Download вдгжеиз © ¢ on every class of ordered finite struc

First-Order Logic vs. Fixed-Point Logic in Finite Set Theory
Albert Atserias
Computer Science Department
University of California
Santa Cruz, CA 95064
[email protected]
Abstract
The ordered conjecture states that least fixed-point logic
LFP is strictly more expressive than first-order logic FO on
every infinite class of ordered finite structures. It has been
established that either way of settling this conjecture would
resolve open problems in complexity theory. In fact, this
holds true even for the particular instance of the ordered
conjecture on the class of BIT-structures, that is, ordered finite structures with a built-in BIT predicate. Using a well
known isomorphism from the natural numbers to the hereditarily finite sets that maps BIT to the membership relation
between sets, the ordered conjecture on BIT-structures can
be translated to the problem of comparing the expressive
power of FO and LFP in the context of finite set theory. The
advantage of this approach is that we can use set-theoretic
concepts and methods to identify certain fragments of LFP
for which the restriction of the ordered conjecture is already hard to settle, as well as other restricted fragments
of LFP that actually collapse to FO. These results advance
the state of knowledge about the ordered conjecture on BITstructures and contribute to the delineation of the boundary
where this conjecture becomes hard to settle.
1. Introduction and summary of results
The main goal of descriptive complexity theory is to investigate the connections between computational complexity and logic on classes of finite structures. As regards firstorder logic, it is well known that every first-order definable
query is computable in LOGSPACE. Moreover, research in
descriptive complexity theory has established that essentially all major computational complexity classes can be
characterized in terms of definability in natural extensions
of first-order logic on classes of finite structures. In particular, Immerman [Imm86] and Vardi [Var82] showed that
Research partially supported by NSF Grant CCR-9610257.
Phokion G. Kolaitis
Computer Science Department
University of California
Santa Cruz, CA 95064
[email protected]
on every class of ordered finite structures, that is to say, if is a class of ordered finite structures,
then the class of polynomial-time computable queries on coincides with the class of queries definable in least fixedpoint logic on . Least fixed-point logic LFP is the extension of first-order logic FO obtained by augmenting the syntax and semantics with a least fixed-point operator for positive first-order formulas. As a general rule, least fixed-point
logic is strictly more expressive than first-order logic. In
particular, this holds true on the class of all ordered finite
structures over a fixed vocabulary, as well as on the class of all (unordered) finite structures over a fixed vocabulary.
There are, however, classes of unordered finite structures on
which LFP collapses to FO, and both are properly contained
in PTIME. McColm [McC90] was the first to focus attention on this phenomenon and to formulate a certain conjecture concerning necessary and sufficient conditions for the
collapse of LFP to FO on an arbitrary class of finite structures. Although in its full generality McColm’s conjecture
was refuted by Gurevich, Immerman and Shelah [GIS94], it
sparked a sequence of related investigations in finite model
theory [KV92, DH95, KV96, DLW96]. Moreover, the following special case of McColm’s conjecture still remains
open:
Conjecture 1 If is an infinite class of ordered finite struc
tures, then first-orderlogic
is properly contained in
least fixed-point logic
on .
This conjecture, which is often called the Ordered Conjecture, was made by Kolaitis and Vardi [KV92]. In view
of the aforementioned result of Immerman [Imm86] and
Vardi [Var82], the
conjecture
is equivalent to the
ordered
assertion that
on every infinite class of
ordered finite structures. Thus, it enunciates an inherent
limitation in the expressive power of first-order logic by
stating that first-order logic can never capture polynomial
time. All empirical evidence gathered to date supports
it. At the same time, researchers have established that either way of settling the ordered conjecture will have sig-
nificant complexity-theoretic consequences. Specifically,
Dawar and Hella [DH95]
if the
showed
that
ordered conjecture is false, then
. Furthermore,
Dawar, Lindell and Weinstein [DLW96] pointed out that if
the
conjecture holds, then
, where
ordered
(the Linear-Time Hierarchy) is the class of languages
computable by alternating Turing machines in linear
time
using a constant number of alternations, and
is
the class of languages computable by deterministic Turing
machines
from
in time. The separation of
can be viewed as the linear version of the separation between the Polynomial-Time Hierarchy PH and the
full Exponential Time EXPTIME; although it is widely believed that both these separations hold, neither has been established thus far.
Intuitively, the difficulty in proving the ordered conjecture arises from the potential presence of powerful builtin arithmetic predicates that may significantly enhance the
expressive power of first-order logic on classes of ordered
finite structures. One especially powerful such predicate
is
on$ the natural numbers ! #"%
the
binary
relation , where holds
if and only if the -th
"
bit of the binary expansion of
is equal to . The expressive power of first-order logic on finite structures with
BIT as a built-in predicate has been investigated in depth
by Immerman [Imm89] and by Barrington, Immerman and
Straubing [BIS90]. In many respects, the presence of embodies the difficulties encountered with the ordered conjecture; as a matter of fact, a particular instance of the ordered conjecture on finite structures with BIT turns out to be
literally equivalent to open
' problems in complexity theory.
More precisely, let
&
0 class of
/all
3
)' (*,+ .be/the
ordered
finite
-structures
$
3
*21 linear order and 54 ,6where
0 is the
standard
8
.
*71
Question 1 Is
on & ? In other words, does the
ordered conjecture hold on & ?
Gurevich, Immerman and Shelah [GIS94] raised this
question and stated that it is a “fascinating question in complexity theory and logic related to uniformity of circuits and
logical descriptions.” Indeed, it can be shown that
:9
on & if and only if log-time
is different
uniform
9
than polynomial-time uniform
(see [BIS90, Lin92] for
the definitions of these circuit-complexity
classes). Moreover, it can also be shown that
on & if and only
if
.
