborhoods of

That is to say, for each

-

Pi

C1 are locally inseperable then they are inseperable

(i.e. no formula o f

in the MAL-logic

."

"3.!k)x..

satisfies

x = xo ,...,xmbl E Mi

,

i = 0,1,

there

m-tuples.

there exists

satisfying :

It is easily seen that any isomorphisms of

(V(m)(x),x)

onto (V(m)(T),y)

carries

< m. 0'' * "xjs-l 0 s- 1 Hence the condition in the lemma is a natural generalization of the condition de-

v(r)

) isomorphically onto

(Xj

V(r)(yj

)

,...,yj

for all

r,s

fining local inseperability. The,lemma asserts that this generalization is already implied by local inseperability. Proof of 2.3. Let

t

:

> 0, k >

the models MO,MI Then M

0

and M

We show the following : 1

and assume that for all

s d n,

realize the same collections of

and all s

1 realize the same t-neighborhoods of

r

< 3k-1t +

I k

~ ( 3-3)

disjoint r-neighborhoods. k-tuples.

This implies the lemma. For, given m, take M O W O , MIECl which satisfy the requirement in the definition of local inseperability for n The proof is by induction on k. If k = 1, then 3

k-l

=

Zm-l .m+$(3m-3).

1 k .t+?(3 -3) = t, hence the

condition is that both models realize the same simple t-neighborhoods, i.e. same t-neighborhoods of

1-tuples.

Assume the conditions to hold for t and

k+l and let x

=

xO,

...,xk

be a

the

Local and non-localproperties

121

(k+l)-tuple in one of the models say Mo. We have to find the corresponding y ' s j in MI. First consider the case where d(x.,x.) > 2(t+l) for all i
j

V(t+l)(xj), Since t+l

=

0,...,n

3

form a collection of

< gk+l. t + 13. (gk+I-3)

k+l

disjoint

(t+l)-neighborhoods.

the condition implies the existence of f * (V(t+')(x.),x.) (V(t+l)(y.)y.), j' J J J J are disjoint. If ~€v(~)(x.), z'€V(~)(X~,)

yo, ...,yk E MI and of isomorphisms

j

re the V(t+l)(y.), j # j'

V(t)(xj),

U f j
J

=

...,k,

0,

J

wheand

> 2. Hence V(t) (x) - is the disjoint sum of the models j = 0, k. The same argument applies to the V(t)(yj)'s. Hence is an isomorphism of (V(t)(T),:) onto (V(t) (Y) - ,Y). -

then d(z,z')

...,

d(xi,x.) G 2(t+l)

Now assume that for some i < j < k

3

; say, without loss of gene-

rality, d(xk-l, x,) < 2(t+l). Replace k + l by k and t by 3(t+l) ; since 1 k k 1 3k- 1 (3(t+l)) + 7 (3 -3) = 3 .t + 2 (gk+I 3) the condition for k + l and t

-

implies the condition for k and (with 3(t+l)

instead of t)

3(t+l).

By the induction hypothesis for k

there is a k-tuple yO,...,yk-lEMI

and an isomor-

phism :

Since xk E v(~(~+')) ( x ~ - ~ ) ,we have V(t)(x,J yk

=

Df f(x,).

that V(t)(yk)

It is easily seen that c V (3t+2)(yk-l).

function inside V(3(t+1))

(xo,.

Mo and similarly for V(t)(yk) in §l).

Hence f maps V(t)(x,)

v(~) (G) onto

yk

Put

C V (3t+2) "(2(t+l))

(yk-l) and, consequently,

Moreover, for members of V(t)(xk)

.. ,xk-1 )

the distance

is the same as the distance function in

and M I . (This follows from the argument for -(I) onto V(t)(yk).

It follows that f maps q.e.d.

vet)(7).

Proof of Theorem 2.2 : Given cp

in L* we have to show that cp does not sepeP(k) in cp. Let ? rate Co and j be obtained from L by adding these predicates and for every model M for L i = 0 , I . Let $ be the Let = {$ : MECi}, let 2 the enriched model for C1. There are alltogether finitely many

2.

Zi

2.

maximum of all k such that P!k) is in For each n, there are by lemma 2.3 J models M EC and MIECl that realize the same (n+g+l) - neighborhoods of n0 0tuples. Let x = xo, x n-l € Mo, Y = yo,. ..,Y,~ E M1 and let

...,

: v(n+g+l)(;)

Vfk)(z)

I

v(n+2+l) (y). If z€V(")(;)

onto Vtk).(f(z)).

