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Horn Complements: Towards Horn-to-Horn Belief Revision

Marina Langlois and Robert H. Sloan

University of Illinois at Chicago

mirodo1 | sloan@uic.edu

Bal´ azs Sz¨ or´ enyi

Hungarian Academy of Sciences & U. of Szeged

szorenyi@inf.u-szeged.hu

Gy¨ orgy Tur´ an

University of Illinois at Chicago,

Hungarian Academy of Sciences & U. of Szeged

gyt@uic.edu

Abstract

Horn-to-Horn belief revision asks for the revision of a Horn

knowledge base such that the revised knowledge base is also

Horn. Horn knowledge bases are important whenever one

is concerned with efficiency—of computing inferences, of

knowledge acquisition, etc.

could be of interest, in particular, as a component of any effi-

cient system requiring large commonsense knowledge bases

that may need revisions because, for example, new contradic-

tory information is acquired.

Recent results on belief revision for general logics show that

the existence of a belief contraction operator satisfying the

generalized AGM postulates is equivalent to the existence of

a complement. Here we provide a first step towards efficient

Horn-to-Horn belief revision, by characterizing the existence

of a complement of a Horn consequence of a Horn knowledge

base. A complement exists if and only if the Horn conse-

quence is not the consequence of a modified knowledge base

obtained from the original by an operation called body build-

ing. This characterization leads to the efficient construction

of a complement whenever it exists.

Horn-to-Horn belief revision

Introduction

Revising a knowledge base in the presence of new, po-

tentially conflicting information is a basic task facing a

commonsense reasoning agent. Belief revision usually ap-

proaches this task by identifying postulates that should be

satisfied by a rational revision operator, such as the AGM

postulates (Alchourr´ on, G¨ ardenfors, and Makinson 1985),

and characterizing operators that satisfy these postulates

(G¨ ardenfors 1988; Hansson 1999). The basic operators are

belief revision, when a new, perhaps contradictory belief is

to be incorporated into the knowledge base, and belief con-

traction, when an undesirable consequence is to be removed

from the knowledge base. One often considers contraction

first, and then defines revision in a standard way in terms of

contraction. Most of the work in belief revision assumes that

the underlyinglogic includes full propositional logic. On the

other hand, motivated in part by efficiency considerations, in

many applications one uses a logic based on only a fragment

of propositional logic.

Copyright c ? 2008, Association for the Advancement of Artificial

Intelligence (www.aaai.org). All rights reserved.

Adapting belief revision to this more general situation has

been initiated by the recent work of (Flouris, Plexousakis,

and Antoniou 2004). They study belief revision in general

logics, and formulate a property called decomposability of

thelogic. Itisshownthatdecomposabilityisanecessaryand

sufficient condition for the existence of an AGM-compliant

belief contraction operator. This framework is then used in

(Flouris, Plexousakis, and Antoniou 2005) to study decom-

position properties of description logics, motivated by appli-

cations to the Semantic Web.

In order to build a powerful agent capable of common-

sense reasoning, one fundamental challenge is to integrate

various capabilities, including belief revision, that have been

studied so far mostly in isolation. A commonsense reason-

ing agent should execute its tasks efficiently, and therefore it

has to use some tractable knowledge representation. In this

paper we explore an approach that takes tractable knowl-

edge representation as the primary constraint, and consid-

ers other requirements, in particular rationality constraints,

as also important, but secondary. For belief revision, this is

a departure from the standard framework.

Horn formulas provide a natural candidate for a general

framework for the integration process. This fragment of

propositional logic is expressive, allows for polynomial time

inference, and indeed is generally computationally tractable,

which explains its central role in artificial intelligence and

computer science. In particular, belief revision with Horn

formulas has been studied extensively (see references in the

next section). However, the problem of belief revision that

maintains a Horn knowledge base throughout the revision

process apparently has not been studied. Here we consider

such Horn-to-Horn revisions. That is, we are interested in

the possibilities and limitations of revising Horn formulas

such that the revised formula is also Horn. This, then, is

a special case of the general framework of (Flouris, Plex-

ousakis, and Antoniou 2004), and thus their general char-

acterization for the existence of an AGM-compliant belief

contraction operator applies. Horn logic as a whole turns

out not to be decomposable. Thus one must ask the more

detailed question of when contraction is possible. More pre-

cisely, one is led to the following problem, which may also

be of interest in itself.

