Using Ideas of Kolmogorov Complexity for Studying Biological Texts Abstract Kolmogorov complexity furnishes many useful tools for studying different natural processes that can be expressed using sequences of symbols from a finite alphabet (texts), such as genetic texts, literary and music texts, animal communications, etc. Although Kolmogorov complexity is not algorithmically computable, in a certain sense it can be estimated by means of data compressors. Here we suggest a method of analysis of sequences based on ideas of Kolmogorov complexity and mathematical statistics, and apply this method to biological (ethological) "texts." A distinction of the suggested method from other approaches to the analysis of sequential data by means of Kolmogorov complexity is that it belongs to the framework of mathematical statistics, more specifically, that of hypothesis testing. This makes it a promising candidate for being included in the toolbox of standard biological methods of analysis of different natural texts, from DNA sequences to animal behavioural patterns (ethological "texts"). Two examples of analysis of ethological texts are considered in this paper. Theses examples show that the proposed method is a useful tool for distinguishing between stereotyped and flexible behaviours, which is important for behavioural and evolutionary studies. Sequences of symbols from a finite alphabet (or "texts") appear as basic objects of study in many scientific fields, including molecular biology and genetics (genetic texts), linguistics (literary and musical texts), zoosemiotics (animal communications), ethology (behavioural sequences) and others. The main problem that researchers face in these domains is finding an adequate model which would allow for assessment of certain characteristics of a text, while using a relatively small number of parameters. One of the most popular approaches is based on the description of sequences by stochastic processes. Modeling of DNA sequences by Markov processes of finite depth, or connectivity, can serve as a good example here. This way of looking at the problem presents some methodological limitations. Indeed, it is difficult to imagine that a text by Shakespeare or a frog's genome could be described adequately by a stochastic process with a relatively small number of parameters. In order to obtain an approximately adequate model of a text of a certain type, it is necessary to increase the number of parameters which, in turn, should be estimated statistically, based on real data. For example, if the frequency of each letter in a genetic text depends on n, n > 0 previous letters, then the number of parameters equals 4 n . When n = 10, the numbers of parameters is 2 20 , that is, about one million, and one needs several times more data to estimate this quantity exactly. Moreover, the finite-memory assumption (with memory say, 5 or 10) is even less realistic for texts in natural human languages.. As it was noted in [31] , if we meet the word "lemma" in a book, we hardly meet the word "love" in it. (This means that, in fact, the memory of human-written texts is unlimited.). One can approach the methodological problems listed above with methods that are close in spirit to the ideas of Kolmogorov complexity. More precisely, the degree of complexity of a "text" could be estimated by its Kolmogorov complexity. Although Kolmogorov complexity is not algorithmically computable, it can be, in a certain sense, estimated by means of data compressors [26, 27] . This approach does not contradict the probabilistic one, because if one looks at a sequence as generated by stochastic process, the length of the compressed sequence can be considered an estimate of the Shannon entropy, which, in turn, equals Kolmogorov complexity. It is worth noting that a quantitative estimation of complexity of sequences in natural "texts" is of interest in its own right. There is a huge body of literature that analyses symbolic sequences by means of Kolmogorov complexity, including diagnostic of the authorship of literary and musical texts [1, 3, 5, 14, 15, 24, 25, 30] . The importance of the use of ideas and methods of Kolmogorov complexity in biological sciences can hardly be overestimated. Such methods were used for estimation of closeness of DNA sequences [1, 3, 14, 15] , construction of phylogenetic trees [3, 5, 14, 15] , and distinguishing between innate and acquired behaviour in ants [19] . Unfortunately, these approaches do not give a possibility to use methods of mathematical statistics and, in particular, hypothesis testing. This limits the applicability of ideas of Kolmogorov complexity to biological studies, because, since Fisher's classic works [6] , statistical testing of biological hypotheses became the main method of quantitative analysis of biological data.. In this paper we suggest an approach which allows us to combine the advantages of methods based on Kolmogorov complexity with classic methods of testing statistical hypotheses. As distinct from other approaches to the analysis of sequences by means of Kolmogorov complexity, we stay within the framework of mathematical statistics. As examples, we consider ethological "texts" of two kinds, namely, territorial behaviour of gulls and hunting behaviour of ants. In the first case we found that the complexity of territorial behaviour in gulls differ in two situations: when a trespasser approaches a resident's nest, and when it escapes from a resident's defended territory. In the second case we found that complete (successful) hunting stereotypes in members of a natural ant colony are characterized by smaller complexity than incomplete hunting stereotypes in naive laboratory-reared ants.. In sum, being applied to members of different branches of Animal Kingdom, the proposed method appears to be a promising tool to distinguish between "basic" stereotypical behavioural patterns and flexible behaviour. We believe that this method is applicable to the analysis of biological texts of different kinds.