Whether F'y)(a)

then for each k < g holds in Mi

(i

f maps

= 0,l)

depends only

H. GAIFMAN

122

on the isomorphism type of (V(k)(a),a). Hence for z€V(")(X) we have (k) 2O )= Pj( k ) ( ~ ) 0 81 I=P.J (fz). Thus f maps V(")(X) isomorphically onto V(n) (Y ), where these are considered as neighborhoods in Go and G1. Consequently to and ?1

are locally inseperable. By Theorem 2 . 1 ,

cp

does not seperate Co and C1. q.e.d.

By analogy to monadic local properties we can consider n-ary that is to say, n-ary relations, R, such that the holding of by the isomorphism type of

(V(k)(a),a),

natural number. Let the AL-logic

local properties ; R(Z)

is determined

where k = kR is an arbitrary, but fixed,

(i.e, arbitrary local logic) be obtained Ly

adding predicates denoting arbitrary local properties and let the n-AL-logic be the one in which only predicates of arity


Does Theorem 2 . 2 . remain true if we replace

are added. "AL-logic" ?

"MAL-logic" by

The argument for the monadic case does not carry over ; for, by adding relations of arity

>

2

one usually changes the distance and a k-neighborhood in the enri-

ched model may contain members which are not in the k-neighborhood in the original model. We do have however :

~

Theorem 2 . 4 .

If Co

and

C1 are locally inseperable and for each k, only fini-

tely many isomorphism types of the models of

Co U C 1 , then

k-neighborhoods of

Ci

and

(n-1)-tuples are realized in

C1 are inseperable in the n-AL-logic.

Actually, it suffices to have a locally inseperable pair of subclasses Ci

C

Co

,

Ci c C1 satisfying the condition of finitely many realizable isomorphism types. The second claim is trivially implied by the first. For if

Ct0 and

CI1 are

inseperable in some logic so are any pair of classes Co,C1 that include them. If, for

some m, all simple 1-neighborhoods in the models of

Ch

U Ci

haveQm

members, then also, for each k,n, the k-neighborhoods around n-tuples are uniformly bounded in size. (This situation obtains in the forthcoming examples). Then, evidently, the condition of theorem 2.4.holds ; but the theorem becomes superfluous, because within

Ch U Ci

, every

formula in the AL-logic is then equivalent to a

first-order formula. Theorems 2.2. 1-neighborhoods realized in

and 2.4. are of interest only when the sizes of

C; U Ci

are not uniformly bounded by some integer.

The proof of 2 . 4 . is given in the appendix

123

Local and non-localproperties Examples

In all cases the property in question, P, is not first-order definable within the indicated class, C. The subclasses IMEC : M satisfies P } not satisfy P }

and

IMEC : M

does

are locally inseperable (except for the last example where they

are only first-order inseperable). This is shown by pointing out a pair

Mo, MI E C r,s

tions of be

of

prototypes such that Mo

satisfies P, M1 does not and, for all

sufficiently small with respect to the size of the prototype,the same collecdisjoint simple r-neighborhoods are realized in both. The models can

s

"blown up"

in the obvious way to any desired size, while

retaining these

properties. 1.

P = Connectedness, C

=

class of finite graphs.

MO

M1 Fig 1

If on each circle we have > 2r+l points then all simple r-neighborhoods are of the form :

and if, on each circle, we have > (2r+l).s

points then s

disjoint simple

r-neighborhoods can be realized. In

[CHI

the method of Fraissc-Ehrenfeucht games is applied

to

this prototype

pair in order to show that connectedness is not first order definable in C. In principle

the method of games is applicable to the other examples. However, if

the models are not as homogoneous, a description of the strategy for n

moves canbe quite involved,

2nd

player's winning

whereas a glance may suffice to

see that the neighborhoods are the same. What is more important is that the method indicates the way in which the prototype models should be constructed. In the following examples we let the drawings speak for themselves.

H. GAIFMAN

124 2.

P

=

a - connectedness

C

=

finite (a-1)-connected graphs.

(A graph is k-connected if the removal of any k-1 edges does not disconnect it). Fig 2.2.

corresponds to the case

L = 4 . In general,

bridges between the two components of M I . For

L- 1

I,-even,

,t

is the number of is the nomber of.edges

issuing from any ordinary vertex (i.e. vertex not connected by a bridge). The case of an odd

I?

is obtained from that of

ordinary vertices in Mo

!,+I

by removing an edge between two

and the corresponding edge in one of the components in

M1.

