GroheSchweikardt LICS04.Ps

7/30/2019 GroheSchweikardt LICS04.Ps

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The succinctness of first-order logic on linear orders

MARTIN GROHE and NICOLE SCHWEIKARDT

Institut fur Informatik, Humboldt-Universitat Berlin

Unter den Linden 6, D-10099 Berlin, Germany

Email: {grohe|schweika}@informatik.hu-berlin.de

Abstract

Succinctness is a natural measure for comparing the

strength of different logics. Intuitively, a logic L1 is more

succinct than another logic L2 if all properties that can be

expressed in L2 can be expressed in L1 by formulas of (ap-proximately) the same size, but some properties can be ex-

pressed in L1 by (significantly) smaller formulas.

We study the succinctness of logics on linear orders that

have the same expressive power as first-order logic. Our

first theorem is concerned with the finite variable fragments

of first-order logic. We prove that:

(i) Up to a polynomial factor, the 2- and the 3-variable

fragments of first-order logic on linear orders have the

same succinctness.

(ii) The 4-variable fragment is exponentially more suc-cinct than the 3-variable fragment.

Our second main result compares the succinctness of

first-order logic on linear orders with that of monadic

second-order logic. We prove that the fragment of monadic

second-orderlogic that has the same expressiveness as first-

order logic on linear orders is non-elementarily more suc-

cinct than first-order logic.

1. Introduction

It is one of the fundamental themes of logic in computer

science to study and compare the strength of various logics.Maybe the most natural measure of strength is the expres-

sive powerof a logic. By now, researchers from finite model

theory, but also from more application driven areas such as

database theory and automated verification, have developed

a rich toolkit that has led to a good understanding of the

expressive power of the fundamental logics (e.g. [3, 10]).

Parts of this work were done while the second author was supported

by a fellowship within the Postdoc-Programme of the German Academic

Exchange Service (DAAD).

It should also be said that there are clear limits to the un-

derstanding of expressive power, which are often linked to

open problems in complexity theory.

In several interesting situations, however, one encoun-

ters different logics of the same expressive power. As an

example, let us consider node selecting query languages

for XML-documents. Here the natural deductive query

language monadic datalog [7] and various automata based

query languages [12, 13, 6] have the same expressive

power as monadic second-order logic. XML-documents are

usually modeled by labeled trees. Logics on trees and and

strings also play an important role in automated verifica-

tion. Of the logics studied in the context of verification, the

modal -calculus is another logic that has the same expres-

sive power as monadic second-order logic on ranked trees

and strings, and linear time temporal logic LTL has the same

expressive power as first-order logic on strings [11].

Succinctness is a natural measure for comparing thestrength of logics that have the same expressive power. In-

tuitively, a logic L1 is more succinct than another logic L2 if

all properties that can be expressed in L2 can be expressed

in L1 by formulas of (approximately) the same size, but

some properties can be expressed in L1 by (significantly)

smaller formulas.

For both expressivenessand succinctnessthere is a trade-

off between the strength of a logic and the complexity of

evaluating formulas of the logic. The difference lies in the

way the complexity is measured. Expressiveness is related

to data complexity, which only takes into account the size

of the structure in which the formula has to be evaluated,

whereas succinctness is related to the combined complexity,

which takes into account both the size of the formula and

structure [15].

Succinctness has received surprisingly little attention so

far; a few scattered results are [16, 2, 1, 4]. In [8], we started

a more systematic investigation. Specifically, we studied the

succinctness of various logics on trees that all have the same

expressive power as monadic second-order logic. While we

were able to gain a reasonable picture of the succinctness

of these logics, it also became clear that we are far from a

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thorough understanding of succinctness. In particular, very

few techniques for proving lower bounds are available.

Indeed, most of the lower bound proofs use automata

theoretic arguments, often combined with a clever encoding

of large natural numbers that goes back to Stockmeyer [14].

In [8], these techniques were also combined with complex-

ity theoretic reductions to prove lower bounds on succinct-ness under certain complexity theoretic assumptions. Wilke

[16] used refined automata theoretic arguments to prove that

CTL+ is exponentially more succinct than CTL. Adler and

Immerman [1] were able to improve Wilkes lower bound

slightly, but what is more interesting is that they introduced

games for establishing lower bounds on succinctness. These

games vaguely resemble Ehrenfeucht-Frasse games, which

are probably the most important tools for establishing inex-

pressibility results.