O O O B I O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Measuring

the

blame

of

each

formula

for

inconsistent

prioritized

knowledge

bases

It

is

increasingly

recognized

that

identifying

the

degree

of

blame

or

responsibility

of

each

formula

for

inconsistency

of

a

knowledge

base

(i.e.

a

set

of

formulas)

is

useful

for

making

rational

decisions

to

resolve

inconsistency

in

that

knowledge

base.

Most

current

techniques

for

measuring

the

blame

of

each

formula

with

regard

to

an

inconsistent

knowledge

base

focus

on

classical

knowledge

bases

only.

Proposals

for

measuring

the

blames

of

formulas

with

regard

to

an

inconsistent

prioritized

knowledge

base

have

not

yet

been

given

much

consideration.

However,

the

notion

of

priority

is

important

in

inconsistencytolerant

reasoning.

This

article

investigates

this

issue

and

presents

a

family

of

measurements

for

the

degree

of

blame

of

each

formula

in

an

inconsistent

prioritized

knowledge

base

by

using

the

minimal

inconsistent

subsets

of

that

knowledge

base.

First

of

all,

we

present

a

set

of

intuitive

postulates

as

general

criteria

to

characterize

rational

measurements

for

the

blames

of

formulas

of

an

inconsistent

prioritized

knowledge

base.

Then

we

present

a

family

of

measurements

for

the

blame

of

each

formula

in

an

inconsistent

prioritized

knowledge

base

under

the

guidance

of

the

principle

of

proportionality,

one

of

the

intuitive

postulates.

We

also

demonstrate

that

each

of

these

measurements

possesses

the

properties

that

it

ought

to

have.

Finally,

we

use

a

simple

but

explanatory

example

in

requirements

engineering

to

illustrate

the

application

of

these

measurements.

Compared

to

the

related

works,

the

postulates

presented

in

this

article

consider

the

special

characteristics

of

minimal

inconsistent

subsets

as

well

as

the

priority

levels

of

formulas.

This

makes

them

more

appropriate

to

characterizing

the

inconsistency

measures

defined

from

minimal

inconsistent

subsets

for

prioritized

knowledge

bases

as

well

as

classical

knowledge

bases.

Correspondingly,

the

measures

guided

by

these

postulates

can

intuitively

capture

the

inconsistency

for

prioritized

knowledge

bases.Keywords:

Inconsistency

measures,

prioritized

knowledge

base,

minimal

inconsistent

subsets,

blame

of

each

formula

in

inconsistency.

Many

(if

not

all)

real-world

applications

are

characterized

by

the

presence

of

inconsistency.

Research

in

inconsistency

handling

has

attracted

significant

attention

in

recent

years

because

of

the

necessity

for

dealing

with

inconsistency

in

applications.

Increasingly,

it

is

recognized

that

measuring

inconsistency

is

a

crucial

part

of

inconsistency

handling

in

computer

science

as

well

as

in

artificial

intelligence,

and

their

applications

A

growing

number

of

techniques

for

measuring

inconsistency

have

been

explored

under

different

scenarios,

such

as

knowledge

merging

ontology

management

software

engineering

and

requirements

negotiation

as

well

as

in

artificial

intelligence

research

in

general

provides

a

good

review

on

some

recent

proposals

for

measuring

inconsistency..

The

overwhelming

majority

of

the

current

proposals

for

measuring

inconsistency

are

concentrated

on

the

degree

of

inconsistency

(or

the

amount

of

contradiction)

for

a

whole

knowledge

base.

In

contrast,

there

are

relatively

few

techniques

for

measuring

the

degree

of

blame

or

responsibility

of

each

individual

formula

of

a

knowledge

base

for

the

inconsistency

in

that

knowledge

base.