MO

3.

P

=

Fig 2.2

being planar

C = class of finite graphs

(A-graph is planar if it is representable in the plane so that each edge is an arc and the arcs do not intersect except at their common end points. In Fig. 2.3. is planar, but

M, is not).

Fig 2.3

MO

125

Local and non-localproperties 4.

P C

=

Hamiltonian (i.e.,

=

class of finite k-regular graphs with a Hamilton path. Here k

3 . For k = 3 , 4 , 5

having a Hamilton cycle)

we can restrict C

should be

further by adding the requirement that

the graphs be planar. (A Hamilton path is a path in the graph passing through each vertex exactly once.

If such a path is also a cycle then it is a Hamilton cycle. A graph is k-regular if every vertex has degree k, where degree (x) = number of edges containing x).

MO

Fig 2 . 4

Figure 2 . 4 . is the construction for C

=

class of 4-regular planar graphs having

a Hamilton path. The construction for k>4 is obtained by replacing each vertex in Fig. 2 . 4 . by a graph

G

graph. Let

(a,b)

(a,a')

and

Hamilton cycle containing

as follows. Let (a,a')

G*

be a k-regular Hamiltonian

be two edges in G* and not containing

such that there is an (a,b)

. Get

G by removing

these two edges (without removing vertices). Now replace each vertex of 2 . 4 . by a copy of

G and use edges issuing from

rent copies. Figure 2.5.

a,a', b

in order to connect the diffe-

indicates how this is to be done. For k = 5 take G*

to be the icosahedron. Since this is planar the construction can be carried out so as to yield a planar graph. (For k>5 the graph cannot be planar, since each planar graph has a vertex of degree 4 5). We leave the case k = 3

for the reader.

H. GAIFMAN

126

Fig 2 . 5

I' (W

4.

=

a, 0

P

=

=

a', O = b. arrows show parts of a Hamilton path) C = connected finite graphs

Eulerian

(A graph is Eulerian if it contains an Euler cycle, i.e. a cycle passing through

each edge exactly once).

As is well known, a (finite) connected graph is Eulerian iff each vertex has an even degree. This is obviously a local property. Consequently, within the class of connected graphs those that are Eulerian are not locally inseperable from those that are not ; in fact

-

they are definable in the MAL-logic. Yet they are first-

order locally inseperable. This is seen by letting C* the form

consist of all graphs of

:

a

Fig 3.7

Let

CocC*

consist of those having an even number of vertices and let

Then, for GEC*, G

is Eulerian or not according as G E C o

place the binary predicate R each M E C*

let M'

or

C1=C*-C0-

GEC'l. Now re-

of our language, L, by a monadic predicate, P. For

be the model for the new language, L', obtained by interpreting

Local and non-localproperties

P as {xEM : x # a and x # b}. Then

This implies that every sentence ( P E L

127

:

is translatable into some tp'fL'

such

that, for all M E C*,

If rp were to seperate C@ C l 0 = {MI : MECo}

models of L'

from

-

C1 then

~ p '

would have seperated

from C'l = {M' : M E Cl}. But this is impossible, because for

we have d(x,y) =

for all

x # y

,

implying that each simple

neighborhood consist of one point and consequently, that

.

Cl0 and CI1 are locally

inseperable

$3.

CERTAIN TRANSITIVE MODELS

Levy's hierarchy classifies formulas in the language of set theory as follows : C -formulas are those in which all quantifiers are bounded, i.e., of the forms 0

3xEy, VxEy. The higher levels are obtained in the usual way by tacking on alternating blocks of unbounded quantifiers. We have : x = 0

x

c=>

VyEx (y#y)

is transitive

x is an ordinal

0

VyEx VzEy (ZEX)

0x

is transitive AVuEx VvEx [u=v V uEv V vEul

x

is zero or a successor

x

is a natural number

0

0

x is an ordinal A(X=OV3UEX Vvfx(v=u V veu))

x is zero or a successor

A

VyEx (y is zero or a successor).

Consequently all these notions are Xo Now in

ZFC

(i.e. expressible in ZFC by

A

z

is finite

x

is finite <=> VZV~EX[ z is not a mapping of x- {y}

0

3z3y [y is a natural number

XI onto XI

maps y onto

x

The first is easily seen to be

C 1 , the second-lI1. Thus finiteness is, in ZFC,

a A -notion. The natural question is whether it is Co. 1

C-formulash 0

finiteness can be defined in either of the two ways :

The answer is negative in

the strongest possible sense : There is a transitive infinite set A , of rank w , such that for any

Eo-

formula

H. GAIFMAN

128

cp(v), A

there exists a finite transitive subset A ' c A ,

iff it is true for A ' . Furthermore A

te is not characterizable

by

such that in cp'

is true for

a Z o - formula in the real world.