In this paper, we study the succinctness of logics on lin-

ear orders (without any additional structure) that have the

same expressive power as first-order logic. In particular,we consider finite variable fragments of first-order logic. It

is known and easy to see that even the 2-variable fragment

has the same expressive power as full first-order logic on

linear orders (with respect to Boolean and unary queries).

We prove the following theorem:

Theorem 1.1. (i) Up to a polynomial factor, the 2 and the

3-variable fragments of first-order logic on linear or-

ders have the same succinctness.

(ii) The 4-variable fragment of first-order logic on lin-

ear orders is exponentially more succinct than the 3-

variable fragment.

For the sake of completeness, let us also mention that full

first-order logic is at most exponentially more succinct than

the 3-variable fragment. It remains an open problem if an

exponential gap between full first-order logic and the 4-

variable fragment is needed, too.

Of course the main result here is the exponential gap in

Theorem 1.1 (ii), but it should be noted that (i) is also by

no means obvious. The theorem may seem very technical

and not very impressive at first sight, but we believe that

to gain a deeper understanding of the issue of succinctness

it is of fundamental importance to master basic problems

such as those we consider here first (similar, maybe, to ba-

sic inexpressibility results such as the inability of first-order

logic to express that a linear order has even length). The

main technical result behind both parts of the theorem is that

a 3-variable first-order formula stating that a linear order

has length m must have size at least 12

m. Our technique

for proving this result originated in the Adler-Immerman

games, even though later it turned out that the proofs are

clearer if the reference to the game is dropped.

There is another reason the gap in succinctness between

the 3- and 4-variable fragments is interesting: It is a long

standing open problem in finite model theory if, for k 3,the k variable fragment of first-order logic is strictly less

expressive than the (k+1)-variable fragment on the classof all ordered finite structures. This question is still open

for all k 3. Our result (ii) at least shows that there areproperties that require exponentially larger 3-variable than

4-variable formulas.Succinctness as a measure for comparing the strength

of logics is not restricted to logics of the same expressive

power. Even if a logic L1 is more expressive than a logic

L2, it is interesting to know whether those properties that

can be expressed in both L1 and L2 can be expressed more

succinctly in one of the logics. Sometimes, this may even

be more important than the fact that some esoteric property

is expressible in L1, but not L2. We compare first-order

logic with the more expressive monadic second-order logic

and prove:

Theorem 1.2. The fragment of monadic second-order logic

that has the same expressiveness as first-order logic on

linear orders is non-elementarily more succinct than first-

order logic.

The paper is organized as follows: After the Preliminar-

ies, in Section 3 we prove the main technical result behind

Theorem 1.1. In Sections 4 and 5 we formally state and

prove the two parts of the theorem. Finally, Section 6 is

devoted to Theorem 1.2.

Due to space limitations we had to defer some of the

proofs of our results to the full version of this paper [9].

2. Preliminaries

We write N for the set of non-negative integers.

We assume that the reader is familiar with first-order

logic FO (cf., e.g., the textbooks [3, 10]). For a natural num-

ber k we write FOk to denote the k-variable fragment of

FO. The three variables available in FO3 will always be de-

noted x, y, and z. We write FO3(


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: {x,y,z} UA (i.e., is an assignment in A). Fora variable v {x,y,z} and an element a UA we write[a

v] to denote the assignment that maps v to a and that

coincides with on all other variables. If A is a set of in-

terpretations and is an FO3(


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Now we are ready to formally state our main lower bound

theorem on the size of FO3(


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For every node v of T the following is true, where is aminimal separator for il(v):

(a) Ifv is a leafthen w() 1.

(b) Ifv has 2 children v1 andv2, and i is a minimal sepa-

rator for il(vi), fori

{1, 2

}, then

w() w(1) + w(2) .

(c) Ifv has exactly one childv1, and1 is a minimal sepa-

rator for il(v1), then w() w(1) + 2 .

The proof of Lemma 3.8 is given in Section 3.3 below.

For a binary tree T we write ||T || to denote the number ofnodes ofT. For the proof of Theorem 3.5 we also need thefollowing easy observation.