Our goal in this paper is to advance the state of knowledge about the ordered conjecture on & by seeking to delineate the “boundary” where this conjecture becomes hard
to settle. To this effect, we study certain fragments of least
fixed-point logic LFP on & and investigate the restriction of
the ordered conjecture on these fragments. We first iden-
tify a natural proper fragment of LFP for which the ordered
conjecture cannot be settled without resolving open problems in complexity at the same time. We then establish
that the ordered conjecture actually fails when further re stricted to certain fairly expressive fragments of
on
& , which means that these fragments collapse
to first-order
logic on & . To isolate these fragments of
, it is more
convenient to conduct our investigation in the context of finite set theory and to use set-theoretic concepts and methods. The starting point is a recent paper by Dawar, Doets,
Lindell and Weinstein [DDLW98], where it was shown that
the standard linear order is first-order
BIT.
'@?BA definable using
More precisely, let &<;>=
A the
C(D*E+ '@?Bbe
class
of
(unordered)
finite
BIT-structures,
where
GH $
./0
. Dawar et al. [DDLW98] showed
*F1
that there is a first-order formula over the vocabulary that defines the standard linear order on the class &;>= . This
rather surprising result was established by exploiting the
existence of a $ well-known
be
JI6K LM$ isomorphism
I6K O(see
N [Bar75])
K I and the
tween and
, where
I
QP
’s are the finite ranks, that is to say, finite initial segments
of the universe of sets
I 9 SR I obtained
JI by$ iterating the power-set
operation:
$
G!I TVK U LM$
. The isomorphism beI K
tween and
is the function W
( YX
defined by the recursion:
W
/$ ZR "[$ J $
! #"%$\
W
W
( This isomorphism enables us to translate questions about
the expressive power of logics on the class &;>'@= ?BA to equiva$
lent
G questions
?BA its$ image
.$\ HLM(/$*5+
'@on
C class
/$0 & ^$B] B W
, where W
. It
W
W
W *_1
is easy to see that the
containments
`
C
]
b
a
&
`
c
]
e
a
d
`
]
G
!I LM$
hold, where ^]
is
the
class
(f*g+
of all finite ranks, and d `] is the class ofLMall
$ near fi!h
nite
ranks,
that
is,
structures
of
the
form
such that
I
h
I
for some natural number * . Dawar
ji
i
GTVU
et al. [DDLW98] showed that there is a first-order definable linear order on d ^] such that its pre-image under
the isomorphism W coincides with the standard linear order
on . It follows that there is a first-order formula that defines the standard linear order on &<;k= . Consequently, the
ordered conjecture on & reduces to the question of whether
on
&<;> = , which, in turn, is equivalent to the
question: is
on &^] ?
The above set-theoretic framework makes it possible to
isolate and
study
variants of the ordered conjecture for fragments of
that are obtained by applying the least fixedpoint operator to collections of set-theoretic formulas with
special syntactic properties. In this paper, we focus on the
collection of l 9 formulas; these/L
are
the first-order formulas
over a vocabulary containing
such
that
oQ$ every
LpoQ$
JmQn Lp
rqsn occurrence of a quantifier is of the form
or
.
The collection of l 9 formulas has played a fundamen-
tal role in the development of set theory for two reasons:
l 9 formulas possess desirable structural properties, known
as absoluteness properties, and they can define many frequently encountered set-theoretic predicates. As Barwise
[Bar75, page 10] puts it, “The importance of l 9 formulas
rests in the metamathematical fact that any predicate defined by a l 9 formula is absolute, and the empirical fact
that many predicates occurring in nature can be defined by
l 9 formulas.” Both these facts will be of use to us in the
sequel. In particular, the proofs of most of our results will
rely heavily on the preservation of l 9 formulas under end
extensions.
$
Let
l 9 be the fragment of least
fixed-point
$
that consists of the
least
fixed-points
of all
$
that are positive in the relation symbol
l 9 formulas . Consider now the following
variant of the ordered con 9 $
jecture: does
collapse to
on & `] ? Clearly, a
l
negative answer to this question will imply that the ordered
conjecture holds on & and, thus, it is at least as hard to establish as the ordered conjecture on & itself. On the other hand,
one may speculate with some reason that the answer to this
question is positive, since l 9 formulas constitute a rather
small (and well-behaved) fragment of first-order
9 $ logic.
Our
l
first main result establishes that if
on
i
&
^
]
,
then
,
which,
in
turn,
implies
that
i
9 $
. Thus, the collapse of
to
l
on & ^] can neither be affirmed nor refuted without resolving long-standing open problems in complexity theory
at the same time.
The above result motivates the further study
of the or 9 $
on &^] . To
dered conjecture for fragments of
l
this effect, we isolate a fairly expressive collection of l 9
formulas $ and establish that the corresponding fragment of
9
l
collapses indeed to
on the class d ^] of
near finite ranks and, consequently, on the class & ^] as
well. This fragment is inspired from the work of Dawar
et al. [DDLW98], who showed that a linear order can be
defined in first-order logic on d ^] . Their proof proceeds by first defining a linear order on d ^] in
n >a$ natural way as the least fixed-point
of a
n >$
certain positive l 9 formula
,
and
then
showing
n >$
that
can actually be expressed in firstorder logic on
.
An
inspection of that particular l 9
d
`
]
n >$
formula reveals that it has the following special
syntactic property: every occurrence of the binary relation
symbol involves bound variables only. We turn this propl 9 as
erty into a concept and define the class
restricted
of
$
the collection of l 9 formulas such
that
U
every occurrence of the relation symbol involves
bound
$
variables only. We then show that if is an
U
arbitrary restricted l 9 formula that is positive
in
B $ , then
its least fixed point
is firstU
order definable on d `] . This generalizes the results in
[DDLW98] and also provides a tool for easily showing that
several other basic queries, such as (finite) ordinal addition,
are first-order definable on d `] .
9 $
l
obtained
After this, we consider fragments of
by restricting the number of free variables in the l 9 formulas under consideration.
