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A Note on Expressive Power of Fixed-Point Logics
GABOR SAGI
Renyi Institute of Mathematics
Budapest H-1053 Realtanoda u. 13-15
HUNGARY
Abstract: Fix-Point logics play an important role in theoretical computer science, particularly in descriptive
complexity theory. We investigate the expressive power FO(PFP) of First Order Logic endowed by Partial
Fixed-Point Operations. We will show that on finite, unordered structures the expressive power of this logic is
not the same as that of second order existential logic. In addition, we prove that the class of finite, unordered
structures whose universe is of even cardinality cannot be axiomatized in FO(PFP), even if our language
contains relation symbols with arbitrarily large arities. This result is a sharpening of Proposition 8.4.4 of [2].
Key-Words: Fixed-Point logics, second order logics, expressive power, cylindric algebras.
1 Introduction
Fixed-Point Logics play an important role in
theoretical computer science and in descriptive
complexity theory because complexity classes can
be characterized in terms of the expressive power of
these logics. Such characterization results say that a
set K of finite, ordered structures can be
axiomatized in a certain Fixed-Point Logic if and
only if the membership problem of K belongs to a
certain complexity class (like PTIME, PSPACE,
etc.).
However, it is also interesting to investigate the
expressive power of these logics on unordered
structures. Let FO(PFP) denote First Order Logic
extended with Partial Fixed-Point operations (this
logic will be recalled later). Below, in Corollary 5
we show that on finite, unordered structures the
expressive power of FO(PFP) is not equal with that
of second order existential logic (cf. Corollary 7.5.9
of [2]). We also note that by Proposition 8.4.4 of [2]
for a relational vocabulary containing unary relation
symbols only, the expressive power of FO(PFP) is
equal with that of usual first order logic. Therefore,
if our vocabulary contains unary relation symbols
only, then the class Keven of all finite structures
whose universe is of even cardinality cannot be
axiomatized in FO(PFP). As a strengthening of this
result, in Corollary 5 we show that Keven cannot be
axiomatized in FO(PFP) using any vocabulary
(containing relation symbols with arbitrarily large
arities).
To prove the above, we will use the theory of
Cylindric Set Algebras. Such an algebra can be
regarded as the “meaning-algebra” of a given first
order model. That is, a cylindric set algebra
describes the complete algebraic structure of
definable k-ary relations of the model (for a fixed
natural number k). All the required details will be
recalled below. For more on cylindric algebras we
refer to [3] and [4].
Our another tool will be the theory of random
structures, essentially in the form presented in
Section 4 of [2]. The idea behind our results is that
finite dimensional cylindric set algebras of random
structures are isomorphic with each other with
asymptotic probability 1 and these cylindric set
algebras determine the action of the fixed-point
operations as well. The main result of the paper is
formulated in Corollary 5.
Throughout ω denotes the set of natural numbers
and for every n in ω we have n = {0,1,...,n-1}. Let
A and B be sets. Then AB denotes the set of
functions from A to B. In addition, P(A) denotes the
power set of A, that is, P(A) consists of all subsets
of A.
2 Construction and Proof
We start by recalling the notion of cylindric set
algebras.
Definition. Let U be any set and let k  ω. The full
cylindric set algebra of dimension k with base set U
is the algebra:
 P(kU); , -, CiU, Di,jU i,j  k
where  and - are set theoretical intersection and
complementation (w.r.t. kU), respectively,
Di,jU = { s  kU: si=sj}
and CiU is defined as follows. For every X  P(kU)
one has
CiU(X) = {s  kU: ( z  X) ( j  k )j i  sj=zj}.
The class Csk of cylindric set algebras of dimension
k is defined to be the class of (isomorphic copies of)
all subalgebras of full cylindric set algebras.
Thus, a cylindric set algebra is a Boolean algebra
with further operations. In the above definition, the
diagonal elements Di,j correspond to atomic
formulas vi=vj while the “cylindrifications” Ci
correspond to existential quantifiers “vi”. Roughly
speaking, a cylindric set algebra is the algebra of
first order definable relations of a given structure.
More precisely, cylindric set algebras can be
naturally associated to relational structures as
follows. Let A be a structure and let k  ω be fixed.
Suppose φ(v0,...,vk-1) is a first order formula whose
variables are belong to { v0,...,vk-1}. Now let
|| φ ||A = {s  kA: A╞ φ [s]}.
Consider the set
D = { || φ ||A: φ is a first order formula containing
variables at most v0,...,vk-1}.
This is the collection of relations of A first order
definable by k variables. D can be extended to a
cylindric set algebra by stipulating that
Di,j
= || vi = vj||A,
A
- || φ ||
= || ┐φ||A,
|| φ ||A  ||ψ ||A = || φ & ψ ||A,
Ci || φ ||A
= ||  vi φ ||A.
This cylindric set algebra will be denoted by Csk(A).
Clearly, Csk(A) can be generated by the set { ||R||A:
R is an atomic formula in the language of A}.
There is some similarity between cylindric set
algebras and the notion of ”s-invariants” in [2]
(Section 3.3.2). However, in our opinion, cylindric
set algebras are much more natural and traditional.