Lemma 3.9. LetTbe a finite binary tree where each node vis equipped with a weight w(v) 0 such that the followingis true:

(a) Ifv is a leaf then w(v) 1.

(b) Ifv has 2 children v1 andv2, then

w(v) w(v1) + w(v2).

(c) Ifv has exactly one childv1, then w(v) w(v1) + 2.

Then, ||T || 12 w(r) , where r is the root ofT. Using Lemma 3.8 and 3.9, we are ready for the

Proof of Theorem 3.5:

We are given an FO3(


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(2)b: 1`

{ub, u

b}

MIN

|diff a(ub), a(u

b)

| , |diff b(ub), b(u

b)

|

and diff a(ub), a(u

b)

= diff b(ub), b(u

b)

.

In what follows we will prove a series of claims which en-

sure that is a separator for I, J . We start withClaim 1. If there is a b UB such that Ia, Jb is sepa-rated by 1({ub, ub}) with z {ub, ub}, then is a separa-tor forI, J . Proof: By definition of we have ({ub, ub}) 1({ub, ub}). Therefore (1)b and (2)b imply that is a sep-arator for Ia, Jb, as well as for I, J .This completes the proof of Claim 1.

Due to Claim 1 it henceforth suffices to assume that for no

b UB, Ia, Jb is separated by 1({ub, ub}) with z {ub, ub}. I.e., we assume that for every b UB, Ia, Jbis separated by 1({min, z}), 1({z, max}), 1({x, z}), or1({y, z}).Claim 2. If a = (u) for some u {min,max, x , y}, thenis a separator forI, J . Proof: Choose b := (u). Therefore, a(z) = a = (u) =a(u) and b(z) = b = (u) = b(u).

We know that Ia, Jb is separated by 1({z, u}), forsome u {min,max, x , y}. Furthermore, since a(z) =a(u) and b(z) = b(u), we have u

= u.By definition of we know that ({u, u})

1({z, u}). Therefore, is a separator for Ia, Jb as wellas for I, J .This completes the proof of Claim 2.

Due to Claim 2 it henceforth suffices to assume that, a =(u), for all u {min,max, x , y}.Claim 3. If 1({min, z}) diff(a,minA), then is aseparator forI, J . Proof: We distinguish between two cases.

Case 1: diff(maxB,minB) diff(a,minA).In this case we can choose b UB with diff(b,minB) =diff(a,minA) (simply via b := a). Obviously, Ia, Jbis not separated by 1({min, z}). However, we knowthat Ia, Jb is separated by 1({z, u}), for some u

{x,y,max

}.

Since diff(a,minA

) + 1({z, u}) 1({min, z}) +1({z, u}) ({min, u}), it is straightforward to see thatIa, Jb, and also I, J , is separated by ({min, u}). I.e.,is a separator for I, J .Case 2: diff(maxB,minB) < diff(a,minA).Since ({min,max}) 1({min, z}) diff(a,minA) >diff(maxB,minB), we know that I, J is separated by({min,max}). I.e., is a separator for I, J .This completes the proof of Claim 3.

By symmetry we also obtain the following

Claim 4. If 1({z, max}) diff(maxA, a), then is aseparator forI, J . In a similar way, we can also show the following

Claim 5. If 1({x, z}) |diff((x), a)|, then is a sepa-rator forI, J .

Proof: We distinguish between three cases.

Case 1: There is a b UB such that diff((x), b) =diff((x), a).Obviously, Ia, Jb is not separated by 1({x, z}). How-ever, we know that Ia, Jb is separated by 1({z, u}), forsome u {min, y, max}.Since |diff((x), a)| + 1({z, u}) 1({x, z}) +1({z, u}) ({x, u}), it is straightforward to see thatIa, Jb, and also I, J , is separated by ({x, u}). I.e., is a separator for I, J .Case 2: (x) a and diff(maxB, (x)) < diff(a, (x)).Since ({x,max}) 1({x, z}) diff(a, (x)) >diff(maxB, (x)), we know that I, J is separated by({x,max}). I.e., is a separator for I, J .Case 3: a < (x) and diff((x),minB) < diff((x), a).This case is analogous to Case 2.

Now the proof of Claim 5 is complete, because one of the

three cases above must apply.