$ observe that if is a unary
We
relation symbol and is a l 9 formula
$ is pos that
itive in , then the least fixed-point of coincides
9
with the
least
fixed
point
of
some
restricted
formula
l
$
9 $
. It follows that unary
collapses to
l
on the class d ^] of near finite ranks and, hence, on the
class & `] as well. Clearly, this raises the question
9 $ whether
l
any similar collapses can be proved for
formulas of higher arities, while keeping
in
mind
that
we
cannot
9 $
l
hope to show
that
formulas
of
arbitrary
arities
on & `] without simultaneously showing
collapse to
that
9 .$ Our main result
along these
lines is that binary
l
collapses to
on & ^] .
Finally, we derive tight polylogarithmic bounds for the
growth of the closure functions of arbitrary positive l 9 formulas on ^] , where the closure function of a positive formula gives the number of iterations needed to reach the least
fixed-point of$ the formula. As a corollary, we show that ev 9
l -definable
ery
query on `] is a member of the
F
complexity class
of queries computable in polylogarithmic time using a polynomial number of processors. This
result appears to be of independent interest and suggests the
pursuit of descriptive complexity in the context of finite set
theory.
2. Preliminaries
Let be a relational
vocabulary, a -ary relation sym $
bol not in , and "^
a first-order formula over
U the vocabulary !
. For every -structure # with uni-$
h
h
verse and every -ary h relation $ on , we write % $
for the -ary relation on
defined by and $ , that is,
$ '& & $ L h +* & B & /
% $
$-,
U
( #)(
U
$
The relation $ is a pre fixed-point of on # if % $
i
$ ; in a dual manner,
$ is a post fixed-point of on #
$
if $
# if$
i % % $$ . Finally, $ is a fixed-point
ofon
$
$ . It is well known that if
U
is
(which means that every occurrence of
positive in
in is within
B an$ even number of negation symbols),
$
then has both a$ least fixed-point ."/ #
U
and a greatest fixed-point 01/ # $ on # . As a matter of
fact, the least fixed-point ."/ #
is the intersection of all$
pre fixed-points, whereas the greatest fixed-point 02/ #
is the union of all post fixed-points. Moreover, they can
6
be computed via transfinite
$ % as
$#$ ev iterations,
!N7follows.
698
For
ery ordinal 3 +
, let .5/ 4 #
. / #
and
4
6
$ % 96 8
0/ 4 #
0 /
$ 4 and 0 / #
0 / #
4 $ N
such 6that ./ #
N 698
$
$#$
$ N
$
. Then . / #
. / #
4
. Furthermore,
there
is
an
ordinal
6
6 8
$
$ and hence . / #
. / #
. / # . The least such ordinal is called$ the closure
ordinal of on # and is denoted by$ / # . Note that
if # is a finite structure, then
h / # is a positive integer
less than or equal than ( ( .
Least fixed-point logic LFP is the extension of first-order
logic FOobtained
augmenting
$ syntax with a new
by
B the
formula
$ every positive
U ,for
in
first-order formula ; naturally, on
U
every structure # this
new
formula
is
interpreted
by the
$
least fixed-point . / # . It is well known that this syntax gives rise to a robust collection of queries on finite
structures. In particular, LFP-definable queries on finite
structures are closed under iterated and nested applications
of the least fixed-point operator for positive formulas (see
[Imm86, GS86]).
HL
is a firstA l 9 formula over the vocabulary order formula
such
that
all
occurrences
of
quantifiers
rq L n $
!m1 L n $
9 $ are of
l
the form
and
. We let
denote
the fragment of
that consists of the least fixed-points
9 $ of
positive l 9 formulas;
this
means
that
every
" l $ formula isof
the
form
, where
B $
U
9
is
a
l
formula
that
is
positive
in
the U
B ary relation symbol and$ has
as free variables.
U
9
Note that every
l -formula involves a single application of the least fixed-point operator, that is, it contains no
nested or iterated least fixed-points. Furthermore, no additional first-order
and second-order parameters are allowed
9 $
in
l -formulas.
In a series of papers,
9 $ Sazonov has studied definability
JI K LM$ in
l
a variant of
on the infinite structure
of
all hereditarily finite sets (see [Saz97] for a$ survey). Here,
9
l
we study uniform definability in
on the collection &^] '^
of?Ball
finite
structures
that
are
images
A
/ $ of the BIT .0
structures
under the
* 1
I K
isomorphism W ( X
.
In
particular,
we
investigate the
9 $
question: does
l
collapse to
on &^] ? In the
process of this investigation,
we also study uniform defin 9 $
ability in
on the classes d ^] of near finite ranks
l
and ^] of finite ranks that envelop &^] from above and
from below.
$#
3. Complexity-theoretic aspects of As
explained in the introduction, separating
from
&
^
]
on
is
literally
equivalent
to
separating
from
, an open problem in complexity theory. $ In
9
l
this section, we show that even the collapse of
to
on & ^] would yield important results in complexity
theory. Hence,
it is difficult to either refute or confirm that
9 $
l
collapses to
on &^] .
i
.
Proof (sketch):
over the
6G Let be a language in
.
We
will
show
that
the
language
alphabet
/8 L
D is in
* over
! ( $0 ! $ $ ( (
#"$
;
then
a
de-padding
argument
puts
! * $0 ! .$#$ $
Theorem 1 If
on & ^] , then
l 9 i
, which in turn implies that
in
. Here, stands for the
*
dual word of obtained by interchanging ’s and ’s. By
Immerman [Imm86] and Vardi&[Var82],
is ),a+ sentence
% 3
there
6('*
of
over
the
vocabulary
, where
%
is a unary relation symbol, that defines on the class
of ordered binary words
with . We can assume that
$#$ /$
with first-order
by the normal3
form theorem for
. Moreover, does not occur in ,
since by [DDLW98]
it is first-order definable from .
'*),+
Similarly, and
need not occur in , because
they
$
are first-order definable aswell.
We
turn
>
$
n n 8
U
into a l $ 9 formula , where is a
J.U
U
variable.