For more information on cylindric algebras and
connection with logic we refer again to [3] and [4].
Throughout k denotes a natural number.
Theorem 1. Suppose A is a relational structure such
that Csk(A) is finite. Then there is a first order
formula φk such that A ╞ φk and for any structure B
one has
B╞ φk  Csk(A)  Csk(B).
Proof. We will construct a kind of diagram of
Csk(A). For any a  Csk(A) let φa be a first order
formula such that a = || φa} ||A and if a = || R ||A for
some atomic formula R then choose φa = R. φk is
defined to be the conjunction of the following
formulas:
• ┐(v0 ... vk-1 φa(v0,...,vk-1)  φb(v0,...,vk-1)) for
every pair of different elements a,b  Csk(A),
• v0...vk-1( ┐φa(v0,...,vk-1)  φb(v0,...,vk-1)) for
every pair of elements a,b  Csk(A) satisfying -a=b,
• v0...vk-1 (vi φa(v0,...,vk-1)  φb(v0,...,vk-1))
for every pair of elements a,b  Csk(A) satisfying
Ci(a) = b,
• v0...vk-1 ( φa(v0,...,vk-1) & φb(v0,...,vk-1)
φc(v0,...,vk-1))
for every elements a,b,c  Csk(A) satisfying
a  b=c.
Since Csk(A) is finite, the above conjunction is
finite, so φk exists.
Clearly, A╞ φk. Now let B be any structure with
B ╞ φk. For every a  Csk(A) let f(a) = || φa ||B. It is
routine to check the following:
• f is injective because of the first clause of the
definition of φk
• f is surjective because the range of f contains the
set { || R ||^B: R is an atomic formula in the
language of B} generating Csk(B),
• f preserves the cylindric operations by the last 3
clauses of the definition of φk.
Thus, f is an isomorphism between Csk(A) and
Csk(B).
Below we will use relatively standard methods from
the theory of random structures. Throughout Gn
denotes the random model, in which the truth of
atomic formulas on tuples are selected
independently, each of them with probability 1/2. In
more detail, such a random structure on n (labelled)
elements can be “constructed” as follows. For each
k-ary relation symbol R and k-tuple a let us
associate a random variable ξR,a taking values 0 and
1 with probabilities 1/2. We assume that these
random variables are completely independent from
each other. Observe the values of these random
variables, and declare A ╞ R(a) iff ξR, a=1. The
resulting structure A is called a random structure.
For convenience, we assume that the universes of
our random structures are of the form n ={ 0,1,2,... }
for some ω  n.
So for fixed n, our probability space is the set of
all structures on n (labelled) elements (in other
words, elementary events are particular structures
on n elements).
If φ is a first order sentence in the appropriate
language, then Pr(Gn ╞ φ) denotes the probability,
that an above described randomly chosen structure
(on n elements) is a model of φ.
By a classical result there is a structure  such
that a random structure on countably infinite
elements is isomorphic to  with probability 1.  is
called “the random structure” and can be
characterized as the “universal countable structure”.
Among others, it has the following remarkably nice
properties.
The first order theory of  can be axiomatized
by a very concrete set  of axioms, such a  is
explicitly given in [2], Example 3.2.11. Here we
don't recall the concrete form of these axioms
because we need only some properties of them.
Namely,  forms a complete theory and, in addition,
the asymptotic probability of each element of  is 1,
that is, if  then
limn  Pr (Gn ╞ ) = 1.
This can be found in Lemma 4.1.2 of [2]. Moreover,
 is an ω-categorical structure which has quantifier
elimination (this can be proved similarly to the
remark after Lemma 6.4.3 of [5], page 177). For
more details on random graphs and random
structures we refer to [1] and [2].
Let us fix an arbitrary relational vocabulary
throughout the paper. The next theorem says that the
cylindric set algebras of the countable random
structure  “can be approximated” by cylindric set
algebras of finite structures, whatever the fixed
vocabulary is.
Theorem 2. For every k   for every large enough
n   there is a finite structure Hk,n such that Hk,n
has exactly n elements and Csk(Hk,n)  Csk( ).
Proof. As we mentioned before, the first order
theory of  has quantifier elimination. Therefore
elements of Csk( ) can be identified by quantifier
free formulas having at most k many variables.
Since up to logical equivalence there are finitely
many such formulas, Csk() is finite. By Theorem
1, there is a formula φk such that for any structure B
we have B╞ φk  Csk(B)  Csk() and  ╞ φk.
Recall that  is a complete theory and ╞  so 
╞φk. By compactness, there is a finite 0   such
that 0 ╞ φk. By Lemma 4.1.2 of [2] the asymptotic
probabilities of elements of  are 1. Since 0 is
finite, this yields
limn  Pr(Gn ╞ 0 ) = 1.
It follows that there is a natural number N such that
for every n > N one has Pr(Gn ╞ 0) > 0. Therefore,
for each n > N there exists a structure Hk,n such that
Hk,n has exactly n elements and Hk,n╞ 0. But then,
by construction, Hk,n ╞ φk so by Theorem 1,
Csk(Hk,n)  Csk( ), as desired.