By symmetry we also obtain the following

Claim 6. If 1({y, z}) |diff((y), a)|, then is a sepa-rator forI, J . Finally, we show the following

Claim 7. If none of the assumptions of the Claims 16 is

satisfied, then is a separator forI, J . Proof: We assume w.l.o.g. that minA (x) (y) max

A.

Since Claim 1 does not apply, we know that also minB

(x) (y) maxB. Since Claims 26 do not apply, wefurthermore know that

1. |diff((u), a)| > 1({u, z}),for all u

{min,max, x , y

}, and

2. minA < a < (x) or (x) < a < (y) or(y) < a < maxA.

We distinguish between different cases, depending on the

particular interval that a belongs to.

Case 1: minA < a < (x).

Case 1.1: diff((x),minB) 1({min, z}).By definition ofwe have diff((x),minB) ({min, x}).Since diff((x),minA) > diff(a,minA) > 1({min, z})

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diff((x),minB), we therefore know that I, J is sepa-rated by ({min, z}). I.e., is a separator for I, J .Case 1.2: diff((x),minB) > 1({min, z}).In this case we can choose b (x) such thatdiff(b,minB) = 1({min, z}) + 1. Then, Ia, Jb isnot separated by 1(

{min, z

}). However, we know

that Ia, Jb is separated by 1({z, u}), for someu {x,y,max}. From the assumptions of Claim 7we know that diff((u), a) > 1({z, u}). Hence wemust have that diff((u), b) 1({z, u}). Therefore,diff((u),minB) = diff((u), b) + diff(b,minB) 1({z, u}) + 1({min, z}) + 1 ({min, u}). Sincediff((u),minA) = diff((u), a) + diff(a,minA) >diff((u),minB), we hence obtain that is a separator forI, J .Case 2: (y) < a < maxA.This case is analogous to Case 1.

Case 3: (x) < a < (y).

Case 3.1: diff((y), (x)) 1({x, z}).By definition of we have diff((y), (x)) ({x, y}).Since diff((y), (x)) > diff(a, (x)) > 1({x, z}) diff((y), (x)), we therefore know that I, J is sepa-rated by ({x, y}). I.e., is a separator for I, J .Case 3.2: diff((y), (x)) > 1({x, z}).In this case we can choose b with (x) < b (y)such that diff(b, (x)) = 1({x, z}) + 1. Then, Ia, Jbis not separated by 1({x, z}). However, we knowthat Ia, Jb is separated by 1({z, u}), for someu

{min, y, max

}. From the assumptions of Claim 7

we know that |diff(a, (u))| > 1({z, u}). Hence wemust have that |diff(b, (u))| 1({z, u}). We nowdistinguish between the cases where u can be chosen from

{y, max}, on the one hand, and where u must be chosen asmin, on the other hand.

Case 3.2.1: u {y, max}.In this case, diff((u), (x)) = diff((u), b) +diff(b, (x)) 1({z, u}) + 1({x, z}) + 1 ({x, u}).Since diff((u), (x)) = diff((u), a) + diff(a, (x)) >diff((u), (x)), we hence obtain that is a separator forI, J .Case 3.2.2: u

{y, max

}.

In this case, Ia, Jb is separated by 1({min, z}), and wemay assume that it is neither separated by 1({z, y}) norby 1({z, max}) nor by 1({x, z}). In particular, we musthave that diff((y), b) 1({z, y}) + 1. Therefore, forevery b with b b < (y) 1({z, y}), the followingis true: Ia, Jb is neither separated by 1({x, z}) norby 1({z, y}), but, consequently, by 1({min, z}) or by1({z, max}). Let b1 be the largest such b for whichIa, Jb is separated by 1({min, z}). In particular,diff(b1,min

B) 1({min, z}).

Case 3.2.2.1: diff((y), b1+1) 1({z, y}).In this case we know that diff((y),minB) 1({z, y}) + 1 + 1({min, z}) ({min, y}).Furthermore, diff((y),minA) = diff((y), a) +diff(a,minA) 1({z, y}) + 1 + 1({min, z}) + 1.Therefore diff((y),minA)

= diff((y),minB), and

I,

J is separated by ({min, y}). I.e., is a separator for I, J .Case 3.2.2.2: diff((y), b1+1) > 1({z, y}).In this case we know (by the maximal choice of b1)

that Ia, Jb1+1 must be separated by 1({z, max}).In particular, diff(maxB, b1+1) 1(z, max).Therefore, diff(maxB,minB) 1({z, max}) +1 + 1({min, z}) ({min,max}). Furthermore,diff(maxA,minA) diff(maxA, a) + diff(a,minA) 1({z, max}) + 1 + 1({min, z}) + 1. Thereforediff(maxA,minA) = diff(maxB,minB), and I, J isseparated by ({min,max}). I.e., is a separator for

I,

J .