Replace each occurrence
o21$ relation
0/,L
o
1
o
o0 / -ary
% o0/J$
in o0 / Lby
, each positive occurrence
% o,/J$
n
in o0 / Lby
,
each
negative
occurrence
o0/ 3 o0/ $
U
n
8
by
in
o0/ by
o0/ n , and
n 8 $ each occurrence
.
Finally,
replace
subformulas
of
the
!m oQ$54
Jm U o L n $647 !m o%L n 8 $64
form
byoQ$54
,
and
subformulas
q
q U o7L n $648 q o[L n 8 $64
of the form
. Using the
by $:9 U !I K LM$
isomorphism W
, it is not difficult to
(
prove
on the construction
L5.byinduction
for every
of $ * that
(
we
have
that
$#$
'^?A 8 $ n n 8 >$ * R <;V $ , $0if and
=;M only
(
W
W
,,
if W
;V $
U
where
stands
for
the
natural
number
represented
in
binary by . Since is a l 9 formula, the hypothesis of the theorem implies that the least fixed-point of is definable by a first-order
$ <;Mon
'@?BA 8 formula
$0& ;M^ ] $#. $ It fol-L
lows
that
the
query
>
(
$
( (
* is first-order definable on &;>= . A result
in [BIS90] implies then that a suitable
$0 ! $ $ encoding of > is
! computable
in
#"$
H 8? @ L * . It turns out that
( (
(
* can serve as
this encoding. This completes the sketch of the proof
8$ that
. Since
A * , the
i
iB
space-hierarchy theorem implies that
.
CD
It should be noted that from
a result of Dawar and Hella
[DH95] it follows
that
if
on & `] , then
i
. The preceding Theorem 1 shows
that the separation of PTIME from PSPACE
9 $ can
be derived
from the weaker hypothesis that
on & `] .
l
i
4. Restricted E collapses to GF
As mentioned earlier, Dawar et al. [DDLW98] showed
HL
that there is a first-order formula over the vocabulary
that defines a linear order on the class d LM`$ ] of near finite
Jh
I
ranks,
that
such that
h
I is, structures of the form
i
k$ this, they considered the following l 9
i GTV U . n For
formula
!mQn L n $ rq L $ n L 8
L n
X
n $#$ $
and showed that its least fixed-point is definable by a firstorder formula on d `] . Observe that the occurrence of the
relation symbol in involves only the bound variables n
and . We now abstract from this observation and introduce
the following concept.
$
Definition 1 A l 9 formula is restricted
U
if every occurrence of the relation symbol involves only
bound variables of .
The main result of this section is that the least fixed-point
of every positive restricted l 9 formula is first-order definable on d ^] and, hence, on & `] as well. In fact, we
show that it is definable by a first-order formula of low syntactic complexity. The class of formulas is the smallest collection of formulas containing the l 9 formulas and
closed!mQunder
$ rqsn L $ disjunction, bounded quantifican L conjunction,
m n
tions
,
, and existential quantification .
The collection of formulas is defined dually
by allowq n
ing closure under universal quantification . We say that
a query > on a class is l -definable if it is definable on by a formula and by a formula.
$
B F
Theorem 2 Let be a restricted l 9 forU
mula that is positive in the -ary relation symbol . The
least fixed-point of is first-order definable on d `] . In
fact, it is l -definable on d `] .
The proof of the above theorem is inspired by the argument of Dawar et [DDLW98] showing
that the least fixed n >$
point of the l 9 formula is first-order definable
on d ^] . We need, however, to establish certain absoluteness properties of arbitrary l 9 formulas, as well as certain
structural properties of arbitrary restricted l 9 formulas that
will be used heavily in the sequel. We begin with a basic
definition from set theory (see [Bar75, page 34]).
/LLet
# and be two structures over the vocabulary . We say that is an end extension of # , and write
& L h
substructure
#
L &of and
L forh every
L % & i , if # is
a L
it is the case that
.
(
(
Lemma1B(Absoluteness
of l 9 formulas [Bar75, page 35])
$
9
If is
a
formula and #
l
& for
'U & & $ L h
i * & , then
every
we
have
that
#
(
,
U
* & B & U
if and only if (
,.
U
$
If
is a l 9 formula,
then the set of $
U
free indices of , denoted by , is the set of indices of
variables that are free in and appear in at least
8 $ one occur . For example, if rence
of
in
the formula
rq L $ 8 $
U $ H is
, then U
$ . Note
R that a
l 9 formula is restricted if and
only
if
. From
L
!h LM$
now on,
we
identify
structures
over
with
their
unih
verse . In the following two lemmas, we establish certain
important properties of l 9 and of restricted l 9 formulas on
near finite ranks.
$
HL
F
Lemma
2 Let
be a l 9 formula over
^
U
h
, where is a -ary relation symbol, and let
be a
I
h
I
near
finite
rank
such
that
.
For
every
" 3
& i
GTVU
4e$h i i h , and
every tuple & * ,& every
$[L Irelation
h *&
/
h TVU * & 4 , we
!I have & that
U
L ( $0.$ $- , ,.
(
$
if and only if
(
h (
In particular,
if is a restricted
l 9 formula,
then
& h
& 4[I
* $ , if and only if
,.
( * $
Proof : We proceed by induction on the construction of
. The only non-trivial case is when is of the form
Jm1 / L 1 $ h & .& In
L h this case, & ( L & 1 * $ , h if and
. if
only
* $ ,,
(
there is some such
that
and
E'&
& / & & / B & $
where
. Therefore, by
U
h U TV* & U (
$
,
induction& hypothesis,
if
and
L h
& L & 1
h only
if* there4
is some
such
that
and
(
$
L
JI $\ $
& 1)L I
.
Since
,
we
have
,
& 1
I(
& L & 1 TVU
that
L it is the$0 case
I i and,L hence,
$0 for
every
I that
( h +* & I & ( L $\ . Consequently,
(
$ ,
& L h
& L & 1
(
if and only
if
there
is
some
such
that
and
h * . 4 JI & / L $0.$
& / ( L $0.$ , , which means that
h ( +* & $ 42!I CD
(
$
, , as required.