By Theorem 2 for each k   there are structures
Fk and Gk such that Csk(Fk)  Csk()  Csk(Gk) and
in addition, the number of elements of Fk is even
while the number of elements of Gk is odd. Let K0 =
{Fk: k } and let K1 = {Gk: k  }.
Definition. Let L be any logic (one can think to an
extension of usual first order logic) and let C0 and
C1 be classes of structures. We say that C0 can be
separated from C1 in L iff there is a formula  of L
such that C0╞  and C1╞ ┐.
Note that the above notion of separation is not
symmetric because L need not be closed under
negation of formulas.
Theorem 3. K0 can be separated from K1 in second
order existential logic.
Proof. Let Φ be a second order existential formula
expressing that “there is an equivalence relation
such that every equivalence class has exactly 2
elements”. Clearly, K0 ╞ Φ and K1╞ ┐Φ.
Now we turn to show that K0 and K1 cannot be
separated by formulas of certain fixed-point logics.
Below we recall from [2] the notion of Partial
Fixed-Point Logic FO(PFP) which is an extension
of First Order Logic. We start by semantics. Let A
be a fixed structure. Suppose φ(R) is a formula in
which R is a relation variable. One can associate an
operation Fφ to φ by stipulating
Fφ(R) = {s  kA: A╞ φ(R) [s]}.
Consider the sequence
(*) , Fφ(), Fφ(Fφ()), ...
and let fpR(φ) be the first element of the sequence
which is equal with its successor (let fpR(φ) =  if
there are no such term in the above sequence). Now
the formulas of FO(PFP) are those of first order
logic together with stipulating that if φ is a formula
and R is a relation variable then FPR(φ) is also a
formula. The meaning (or interpretation) of such a
formula in a given model A is
|| FPR(φ) ||A = fpR( || φ ||A).
Observe that the terms of the sequence (*) are
always belong to Csk(A) if φ contains at most k
many individual variables. To prove this, it is
enough to show
(**) if x  Csk(A) then Fφ(x)  Csk(A).
This can be done by induction on the number l of
fixed-point operations in φ. If φ doesn't contain
fixed-point operations then Fφ(x) is the result of a
term function of Csk(A) (this term function can be
obtained from φ by a straightforward way). So in
this case (**) holds. Now suppose (**) holds for φ
which contains l many fixed-point operations. Then,
by induction, every member of (*) belongs to
Csk(A) so || fpR(φ) ||A  Csk(A) as well. Since
Csk(A) is closed under ┐, & and under
cylindrifications, it follows that (**) holds for any
formula with at most l+1 many fixed-point
operations. This completes the induction.
Thus, we have proved that for each formula φ of
FO(PFP) every member of (*) belongs to Csk(A). It
follows that for each formula φ of FO(PFP) and for
each finite structure A there is a first order formula
t(φ) such that || φ ||A = || t(φ) ||A. Of course here t(φ)
depends on A as well. Suppose φ contains at most k
many individual variables only. Then one can show
by a similar easy induction on the complexity of φ
that if ||φ||A = || t(φ) ) ||A and Csk(A)  Csk(B) then
||φ||B = || t(φ) ||B. Particularly, if Csk( A)  Csk(B)
and A ╞ φ then B ╞ φ.
Theorem 4. K0 cannot be separated from K1 in
FO(PFP).
Proof. Suppose φ is a formula of FO(PFP) such that
K0 ╞ φ. Let k be bigger than the number of different
individual variables occurring in φ. Then on one
hand, Fk╞ φ. On the other hand, Csk(Fk)  Csk() 
Csk(Gk) so Gk ╞ φ. Thus φ doesn't separate K0 from
K1. This completes the proof.
Now we can prove the main result of the paper. As
we mentioned, it is related to Corollary 7.5.9 of [2]
and generalizes Proposition 8.4.4 of [2] to
vocabularies containing relation symbols with
arbitrarily large arities.
Corollary 5. (1) The expressive power of FO(PFP)
on finite unordered structures is not the same as that
of second order existential logic.
(2) The class Keven of all finite structures whose
universe is of even cardinality cannot be
axiomatized in FO(PFP) for any relational
vocabulary.
Proof. Immediate from Theorems 3 and 4 because
Keven separates K0 from K1.
Acknowledgment.
Research supported by
Hungarian National Foundation for Scientific
Research grants D042177 and T035192.
References:
[1] B. Bollobás, Random Graphs, Academic Press
Inc, London, (1985).
[2] H.D. Ebbinghaus, J. Flum, Finite Model Theory,
Springer-Verlag, (1999).
[3] L. Henkin, J. D. Monk, A. Tarski, Cylindric
Algebras Part 1, North--Holland, Amsterdam
(1971).
[4] L. Henkin, J. D. Monk, A. Tarski, Cylindric
Algebras Part 2, North--Holland, Amsterdam
(1985).
[5] W. Hodges, A shorter model theory, Cambridge
University Press, (1997).