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Problem A. For Horn formulas ϕ and ψ, where ψ is a con-

sequence of ϕ, when does there exist a proper Horn conse-

quence χ of ϕ, such that ψ ∧ χ is equivalent to ϕ?

Such a formula χ is called a ϕ-complement of ψ, and it

corresponds to the result of a contraction operator, when the

formula ψ is contracted from the knowledge base ϕ. For the

decomposability framework, it would be desirable always to

have a complement, but unfortunately this is not the case.

(See Example 9.) When a complement does not exist, one

may try to find an approximate complement.

Our main result (Theorem 7) gives a complete answer to

Problem A by giving two characterizations of all those pairs

ϕ and ψ for which ψ has a ϕ-complement. The character-

izations give efficiently decidable criteria and lead to effi-

cient algorithms to construct a complement, if it exists. The

complements constructed are only polynomially larger than

the original knowledge base (although repeated application

could cause an exponential blowup). As a corollary, one

obtains a complete description of decomposable Horn for-

mulas as well, where a Horn formula is decomposable if all

its Horn consequences have a complement. We also present

some computational results on the fraction of Horn impli-

cates of a random Horn formula having a complement.

The rest of the paper is structured as follows. The next

section gives a brief overview of previous related work. The

section after that gives a more detailed review of Flouris,

Plexousakis, and Antoniou’s AGM-based framework and

their basic results together with the formal connection be-

tween belief revision theory and the rest of this paper. Then

background is provided on Horn formulas. The next three

sections contain our results, a proof sketch of the main theo-

rem and the sketch of another proof of the first characteriza-

tion. The last two sections present the experimental results

and some remarks on further work.

Related work

Horn formulas have been considered previously in several

papers dealing with the complexity of belief revision, e.g.,

(Eiter and Gottlob 1992; Gogic, Papadimitriou, and Sideri

1998; Jin and Thielscher 2005; Liberatore 1997; 2000;

Nebel 1998). The results obtained in these papers are mostly

complexity-theoretic negative results, and they deal with re-

visionmethodswheretherevisionofaHornknowledgebase

is not necessarily Horn, or they propose revision methods

that may be inefficient. A more detailed comparison with

our work will be given in the full version of this paper.

If ψ is a single Horn clause implicate C, then Problem A

can be reformulated as follows: does ϕ have an irredundant

conjunctive normal form expression containing C? The re-

lated question where C is a prime implicate and the irredun-

dant conjunctive normal form expression is also assumed to

consist of prime implicates only, has been studied by (Ham-

mer and Kogan 1995). They call such a prime implicate

non-redundant, and show that non-redundancy is polynomi-

ally decidable for negative clauses, but it is NP-complete for

definite clauses.

General logics and belief contraction

In this section we give a brief outline of the formal frame-

work for belief revision in general logics, following (Flouris,

Plexousakis, and Antoniou 2004). We also indicate how this

very general and abstract framework can be specialized to

Horn logic.

A logic is specified by a set of expressions L and a con-

sequence operator Cn : P(L) → P(L), where P(L) is the

power set of L. The consequence operator is assumed to

satisfy the properties of inclusion (A ⊆ Cn(A)), iteration

(Cn(A) = Cn(Cn(A))) and monotonicity (A ⊆ B implies

Cn(A) ⊆ Cn(B)) for every A,B ⊆ L.

For Horn logic, L is the set of all Horn clauses over a fixed

finite set of variables, and for a set of Horn clauses ϕ, the set

Cn(ϕ) contains all Horn clauses implied by ϕ. The required

properties are clearly satisfied.

A theory or knowledge base K is a subset of L such that

K = Cn(K). For Horn logic, a theory can be specified

by a set of Horn clauses. The corresponding theory then

consists of all their consequences. A contraction operator,

denoted by −, is of the form − : P(L) × P(L) → P(L),

and maps a knowledge base and a set to be contracted to

a new knowledge base, the result of the contraction. The

(generalized) AGM postulates for contraction are1

closure: K − A = Cn(K − A)

inclusion: K − A ⊆ Cn(K)

vacuity: A ?⊆ Cn(K) implies K − A = Cn(K)

success: A ?⊆ Cn(∅) implies A ?⊆ Cn(K − A)

preservation: Cn(A) = Cn(B) implies K − A = K − B

recovery: K ⊆ Cn((K − A) ∪ A).