(
The next lemma yields an absoluteness property of l 9
inductions on near finite ranks and also reveals that every
positive restricted l 9 formula has a unique fixed-point on
near finite ranks.
$
HL
Lemma
3 Let be a l 9 formula
over
^
U
that is positive in the -ary relation
symbol
h
I
h
I , and let
be " a near
finite
rank
such
that
3
i
i Jh $ GTV
4@U I . For
!
.
#
"
every
and
every
,
we
have
that
/
*
JI $
Jh $ 4 I + JI $
is
. /"
0 /"
and 0 / "
$
!h $ . If, inJh addition,
9
01/
a restricted l formula, then ."/
.
Proof (sketch): The first statement is proved by induction
on ! using absoluteness. For the second statement, if is
9
a positive
L that
Jh $ restricted
Jh $ l formula, then it can be shown
05/
.
/
by
induction
on
the
maximum
-rank
& i B &
'& & $ZL
Jh $
05/
of
, where
, and using
U
U
CD
Lemma 2.
We now
have
9 $ all the necessary
tools to show that restricted
l
collapses to
on the class d ^] of
near finite ranks.
Proof of Theorem 2: The
keyidea
the proof is that there
4 of
$
is a first-order formula Jh $
that approximates
the
h
U
1
0
/
greatest
fixed-point
on
near
finite
ranks
;
the
for4
Jh $
mula is based on the characterization of 0/
as the
union of all post fixed-points of . This approximation
can be$ improved to yield an exact first-order definition of
4
!h
05/
by first replacing every occurrence of in by ,
and then iterating a constant number of times. The result will then follow from Lemma 3, which asserts that the
greatest fixed-point of the restricted l 9 formula coincides
with its least fixed-point. This type of argument was used by
Dawar et al. [DDLW98]
n kto$ show that the least fixed-point of
the l 9 formula in the beginning of Section 4 is
first-order definable on d `] . Here, we have to deploy the
machinery of Lemmas 1, 2 and 3 to show that the argument
can be extended and applied to every restricted l 9 formula.
$
We will construct a formula
Jh $
U h L that defines
the greatest fixed-point 0 /
on every d `] such
I
h
I
that
for some
(for near finite$
h i
*
1
i I TVU
!h
ranks below 8 we can obtain a l 9 definition of 0 /
by iterating a sufficient number of times). The construction of is carried out in
n $ three steps. For nthe first step, define a l 9 formula "
expressing that is the encoding
h
on . We use the standard encodof a post fixed-point
of
HB n n
ing
of a pair by the set
, and of
B n $ a tuple
U
"
by$?8 o B o
.
Then
is the
n
rq U
U L n $ o o n $ $
formula $
,
n
U
U
n
where is a l 9 formula
expressing
that
is
a
set
of
encodings of -tuples,
from
and
o0/ $
o0/ by
o0/ replac o,/ is obtained
L n
ing atomich formulas
by & & .
Let $ $ L be such
U
thath the set $
(
&
i & is in ; it is not hard to show that
$
h U *
fixed-point of
( $ " h $ , if and only if $ is a4 post
$
$ on . For the second step, let
be the
Jm n $ n $ 8 B U L n $
formula
expressing
"
U
B $
that
belongs to some post fixed-point of .
U
4
Jh $ 4[I 8 !h $
Claim 1: 0 /
0 /
.
TVU i
i
is l 9 , Lemma 3 implies that
Proof of
!h $ 4pClaim
I 8 1: Since
JI 8 $
0 /
0 /
Thus,
first
Bthe
& $p
L
TVU & . g
'& for
V
T
U
inclusion
it
suffices
to
show
that
if
U
!I 8 $
h 4 *&
0 /
(
for the
8 a witness
$
TV!mQU n ,$ then4
, . We
!I need
quantifier
and
in . Put $
0 /
$ .
L TVI U
I 8
h
Since $
. Morei & L TVU , we have that & L i
h over,
since
,
we
have
that
;
hence,
$
L n $ * & h n $ *(
", . We can now show
(
"
",
that
I 8
h
using Lemma 1 and the fact that
.
For
the
TVUGJi h $ second inclusion, we use the fact
that 0 /
is the union
h
of all post fixed-points of on .
4
Thus, yields
fixed!h $ an approximation of the greatest
point 01/
of . For the third step, we iterate a number
of times (in fact, times)
to obtain
4 progressively
$ 4 better
$
approximations. Define 9 U
(
U
B 0 4,/ $
L
4,/ U
U , for
. Note that each is a formula.
4 /
!h $ 4 I 8 !h $
/
1
0
/
05/
, for every
Claim
2:
B
i
T<UT i
.
is l 9 , Lemma 3 implies that
Proof
$ 45Claim
/ $
Jh of
I 8 2: Since
JI 8 / 05/
05/
. We proceed by
V
T
#
U
T
.
TVU#T
induction on . Claim 1 takes care of
the
case
Assume that& Claim
in & for
$ L 1 J.h For
$ 4Othe
& 2holds
I first
/
8
clusion, if
0 /
,
h * & JU h $
& L I / 4p
h
V
T
U
T
8
then
(
0 /
, . Since
TVUT
and is a restricted
Lemma 2 implies that
h
* & / Jh $ 4 l 9 I formula,
8 /
(
0
, . Note that this is the
T
first place in this proof where we use the assumption that
the l 9 $ formula
By induction hypothesis,
4 /
Jh 4 I 8 / is restricted.
h * & 4 / ; therefore,
0 /
(
,,
U that h ( 04 / $ * U & ,
T i It follows
since is monotone.
h 4 / *&
U
(
, . The second inclusion can be proved usand
ing the induction hypothesis and the monotonicity of ; the
details are left to the reader.