We now have shown that is a separator for I, J , ifCase 3 applies.

Together with the Cases 1 and 2 we therefore obtain that

the proof of Claim 7 is complete.

Since at least one of the Claims 17 must apply, the proof

of Lemma 3.11 finally is complete.

Proof of part (c) of Lemma 3.8:

Let v be a node ofT that has exactly one child v1. Let be a minimal separator for il(v), and let 1 be a minimalseparator for il(v1) =: A1, B1. Our aim is to show thatw() w(1) + 2.

From Definition 3.7 we know that either sl(v) = orsl(v) = Qu, for someQ {, } and u {x,y,z}.Case 1: sl(v) = In this case we know from Definition 3.7 that il(v) =B1, A1. Therefore, 1 also is a (minimal) separator foril(v). In particular, w() = w(1) w(1) + 2.

Case 2: sl(v) = QuIn this case let be the separator for il(v) defined inLemma 3.11. Since is a minimal separator for il(v), itsuffices to show that w() w(1) + 2.

Let u, u be chosen such that {x,y,z} = {u, u, u}.Using Definition 3.3 and the particular choice of , it isstraightforward to see that

c() = ({u, u}) c(1) + 1 (1)

and that({min,max}) b(1) + 1 . (2)

Furthermore, for arbitrary u, u {x,y,z} we have({min, u}) + ({u,max}) b(1) + 2c(1) + 2 , (3)

which can be seen as follows: If u = u or u = u, then({min, u}) = 0 or ({u,max}) = 0. Consequently,

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({min, u}) + ({u,max}) MAX {b(1), b(1) +c(1) + 1}. Ifu and u both are different from u, then

({min, u}) = MAX

1({min, u}),1({min, u}) + 1({u, u}) + 1

and

({u,max}) = MAX

1({u,max}),

1({u,max}) + 1({u, u}) + 1

.

Therefore,

({min, u}) + ({u,max})

MAX

8>>>>>>>:

1({min, u}) + 1({u,max}),

1({min, u}) + 1({u,max}) + 1({u, u}) + 1,

1({min, u}) + 1({u, u}) + 1 + 1({u,max}),

1({min, u}) + 1({u, u}) + 1+1({u,max}) + 1({u, u

}) + 1

9>>>>=>>>>;

which, in turn, is less than or equal to

MAX

b(1), b(1) + c(1) + 1, b(1) + 2c(1) + 2

.

I.e., we have shown that (3) is valid.From (2) and (3) we obtain

b() b(1) + 2c(1) + 2 . (4)

From (1) and (4) we conclude that

w()2 = c()2 + b()

`

c(1) + 12

+ b(1) + 2c(1) + 2= c(1)

2 + b(1) + 4c(1) + 3 w(1)

2 + 4w(1) + 4

=`

w(1) + 22

.

Therefore, w() w(1) + 2.This completes the proof of part (c) of Lemma 3.8.

Remark 3.12. From the above proof it becomes clear,

why Definition 3.3 fixes the weight of a separator by

using the

-function. Let us consider the, at first

glance, more straightforward weight function w() :=MAX {c(), b()}. In the proof of part (c) of Lemma 3.8we then obtain from the items (1) and (4) that w() 2c(1) + b(1) + 2 3 w(1) + 2. Therefore, a modi-

fied version of Lemma 3.9, where item (c) is replaced bythe condition If v has exactly one child v1, then w(v) 3w(v1) + 2, leads to a (much weaker) bound of the form||T || c lg(w(v)). This, in turn, leads to a weaker ver-sion of Theorem 3.5, stating that |||| c lg w().However, this bound can already be proven by a conven-

tional Ehrenfeucht-Frasse game and does not only apply

for FO3(


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Lemma 5.1. For every FO(


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Proofs of Lemma 6.1 and Theorem 6.2 can be found in [9].

Let us remark that, by modifying the proof of the above

result, one can also show that the FO(

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