A contraction operator is AGM-compliant if it satisfies these

postulates. A logic is AGM-compliant if there exists an

AGM-compliant contraction operator for it.

Given a logic, and subsets K,A ⊆ L, the set of comple-

ments of A with respect to K is

A−(K) =

{B ⊆ L : Cn(B) ⊂ Cn(K) and Cn(A ∪ B) = Cn(K)}

if Cn(∅) ⊂ Cn(A) ⊆ Cn(K), and A−(K) = {B ⊆ L :

Cn(B) = Cn(K)} otherwise.

A set K ⊆ L is decomposable if A−(K) ?= ∅ for every

A ⊆ L. A logic is decomposable if every K is decompos-

able. For Horn logic, these definitions specialize to Defini-

tions 3 and 4.

Theorem 1 ((Flouris, Plexousakis, and Antoniou 2004)). A

logic is AGM-compliant iff it is decomposable.

In particular, Flouris, Plexousakis, and Antoniou show

that if a logic is decomposable, then by selecting any com-

plement one obtains an AGM-compliant contraction opera-

tor. More generally, if K is decomposable, then by selecting

1(Flouris, Plexousakis, and Antoniou 2004) refers to these as

generalized versions of the properties, but for simplicity we omit

the term ‘generalized’.

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any complement with respect to K, one obtains a contrac-

tion operator that handles contractions from K in an AGM-

compliant manner. It also follows from their arguments that

if A−(K) ?= ∅ then selecting any complement of A with re-

spect to K handles the contraction of A from K in an AGM-

compliant manner.

Preliminaries

Let U be the set of propositional variables in our universe.

Two clauses collide if they contain a pair of complementary

literals. AclauseisHornifitcontainsatmostoneunnegated

literal, definite if it contains exactly one unnegated literal,

and negative if it contains only negated literals. A (definite)

Horn formula is a conjunction—or a set, whichever view is

more convenient—of (definite) Horn clauses (an empty con-

junction is always true). A Boolean function is a (definite)

Horn function if it has a (definite) Horn formula.

LowercaseRomanalphabet lettersf,g,hdenoteBoolean

functions; lower case Greek letters ϕ,ψ,χ denote Boolean

formulas. If we use ϕ in a place where a function is ex-

pected, then ϕ stands for the Boolean function represented

by ϕ. Two formulas ϕ1 and ϕ2 are equivalent, denoted

ϕ1∼ ϕ2, if they represent the same Boolean function.

For a Horn clause C, let Body(C) be the set of variables

corresponding to the negated literals in C, or their conjunc-

tion (which will be clear from context). Also, let Head(C)

be the unnegated variable of C if C is a definite clause, and

0 if C is a negative clause. We use → to denote the Boolean

implication operator, so Horn clause C can be written as

Body(C) → Head(C). For example, if C is the Horn clause

¯ x∨ ¯ y∨z, then Body(C) = {x,y}, Head(C) is z, and C can

also be written as x,y → z or (x∧y) → z. If C is the Horn

clause ¯ x∨¯ y then it can also be written as x,y → 0 or simply

x,y →.

A Boolean function g is a consequence of Boolean func-

tion f, denoted f ⇒ g, if every assignment that satisfies f

also satisfies g. A function g is a proper consequence of f,

denoted f

implicate of f if f ⇒ C.

The set of satisfying (resp., falsifying) truth assignments

of f is denoted by T(f) (resp., F(f)). A function f is anti-

monotone if T(f) is downward closed, i.e., f(a) = 1 and

b ≤ a imply f(b) = 1.

We will use a slight generalization of anti-monotone func-

tions.

Definition 2 (Almost anti-monotone function). A function

is almost anti-monotone if it is either anti-monotone, or

there is an anti-monotone function g such that T(f) =

T(g) ∪ {1}, where 1 is the all 1’s assignment.

Every almost anti-monotone function is Horn. Now we

formulate the central concept discussed in this paper.