Note that for any cp(v)ECO, with v cp' (v)ECO,

such that cp(v)

is primitive recursive. Thus being fini-

as its only free variable, there exists

all quantifiers are of the form QxEv

and we have,

in pure logic : is transitive -+ (cp(v) <-->

v cp' (v)

is obtained by replacing every

..."

every "VxEu V'(v),

by

"VxEv (xEu

--+

cp'(v))

"3xEu.. ." by "3xEv (xEu ...)" and ...)". Furthermore, we can assume that in A

v occures only as a bound for quantifiers. For if x

ferent from v

is any variable dif-

then its occurences inany quantifier-free part are within the

scope of some QxEv, hence the occurences (i.n a quantifier- free part) of VEX and v = x

can be replaced by

ly v = v and

vEv

vely. Now let cp"

x # x , and the occurences of xEv- by

are replaceable by, say Vy€v (y=y)

x =

X.

Similar-

3yEv(y#y),respecti-

be the sentence obtained by removing the bound on the quanti-

fiers, i.e., replacing each QxEv by A

and

Qx. It is easily seen that for a transitive

:

we have

Vice versa the satisfaction of any sentence in

(A,ErA)

'

is equivalent to the

A

of some Z --formula cp(v). Hence we have to construct a tran0 sitive infinite model M such that for any sentence cp there exists a finite satisfaction by

transitive M'cM

such that

The construction to be given here is similar to the one used in

[GI.

To show

that M has the desired property a rather involved argument was used there,which relied on the generalization of Marcus's result. Here we get it at a glance by realizing that the classes {MI

and

{M'cM:M'

is finite and transitive) are

locally inseperable. Construction of M. Let

l
Let

(O,l}* = set of all finite hinary sequences. For w€{O,l}*, let

0

1

of w, A = empty sequence. For X c { O , l } * , Xi = {wi : wfX}

(where i€{O,l}).

put

1x1

=

cardinality of

!L(w)=length

X,

We construct, level by level, a tree

129

Local and non-localproperties T = U L. c {O,l}*, where L. iEw Lo

=

=

tn

i-

level of T

=

{WET: P(w)

=

i}.

=

7 1L.I and

{A}

For j not of the form n.

Lj+1

9

=

Lj0

, Lj+l = L.0 u L.l J J

For j

=

n2k

For j

=

n2k+l

, choose any L?~jc L Put Lj+l

=

such that

* J

IL.1

1

J

L t O ULfl.

Thus, branching occures only at the n.-levels ; but for i odd half of the branches reaching this level are cut off, namely the branches whose ends are not in L* It follows that for j = nZk ILj+ll=ZILjl and for j not of this form j' J = 1 and, for n2k < j < n 2 (k+ 1 ) ' ILJ.+ l I = I L J. L Consequently, for j 4 no, IL.1 th ILjI = 2 k + 1 . For j = n2k+l , the j- level contains 2k+1 leaves. A l s o , every leaf of T is in some n2k+l-level. The choice of L* can be made effectively, J

giving rise, say to a primitive recursive T. The model, M, constructed from T will be as effective as T. With each WET we associate a set h(w) h(A)

=

0,

h(w0)

=

{h(w)).

as follows : If wlET, then, as is easily seen, w must

be of the form w 0 and we put h(w1)

=

{h(w),h(wo)l.

Let M be the model whose domain is {h(w)

: w€TI. Evidently M

is transitive.

Fig 3 . 1 . describes a portion of M. Members a,bEM such that aEb are exactly those connected by an edge going upwards from a to b. Filled circles correspond to leaves of T. It is clear that the shortest path connecting h(wl)

and h(w2)

in M corresponds to the path connecting w1 and w2 in T, except that at branching points we can use a small short cut : instead of going from w then to wO1 Let M,

=

we can go directly from h(w)

the m-level trancation of M

=

to h(wO1). {h(w)EM:

E(w) 4 m). It is not difficult

to see (by actual looking) that, for given r,s, if m sufficiently large then M and M,

to w0 ana

is of the form

realize the same collections of

s

% and disjoint

simple r-neighborhoods. Here is a more formal proof : nj+2 and Suppose that nj t l < E(w) there exists exactly one h(w')EV(r)(h(w))

2 r < 1 1 ~ + ~ - nIf~ .E(w)-nj+l

< r then

such that E(w') E {no, n l , . . . I

and

H. GAIFMAN

130

this is a branching point at level nj+l. If nj+2- C(w) < r then, again there is one such w'. It is at level nj+2 and is either a branching point o r a leaf.