4 Q
$
Finally, let be the formula 8
$ 2
U
J.h Claim
0 /
implies that defines the h greatest fixed-point
I
h
I of
on every near finite rank
such
that
F$
i
i GTVU
1 . Let for some
be the dual formula
* $
3
3 of . Note that
l 9 for$
Jh $ h is also !ah restricted
mula. Moreover,$ 0/ . /
(see
[Mos74]).
1
!h
Therefore, . /
is definable
$ taking the negation of
!h by
the $ formula that defines 0/ . The l $ definability
Jh
Jh
/ Jh $ of
follows immediately, since . /
by
. /
0
CD
Lemma 3. This completes the proof of Theorem 2.
4,/ and
//0
$ U (
Example 1: In the full paper we show that, in spite of the
stringent syntactic requirements imposed on restricted l 9
formulas,
several
9 $ natural queries can be expressed using rel
stricted
formulas. Thus, Theorem 2 provides a
versatile tool for showing that such queries are first-order
definable on d `] . Here, we illustrate this technique by
considering the query (finite) ordinal addition
>, where
$ L
h
Jh $ >
if
is
a
near
finite
rank,
then
>
3
h
- k
, , 3 are ordinals such that 3
. The stan(
dard recursive specification of ordinal addition does not lead
to a restricted l 9 formula. Consider, however,
the follow >
ing alternate recursive
specification:
if
are
ordinals,
3
- then 3
if and only if
$ 7 8
8 V
$(7
3 L!V
3
L
$
$
L
$
Jm
!m
Jm
!m" L $
3
3
!
8
8
- 3 &8
3
3 " - 8 " - $B
3 This recursive specification can easily be transformed into
the least fixed-point of a positive restricted l 9 formula that
defines ># ; to this end, we use the
fact that
an or being
- dinal is l 9 definable, and that
.
5. Unary E and binary E
$
l 9 denote
For every positive
, let -ary
$
integer
the fragment of
l 9 that allows the
formation
B of$ least
9
fixed-points
of
positive
l
formulas
such
U
that is a relation symbol of arity . The following simple observation reveals that the smallest of these fragments
collapses to
on d `] .
$
Proposition 1 If is a unary l 9 formula that is
positive
in
,
then
there
is
a unary restricted l 9 formula
$
$ that
is
positive
in
and such that . / #
$
. # . Consequently, unary
/ # 9 $ on every structure
l
collapses to
on d ^] .
$
Proof: Let
be the restricted l 9 formula obtained
$
$
by replacing every occurrence
of$
from Q
o
o by
, while preserving all occurrences
with a
bound variable. An easy induction shows that for every
$ structure
3 $ we have that .1/ 4 #
$ # and every
$ ordinal
/ # . Theorem 2 implies then
. . / 4 # ; thus
. / # 9 $
CD
that unary
collapses to
on d `] .
l
$
According toTheorem
1, if l 9 collapses to
on & `] , then
. The proof of this
theorem makes
a
crucial
use
of
the
hypothesis
that -ary
9 $
l
collapses to
for every
$
+ . In view
of
Proposition 1, one may investigate whether -ary
l 9
collapses to
on & `] for particular values of bigger than $ . The main result of this section is that binary
9
also collapses to
on & `] .
l
$
8 F
Theorem 3 Let be a l 9 formula that is posU
itive in the binary relation symbol . The least fixed-point
of and the greatest fixed-point of are first-order definable on &^] . In fact, the least fixed-point is -definable,
whereas the greatest fixed-point is -definable on & ^] .
Proof (sketch): For simplicity, we
9 $ only
sketch the proof for
l
the collapse of binary
to
on the smaller class
`] of finite ranks. After presenting the sketch, we outline
how the proof can be modified to establish the collapse on
the class &^] .
As in the proof of Theorem$ 2, the first key idea is that
JI
I
the greatest fixed-point 0 /
of on can be approximated by a 4 first-order
8 $ formula. In fact, we can start with the
formula
featured in that proof, because Claim
U
1 uses only the hypothesis that is a l 9 formula (and not
the additional
hypothesis
JI $ 4[
I 4in Theorem
JI 2$ that is restricted).
Thus 01/
it is not
JI i 05/ $ . Moreover,
I
4 i 05/
hard to verify that , because
is an actual
finite rank (instead of a near finite rank).
Our goal is to im!I $
prove
on this approximation
4,/
JI / $ of 0 4 / by defining
/ .formu/
las such
that
0
,
for
Since
T
8 $
may
not
be
a
restricted
formula,
a
difficulty
U
arises from/# the
in /!$of subformulas
/ oH$ presence
o6
1.$ potential
L2/of the
formo
,
, and
, where
and is4 / a bound variable of . Actually, in building the formulas , the
is caused by the subfor-o
$ serious
o difficulty
/ $
/ oHmost
is l 9 and
mulas
and
. Note that, since
o
I
, every element of
is a bound variable
of
I witnessing
/
must be in
choice
, the set
/ oQ$
o6 of
/!$
I U . Therefore, for every
of elements
of
witnessing
(or
)
is
a sub
I I
set of U and, hence, a member of . In turn, this makes
I
it possible to use first-order existential quantifiers over
o
to quantify the set of all such elements witnessing . Note
that this would not be possible, if we had to work with an
h
arbitrary near finite rank , as the above set may not be in
h
.
4/
4,/
We
will
build
the
desired
formula
from
4
4
U , for
9
,
where
we
take
to
be
.
Let
be
the
set
// . " For
each
containing
,
we
will
4 / 8$
k 8 $ i
define a formula
. Then each
will be
U
U
the formula
/ $ 8! / 8 $ 8
" 8 $
U
U
P
/ $
Jm o s$ !m o $ / L%o 8
where
is the formula
each
U
U
o<L[o $
of containing
UU
TVU and is the set of4/ subsets
8 $
subformulas of
. The first two
8
I U / are introduced
!
to ensure
that
and
belong
to
H6 / U n 1 1
T . Let be the
be a new variable for each
and let
set 8 L ! .