Definition 3 (f-complement). For Horn functions f and g

such that f ⇒ g, a Horn function h is an f-complement of g

iff f⇒

According to the definition, no complements exist if f ∼

1 (where 1 denotes the identically 1 function). Also accord-

ing to the definition, g ∼ 1 can never have a complement,

⇒

? g, if f ⇒ g but not g ⇒ f. A clause C is an

? h and f ∼ (g ∧ h).

so this case is excluded from consideration in the following

definition.

Definition 4 (Decomposable Horn function). A Horn func-

tion f is decomposable if every Horn consequence g ?∼ 1 of

f has an f-complement.

One usually works with formulas as opposed to functions,

but as the notions of complement and decomposability de-

pend only on the function represented by the formula, the

definitions are given in a syntax-independent way.

Results

For a function f and a set of variables X ⊆ U, we define the

f-closure of X to be the set of variables

Clf(X) = {v ∈ U : f ⇒ (X → v)} .

A direct consequence of this definition is that if a negative

clause C is an implicate of f, then Clf(Body(C)) = U.

In order to formulate our main result, we need two defi-

nitions. The formula ˆ ϕ is obtained from ϕ by adding to the

body of each definite clause in ϕ a variable not contained in

the closure of its body, in all possible ways.

Definition 5 (Body-building formula ˆ ϕ). For a Horn for-

mula ϕ let ˆ ϕ be the formula

?

We could have defined ˆ ϕ as a conjunction over all clauses

of ϕ, as negative clauses would make no contribution. Every

clause of ˆ ϕ is definite. It may be the case that ˆ ϕ is the empty

conjunction. This happens, for example, when ϕ consists of

negative clauses only.

GivenaHornformulaϕandaHornclauseD, wepartition

the clauses of ϕ not colliding with D into two classes.

Definition 6 (Formulas Aϕ(D) and Bϕ(D)). Given a Horn

formula ϕ and a Horn clause D, let

Aϕ(D) =

{C ∈ ϕ : C,D don’t collide, Body(D) ⊆ Clϕ(Body(C))},

Bϕ(D) =

{C ∈ ϕ : C,D don’t collide, Body(D) ?⊆ Clϕ(Body(C))}.

The existence of a complement can now be characterized.

Theorem 7 (Main theorem). Let ϕ ?∼ 1 be a Horn formula,

and ψ be a Horn consequence of ϕ. Then the following are

equivalent:

1. ψ has a ϕ-complement,

2. ˆ ϕ ?⇒ ψ,

3. for some clause D of ψ it holds that Bϕ(D) ?⇒ D.

Although the definition of ˆ ϕ is given in terms of a for-

mula, it follows from this characterization that it depends on

only the function (see also Lemma 14 below). The following

corollary gives the algorithmic aspects of Theorem 7.

Corollary 8. There is a polynomial time algorithm which,

given a Horn formula ϕ and a Horn consequence ψ of ϕ,

decides if ψ has a ϕ-complement, and if it does, then con-

structs such a ϕ-complement.

C∈ϕ definite

?

v?∈Clϕ(Body(C))

(Body(C),v → Head(C)).

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The following simple example illustrates the results.

Example 9. Let U = {x,y,z}, ϕ = C1∧ C2, where C1=

(x → y) and C2 = (y → z). Then Clϕ(x) = U and

Clϕ(y) = {y,z}. So ˆ ϕ = (x,y → z).

The clause (x,y → z) is implied by ˆ ϕ, and so it has no ϕ-

complement. This is also shown by the fact that Bϕ(x,y →

z) = {y → z}, which implies (x,y → z).

On the other hand, the clause (x → z) is not implied by

ˆ ϕ, so it does have a ϕ-complement. This is also shown by

the fact that Bϕ(x → z) = {y → z}, which does not imply

(x → z). Both constructions mentioned in the paper give

the ϕ-complement (x,z → y) ∧ (y → z).

Decomposable Horn functions have the following charac-

terization.

Theorem 10. For every Boolean function f the following

are equivalent:

1. f is a decomposable Horn function,

2. there is a Horn representation ϕ of f such that ˆ ϕ ∼ 1,

3. for every Horn representation ϕ of f it holds that ˆ ϕ ∼ 1,

4. for every Horn implicate C of f

Clf(Body(C)) = U,

5. f is almost anti-monotone.

it holds that

Proof sketch for Theorem 7

We take care of the case of negative clauses first.