Fig 3 . 1

$

I

I

1

I

I

I

t

I

0 0

In all other cases there is no such w'. Hence we can divide the r-neighborhoods of h(w) (I)

into the following 3 types.

A simple chain with

some leaf w',

d(h(w),h(w'))

(11) A simple chain with

not in the centre. This is the case where, for

h(w)

h(w)

<

r. at the centre. This is the case where the

neighborhood contains no leaf of distance < r and no branching point. (111) There is a branching point of distance


neighborhood is either of the king of Fig 3.2.

from w. In this case the

(111) or a substructure obtained

Local and non-localproperties by omitting one of the 3 branches. h(w)

131

may be either the branching point or on

one of the branches. It is not indicated and does not matter in the argument.

P

9

b

Fig 3 . 2

Assume that

j

% < X(v)<%+l, < n j + * , such that isomorphic. For if l . ( V ) - %
is odd and nj+l-n. > 2r+l. Let J

L(v) > n j + l , say

where k>j+l. Then there exists w, satisfying nj+l< k(w) the r-neighborhoods of choose w

h(w)

and h(v)

are

satisfying L ( W ) - ~ ~ + ~= L(v)-%.

satisfying n j + 2 - L(w) =

?ctl -

L(v)

and such that the n j + 2 -

is of the same type (leaf or branching point) as the

of w dant of

V.

In all other cases choose w

such that

L(w)

-

level descendant level descen-

II,+~-

nj+l and n j + 2 - E(w)

are both > r. Now consider Mm, for m = nk

where k Z j + 3

neighborhoods around points whose level is in

and

2r+l < nj+l "j+2 )

-

nj. The r

("j+l and Mm. If furthermore, j is odd, then every simple r-neighborhood in M some h(v)

-

are the same in M around

at level > nj+l is isomorphic to the r-neighborhood of some h(w)

at level E (n , I I ~ + ~ IThe . argument given for M works also for Mm. The essenj+l tial point is that by truncating M at level % one converts all branching points at that level into leaves and consequently neighborhoods may change from type (111) to types (11) and (I). But these can be simulated in the lower region. using the leaves at level nj+2. (It is for this reason that many

leaves).

T

should contain

H. GAIFMAN

132

Given If m

r

and

=

and

hoods in M

s

j

choose k>j+3

2 j +I 0

such that nj+l - n . > 2r+l

!?,(v)
,

the

2'

such that

>

2s.

h(v)

is matched by

nj+l< !l(~)
disjoint branches going from level nj+] + 1

terminate in leaves and ''2

to nj+2

, of

which

in branching points ; also each neighborhood

< n. intersects at most J+l enough disjoint branches to realize the rest.

around a point a1 level

r

of these. Hence there are Q.E.D.

...

in which m e {no, n l , }, the claim need not be true, even M, is arbitrarily large. For M, may have r-neighborhoods of type I11 whose

Note that, for if m

and ''2

J

disjoint simpler r-neighbor-

r-neighborhood of

itself. The rest can be realized around h(w)'s There are 2

s

can be matched by an isomorphic collection in the other

or in M,

model as follows : If

=

then any collection of

upper end nodes are of distance < r from its distinguished member. Such neighborhoods are not realized in M. However, using a somewhat more complicated tree we

can construct a transitive

M of rank u having the following curious property

For every first-order sentence cp M cM c 0 1

Mi

I= cp

... c M

c

...

and every ascending infinite chain

of transitive models such that

f o r all but finitely many

i's.

UM. = M,

M I=cp

iff

:

Local and non-localproperties

133

APPENDIX Proof of Theorem 2 . 4 Consider a predicate P(k) (T)

-

which denotes a property of the k-neighborhood of

-

v , where v = v~,...,v~-~. We show that, as far as models of

ned, LU

P(k)()

1

is equivalent to a first-order formula in a language

{RY')

: i
such that each R!k+l) 1

a property of the

,...,um- 1 )

u

--+ M

and

$

J

, we

I.

that, that for some constant

?

denotes

(2k+l)(n-1)).