From
now
on,
we
use
the abbreviations
o o o 8 $
U 8 $ n n n n 8 n 8 $
9 9
9 9
. Let
8 " n $ $ U
" U $ ,
U !m U n $ " n , $ and
be the formula
" " and
1
1
where
n $B
1 1
/ / P / /G oH$ o n $#$B
P
n $
1
1
is the formula
n 1 1$ 1 1 q 9 L
oH$ o n $ $0
4 /
o n $
oQ$ 7 o6 n $
$
while
is the formula
6
o
n
U
where
Jm1 9 L n 1 1 $ o 1 !/ / P o / 8 o 8 U /"$B
U
P
!1
1
8
7
o 8 1 . $0
1 1 n 1 1
with 9 a fresh variable. Note that says that is
the set of witnesses
for bound variables that relate to
4
4 /
U or
to 8 . Since 9 is a formula, it is easy to see that each
is also a formula.
!I
!/ $ 4 /
B
Claim 3: 0 /
for
.
T
4
Proof
of Claim
, since 9
4
LC/3:/ The
# claim holds for
is . Fix
assume that the claim holds
; we show that itandholds
for
for . We show first that
4 / 1
JI / $
& '& & 8 $@L I 8
0 /
be such that
T . Let
U
i
4 /
&
& L
*
L that
* & definition of , it follows
I 8 / , . I From
" the
, and
with (
, for some . For
i
T L ! , let 1 1 @L I be a witness
every 1 1 8
for
the
quanti mQn U " fier
for
in L , and let be theI corresponding
n
* & & 8 witness
& 0$
, we have
(
,.
. Since that
U
& B$ It suffices to show that is a post fixed-point of I
on & ; then, the
of and Lemma 3 will
imply
/$
JI monotonicity
& 0$
that is in 0 /
. We need to verify that & 0$ $ T L I 8
I * & i
. Let
be such that
,.
( the three disjuncts
Then must satisfy one4,of
of
.
As
I
/ *
sume first that
(
4 / $3
/
Lemmas
JI 2 and
U , . Using
05/
and the induction hypothesis that T IU ,
4 / U
it can be verified that
is a post fixed-point of on .
U 4 / I * (4 / & 0$
, . But
Thus
and hence,
(
i
I U * U & 0$
, , as required. Asby monotonicity,
(
sume now
that
satisfies
the
second
disjunct.
8 L !
L 11
&
Then,
& 1 there
& 1 $
exist
and 1 9 1 such that
.
1 1 U
From the choice of and the definition of , we
I
& 1 & 1 & 0$
I
know that ( * , ; therefore,
(
B
$
&
+* , , as required again. The case of the third disjunct is handled in a similar manner.
4 / !I / $
. If
Consider next
& L
JI / $ the inclusion
I * & 0 / ! I T /!$ i
/
, then
.
Let
be
0 /
(
0
,
T
T
the biggest
subset
of
containing
and
such
that
for
every
8 L it is the case that I ( * & / & / 05/ JI / $ , .
& & 8 T .$#8
U
!I / $ 4 !I / Let $ be the set 01/
.
T 8 L T U
U I
(
By / Lemma
2,
for
every
we
have
that
1 1
& & / G
U
n 1 1 +* $ , . We construct witnesses for in
" & B$ such
that
$
,
which
will
prove
the
claim.
L !
1 1
Fix
1 1 U 8 & L and
$ L if 8U , then
I define
/
& & as1 follows:
, then
1 1 & L I T / U[( & 1 & $L $ ; if
$ . Using the induc T U^(
tion hypothesis
?1 1and the definition of $ , it can be checked
that the sets
have the aforementioned properties.
4 8$
This concludes the proof that the $ formula JI
8 U $
defines the greatest fixed-point 0/
of on
U
` ] . By
applying
the
same
argument
to
the
dual
formula
8 $
JI $
./
,
we
establish
that
the
least
fixed-point
$
U 8
of is definable on ^] .
U
Finally, we comment on the modifications needed to extend the proof to the class & `] . The first modification
is that we have to add an extra iteration in the construction of the first-order
8 $ formula that defines the greatest fixedpoint of on &^] . Specifically,
instead of four
4 (4 4 (4 4 U
8
formulas, we will need five formulas 9
;4 the
U
greatest fixed-point will be defined$ by the last formula .
9
l
In proving that binary
collapses to
on ^] ,
we used twice the assumption
that
we
were
working
with
I
structuresL of the form
for some *)+
, instead of struc
h
I
h
I
& `] with 5i
tures
for some
.
4 i
$
*
+
G
<
T
U
J
I
05/
.
This
can
The first time was to ensure that 4 8 $ be taken care of by considering the formula
I
(
4,/
U
/
/ 8 $ 8 4 8 $
/ $
.
Here,
is a formula
/U $
U
similar to the formula defined
in
the
proof,
inL
h whose
SI /
/
&
^
tended meaning is that for every
/ ] $
I
h
I
a
such that
. / For$ the rest of the proof,
[i
TVU should be used in place of
.
h I
The second time we used the assumption that
for some * +
that the existentially quantito n ensure
n 9 was
9 8 can
fied variables
h be witnessed by elements of
U
these
the universe of the structure . As mentioned earlier,
h L
d
`
]
witnesses may noth exist
within
an
arbitrary
, or
L
even an arbitrary
&
^
]
.