Lemma 11. Let ϕ,ψ ?∼ 1 be Horn formulas such that ϕ ⇒

ψ, and ψ contains a negative clause D. Then

• ψ has a ϕ-complement,

• ˆ ϕ ?⇒ ψ,

• Bϕ(D) ?⇒ D.

Fortherestoftheproofwemayassumethatψ isadefinite

Horn formula.

The (1 ⇒ 2) part of the proof is based on the following

lemma.

Lemma 12. Let f be a Horn function and let D1= (B →

z) and D2 = (B → u) be definite Horn clauses with the

same body B such that f ⇒ D1and f ?⇒ D2. Then

D = (B,u → z)

has no f-complement.

The (2 ⇒ 3) part of the proof is omitted.

(3 ⇒ 1) part of the proof, let D be a clause in ψ such that

?

C ∈ Aϕ(D) let

χ?

C

=

z∈Body(D)

χ??

C

=(Body(C),Head(D) → Head(C)),

and finally put

C∈Aϕ(D)

For the

C∈Bϕ(D)C ?⇒ D. It can be shown that Aϕ(D) ?= ∅.

Now we can define a ϕ-complement of ψ. For each clause

?

(Body(C) → z),

χ =

?

χ?

C∧ χ??

C

∧

?

C∈(ϕ\Aϕ(D))

C

.

Thus χ is formed from ϕ by replacing clauses C ∈ Aϕ(D)

by χ?

Note that in the definition of χ??

then Head(C) = 0.

Example 13. Consider ϕ = (x → y) ∧ z and ψ = z. Then

both clauses of ϕ are in Aϕ(z), and so the ϕ-complement

of ψ provided by the construction (after deleting redundant

clauses) is (x,z → y).

Singleton Horn extensions and ˆ ϕ

A different proof of the equivalence (1 ⇔ 2) in Theorem

7 provides a semantic characterization of the body build-

ing formula. The proof is based on the following lemma.

It shows that T(ˆ ϕ) \ T(ϕ) consists of precisely the single-

ton Horn extensions of ϕ, i.e., of those points which can be

added to the set T(ϕ) maintaining the Horn property.

Lemma 14. Let ϕ be a Horn formula and a ∈ F(ϕ). Then

T(ϕ) ∪ {a} is a Horn function iff ˆ ϕ(a) = 1.

The ⇐ direction of Lemma 14 can be proved by con-

structing a Horn formula χafor T(ϕ) ∪ {a} for every truth

assignment a ∈ T(ˆ ϕ) \ T(ϕ), and this, in turn, gives an

alternative construction of complements.

Both constructions for the complement may increase the

size of the formula by a linear factor, and it is not known

whether this increase is necessary. Similar questions for

DNF are studied in (Mubayi, Tur´ an, and Zhao 2006).

C∧χ??

C, and leaving the rest of the formula unchanged.

C, if C is a negative clause

Experimental results

Recently there appears to be growing interest in exploring

the computational properties of belief revision methods by

running experiments (Benferhat et al. 2004; Bessant et al.

2001). The results presented in this paper raise the related

question of what fraction of implicates of a random Horn

formula have complements. Properties of random CNF ex-

pressions, such as their phase transition from almost surely

satisfiable to almost surely unsatisfiable have been, and are,

much studied (Martin, Monasson, and Zecchina 2001). Sim-

ilar work has also been done for random Horn formulas

(Moore et al. 2007). The results indicate that the choice of

the probability distribution on Horn formulas requires care.

We have considered the following probabilistic model

to generate a random Horn formula. The parameters are

n,m,p and q. Here n is the number of variables, m is the

number of clauses and p is the fraction of definite clauses.

For each definite clause we pick the head from the uniform

distribution over the variables. The bodies of the clauses are

generated by determining the clause length using a geomet-

ric distribution of parameter q, and then picking the right

number of variables without replacement, again using uni-

form distribution. This model produces Horn formulas with

clauses of small, but not uniformly bounded size.

After having generated a Horn formula ϕ, we used ex-

haustive testing of all implicates for having a complement.