$,

such that M

= L U {R.( k + l ) : XI},

d(u.,u. ,) 3

It follows that for every model M

(z)

R.(k+l)

1 CP

5

<-7

I-finite, such

have :

m- 1

R!k+l)(uo,...,u

<

we can get

is in some

< n,

are cancer-

and the following is valid :

(d(ui,u.)

i
is a sentence in the n-AL-logic

for all b l E C ' O U C f l

is of arity

-

(k+l)-neighborhood of

R!k+l)(uo

If cp

Ch U C'

for L, if

is the distance in the enriched model

M"

J

I.)

is the distance in M

d 'i,

(for L)

<

and

2

we have :

d(a,b) G !L.a(a,b).

"V"

Hence if

"?"

and

?(j)(g) c

have :

denote, respectively, neighborhoods in M

V("j)(;).

now used to show that no sentence of The predicate Let

M

Iq.(T) J

i
R(k+l)

:j < p >

C (d(v:,v:) I

J

'

2 k + l ) , where ( i , j ) either

k p(k)(v)

+ I)'

+ or

<->

w

(P(k)(G)

j


determines a graph Gj(T)

for which the formula d(vi,v.) 3

v.

and v

k cpj(T)

-+

j

seperates C h P(k)(F)

and

we

Ci.

are obtained as follows :

denotes the formula

-

is chosen. Here

is a first-order formula of the form

3

2

be the set of all conjunctions of the form

gation and for each that each c p . 6 )

2

and the translation of

.

<

and

The technique of the proof of Theorem 2 . 2 . can be

<

2k+l

A

+,

'-I)'1

Tn(n-1)

p = 2

its ne-

. Note

+(k+l)(v).Evidently

Q.(T)). 3

in which the edjes consist of all

(vi,vj)

is a conjunct. It is easily seen that if

G . ( G ) then J (n-1) ( 2 k + l ) . Hence for those j for which

are in the same connected component of d(v. ,v.) 1

3

<

G.(T) is 3

H. GAIFMAN

134

connected we can take P(k)(;)

Q(y)

sider

(y)

= cp

4

A

for which

its components. Then each

Ti J

(T),a)

...,

is not connected and let Fo, v be t- 1 n variables and for each vi,v in the same j (2k+l)(n-l). If M I= cp(a) then V(k)(a) is c

i
the disjoint sum of the V(k)(zi), (V(k)

-

G (v) 4

has

~ ( 7-+) d(vi,v.) <

component,

(T). Con-

cp.(v) as one of the predicates J -

is completely determined by the isomorphism types of

(V(ai) ,Ti), iet.

Consequently an equivalence of the following form is valid

(V(k)

(si)is a new predicate describing an isomorphism type of

(k)

where

'i. j

(Ti),yi) and

the disjunction ranges over all combinations which, given

cp(y) , imply P(k)(y).This disjunction can be infinite. But for n'
C' U

Ci. Hence for some finite J' c J, the equivalence, with

v.,v J

.J

j'

€

v.

1

(v.) 1

A

, JlCk+') i

CA U Ci. Let

$(PI) (Ti) be the con-

-

v ) < 2k+l which occur as conjuncts in Q ( v ) , where j' j ' and let R!kcl) (Ti) be a new predicate equivalent to 1,j

junction of all

S:k!

0

replaced by J', holds in all models of d(v

(vi). Then the

RlkT1) have the desired property and ,J

Local and non-localproperties

135

REFERENCES [MU1

Aho, A.V.,Y. Sagiv and J.D. Ullman,

Equivalences among relational

expressions. SIAM J. Computing (1978). Aho, A.V. and J.D. Ullman, Universality of data retreival languages. Proceeding 6thACM Symp. on Principles of Programming languages. San-Antonio, Texas (Jan 1979) pp. 110-117. [CHI

Chandra A.K. and D. Harel, Computable queries for relational data bases, Proc. 2ISt FOCS, Syracuse, New York (Oct 1980) pp. 333-347. Feferman, S. and R . L . Vaught, The first order properties of algebraic systems. Fund. Math. 47 (1959) pp. 57-103. Gaifman, H.

Finiteness is not a

X -property. Israel J. of Math. 19

(1974) pp. 359-368.

Marcus, L.

Minimal models of theories of one function symbol. Israel

J. of Math. 18 (1974)

pp. 117-131 (also Doctoral Thesis, The Hebrew

University, Jerusalem 1975).