When
showing
that
binary
9 $
l
collapses to
on & `] , this difficulty is only
encountered in4 the
last
iteration,
that is, in the correctness
of the formula . This obstacle, however, can be overcome
by using the fact that the witnesses can be restricted to be
8
subsets of the transitive closure of the arguments
,
&
U
where the transitive
closure
of
a
set
is
defined
inductively
& $ & N B$ fL & . The reaas follows:
(
son is that, since is a l 9 formula, the interpretation of
every bound variable of can be restricted to the transitive closure of the sets that interpret the free variables
/1
of . Consequently, we may split
de / 1 the
08/ 1 witnesses
/ 1
fined in the proof into three sets U
with the fol / 1 / 1 4 & 08/ 1 / 1 4 & 8
lowing
U
, and
/ 1 interpretations:
/ 1 4 # & $
'& 8 $#$ U ,&
& 8 $$
. Observe
U
U
L h1
/ 1 08/ 1 L h
h L
& &
& `] and U 8
that if
, then U
,
h
&
'& $ 3 is closed
because
under
taking
subsets
(if
,
then
/1 L h i
B$
W U
W U ); observe also that
, because
its
&
rank is less
than
the
maximum
of
the
ranks
of
and
. The
08/ 1
/1
/1
U
sets
,
,
and
capture
all
relevant
information
about
/1
; therefore, only these elements need to be existentially
quantified. Clearly, several changes in the formulas have to
be made; the technical details will be included in the full
CD
version of the paper.
$
8
The inquisitive reader may wonder whether the argument
of Theorem 3 extends to higher arities. The answer is that
it does not, for the reason that we cannot encode binary reI
I
lations on
it is possible
U by elements of I , whereas
I
to encode unary relations (sets) on
by elements of .
U
We note, however, that Theorem 3 extends to binary l 9 forL
mulas over an expanded vocabulary that, in addition to ,
contains other relation symbols, as long as they are interpreted by l -definable queries on & ^] .
Example 2: The aforementioned extension of Theorem 3
can be used to show that the Even Cardinality query
>
Jh $ & L h
&
( the cardinality of is even
is first-order definable
L write
`
n on$ & ^] . For this, oneHcan
a l 9 formula over the vocabulary
,
n
whose least fixed-point defines the binary query “ is an
even element of with respect to the linear order ” on
&^] . Here, is interpreted by the l -definable linear order on & `] described in the beginning of Section 4.
6. Closure functions of formulas
B $
Let be an arbitrary positive first-order
U
formula. From the preliminaries,
recall that if # " is a finite
$
structure, then / #
is the smallest integer
that
. # $ N 8 . # $ . In general, the rate ofsuch
growth
/
/
$
of / #
can be as high as a polynomial in the cardinality
of the universe of # . Here, we analyze the rate of growth
of the closure
on finite
of positive l 9 formulas
JI LM$ function
is a positive
ranks
,* +
.JI We
establish
that
if
$
l 9 formula, then I / I is bounded by a polylogarithm
of the cardinality ( ( of . Moreover,$ we show that if !I
is a restricted l 9 formula,
then 2/
I
is bounded by the
iterated logarithm of ( ( .
$
B Theorem 4 Let be a l 9 formula that is
U
positive in the -ary
relation
symbol
. Then, for all suffi
ciently large *
, we have
8 I 3
!I $ 3
I $0
V
T
U
2 /
(
( U
#"$
( (
*
U
J I $ 3
l 9 formula, then 2 / Moreover,
3O - if I is a$ restricted
#"$
( ( .
*
Proof (Sketch): We outline the proof of the first statement only;
an easier proof.
JI $ 3 the second
!I $ has
-Y 8
I We show that
2 /
2 /
V
T
U
(
U
U ( U holds for all
8 by expanding
. The result will follow
3 recurrence
*e+
I this
I $
8
V
T
U
U
and using the fact that
(
(
"$
I ( (
*
U
TVI U ( U ( U ,
for all sufficiently
large * . Put !
"
on
and let
be the closure ordinal
of
!I $
T " JI U $. . ItSo,is
.
enough to show that L . / T " TVU
/
&
!I $ i
let us assume
I
* & thatJI $ . / T " TVU , which means
I that
(
. / T "
, . Lemma 2 implies
that
& JI $ 4 !I (
& B &1 .$
+* . / T "
that
Jclaim
I $
JI $ 4f JI U& U & $ . , .T We
. / T "
"
U
.
This
U
U
i /
will prove I our goal, & because
the
monotonicity
of & im
$
L
!
I
plies that$
. T " JI . ( * . / T " U , and, therefore,
/
ofJI " $ , it is easy to show
Using Lemma
the choice
JI $ 4e3I and
that . / T " . T " U
. So, to prove
!I $ the
4
U i / those tuples
claim, it remains to consider
in
.
T
"
& & .$ JI !I / $ 4 I that are not in . / JT I " $ 4ZJI U .
U
U For each + , let $ be the set
T
U 3
& & .$
JI $ 4 I .. /First
. / T note that ( $ " (
1
U
B
.
$
!I
&
&1
I U
(
1 U ( . A simple Jcounting
U
U
I the cardinality of the set
argument
& & shows
.$ that
I U
is
U
1
U
/
/
I 3 I (
(
(
(
/
/
U
U U !
U
U
3
Therefore ( $ " (
JI $
! . Now let + be the smallest integer
such that $
because
$ TVU . Such an exists,
.
T
/
JI $
eventually reaches the fixed-point ."/
R
9 . If + ! , then
a $ U a
a
the sequence of proper inclusions a $
"
$ " holds.
(
$
(
!
Hence
,
which
contradicts
the fact that
3
( $ "9(
!
!
. Thus, must
hold, from
which
it
JI $ CD follows that
$ "
$ TVU
$ $ " U
. / T " U
i
.
In the full paper, we provide examples showing that the
9
above bounds are tight.
n >$ As regards restricted l formulas,
the formula
for the linear order is such an example.
We conclude by pointing$ out that the above Theorem 4
9
implies
that every
l -definable query on `] is in
, the parallel complexity class consisting of all queries
computable in polylogarithmic time using a polynomial
number of processors (see [Pap94, GHR95, BDG90] for a
thorough coverage of
). This suggests the systematic
pursuit of descriptive complexity in the context of finite set
theory.
Acknowledgment: We are grateful to Moshe Y. Vardi for
his constructive criticism and comments on an earlier draft
of this paper.
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