This was done by constructing the formula ˆ ϕ, and checking

whether a candidate clause is a consequence of ˆ ϕ. By The-

orem 7, these are the implicates that do not have a comple-

ment. Because of the exhaustive testing, we present results

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20

25

30

35

40

45

50

55

60

0.1 0.2 0.30.4 0.5 0.6 0.70.80.91

Fraction of definite clauses

% of implied definite implicates

Figure 1: Percent of definite Horn implicates of ϕ implied

by ˆ ϕ as a function of p. Measured on 100 random formulas

with n = 12 variables, m = 24 clauses, and q = 1/3.

for 12 variables. As negative implicates always have com-

plements, the figures show the fraction among definite im-

plicates. For definite implicates, we want to know when we

have an exact contraction operator, and when we must turn

to some form of approximation or violation of some AGM

postulate. The computational results suggest that in certain

ranges of the parameters complements are likely to exist.

For example, Figure 2 shows that for a random formula of

50 or more clauses over 12 variables (with the clause dis-

tribution given in the figure), at least 3/4 of the implicates

have complements. It appears to be an interesting problem

to obtain theoretical results on the fractions.

20

25

30

35

40

45

1015 20 25 30354045 5055

Number of clauses

% of impled definite implicates

Figure 2: Percent of definite Horn implicates of ϕ implied

by ˆ ϕ as a function of m. Measured on 100 random formulas

with n = 12, p = 1/2, q = 1/3.

Remarks and further work

The results of this paper provide a first step towards Horn-

to-Horn belief revision, but much remains to be done. It

would be interesting to get a characterization of all com-

15

20

25

30

35

40

45

50

55

60

23456789 10 1112

Length of Implicates

% of implied definite implicates

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

Figure3: PercentofdefiniteHornimplicatesofϕimpliedby

ˆ ϕ, forspecificimplicatesizes, asafunctionofimplicatesize.

Measured on 100 random formulas with n = 12 variables,

m = 24 clauses, various p, and q = 1/3.

plements, in the cases when a complement exists.

other next step is to investigate the two supplementary AGM

postulates for contraction in the Horn case; Flouris refers

to contraction operators also satisfying these postulates as

fully AGM-compliant (Flouris 2006). The problem of ex-

tending the framework of Flouris, Plexousakis, and Anto-

niou to revision, as opposed to contraction, at least in the

Horn case, is also open.But perhaps most importantly,

one should study the possibilities of approximating comple-

ments in cases when a complement does not exist, giving up

on adherence to the postulates in order to gain efficiency. As

noted in the introduction, this would constitute a departure

from current approaches to belief revision theory.

Our long term goal towards the construction of a com-

monsense reasoning agent is the integration of belief revi-

sion and learning. A commonsense reasoning agent should

not only be able to do both, but to do both efficiently. An

important motivation to study this problem is the interactive

acquisition of large commonsense knowledge bases, such as

the Open Mind Common Sense (Singh 2002) project. Here

it seems reasonable to assume that the knowledge base re-

ceives contradictory information from the users, and thus it

has to revise its contents, and at the same time should im-

prove its quality in the long run. Also, Horn logic seems

to be a reasonable knowledge representation, as inferences

needtobedonewiththeknowledgeacquired. Aninteresting

application of such knowledge bases is given by the recent

work of (Pentney et al. 2007), showing that such knowledge

bases could be combined with sensor data in health care and

other areas.

The combination of belief revision and learning has

been studied by, e.g., (Kelly, Schulte, and Hendricks 1995;

Martin and Osherson 1997; Pagnucco and Rajaratnam 2005;

Wrobel 1994). We plan to approach the problem of combin-

ing belief revision and learning in a formal model of compu-

tational learning theory. Due to lack of space, we do not de-

scribe the formal modeling details, but simply state the prob-

An-

Page 6

lem one may refer to as Knowledge Base Learning (Know-

BLe):

Problem B. Find an efficient algorithm that learns a propo-

sitional Horn formula in the model of learning from entail-

ment (with or without queries), and updates its hypotheses

in a rational manner.

Acknowledgment

This material is based upon work supported by the National

Science Foundation under Grant No. CCF-0431059. The

third author was also supported in part by the NKTH grant

of Jedlik´Anyos R&D Programme 2007 of the Hungarian

government (codename TUDORKA7).

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