However,

in

many

applications,

it

is

desirable

to

resolve

inconsistency

by

measuring

and

identifying

the

responsibility

of

each

formula

for

that

inconsistency.

For

example,

in

requirements

engineering,

developers

or

requirement

analysts

need

to

know

the

responsibility

of

each

requirement

involved

in

inconsistency

to

take

an

appropriate

action

for

resolving

the

inconsistency..

The

current

proposals

for

measuring

the

blame

of

each

formula

for

inconsistency

of

a

knowledge

base

can

be

classified

into

two

categories.

The

first

category

focuses

on

the

distribution

of

an

existing

(syntax-based

or

model-based)

measure

of

the

degree

of

inconsistency

of

a

knowledge

base

among

formulas

belonging

to

that

knowledge

base

by

using

a

cooperative

game

theory

model

The

proportion

of

the

existing

measure

distributed

to

an

individual

formula

of

a

knowledge

base

is

considered

as

a

measurement

of

the

blame

of

that

formula

for

the

inconsistency

of

that

knowledge

base.

The

Shapley

inconsistency

value

presented

by

Hunter

and

Konieczny

is

the

most

representative

work

of

the

first

category,

in

which

the

Shapley

value

the

well-known

coalitional

game

model,

has

been

used

to

distribute

an

existing

measure

of

the

overall

inconsistency

of

a

knowledge

base

to

each

formula

belonging

to

that

knowledge

base.

In

contrast,

the

second

category

such

as

argues

that

it

is

natural

and

intuitive

to

define

syntactic

measures

of

the

blame

of

each

formula

of

a

knowledge

base

for

its

inconsistency

from

minimal

inconsistent

subsets

of

the

knowledge

base

directly.

Note

that

it

follows

the

viewpoint

that

minimal

inconsistent

subsets

of

a

knowledge

base

could

be

considered

as

the

purest

form

of

inconsistency

of

the

knowledge

base

Moreover,

Hunter

and

Konieczny

argued

that

syntax

sensitive

measures

are

necessary

for

some

applications

such

as

requirements

engineering

The

MinInc

inconsistency

values

presented

in

and

the

scoring

function

of

singleton

sets

presented

in

can

be

considered

as

the

most

representative

works

of

the

second

category.

In

particular,

an

interesting

link

between

the

two

approaches

is

that

the

MinInc

inconsistency

value

MIV

C

is

also

a

special

Shapley

inconsistency

value

Note

that

approaches

in

both

the

first

and

the

second

categories

are

concerned

with

classical

knowledge

bases

only.

Proposals

for

measuring

the

blame

of

each

formula

belonging

to

a

prioritized

knowledge

base

for

the

inconsistency

of

that

base

have

not

yet

been

given

deserved

attention.

That

is,

the

current

proposals

for

measuring

the

blames

of

formulas

for

inconsistency

of

a

knowledge

base

do

not

take

the

priority

of

each

formula

into

account.

However,

as

a

description

of

the

importance

or

reliability

of

the

corresponding

knowledge,

the

priority

of

each

formula

always

plays

an

important

role

in

inconsistency

handling

in

artificial

intelligence

and

many

applications,

such

as

belief

revision

knowledge

base

merging

and

inconsistency

handling

in

requirements

engineering

It

is

natural

to

take

the

priority

of

each

formula

into

account

when

identifying

the

blames

of

formulas

for

inconsistency

of

a

prioritized

knowledge

base..

The

Shapley

value

model

of

the

first

category

is

not

appropriate

for

characterizing

the

blame

of

each

prioritized

formula

for

the

inconsistency

of

a

prioritized

knowledge

base,

even

if

we

consider

the

priority

of

each

formula

when

measuring

the

amount

of

inconsistency

of

that

prioritized

knowledge

base.

To

illustrate

this,

let

us

consider

a

scenario

of

two

conflicting

witnesses.

Suppose

that

Alice

and

Bob

are

witnesses

about

the

case

of

John

being

accused

Measuring

the

blame

of

each

formula

for

inconsistent

prioritized

knowledge

bases

of

breaking

the

speeding

limit

in

an

traffic

accident.

Alice

claims

that

John

was

breaking

the

speeding

limit

when

the

traffic

accident

happened.

But

Bob

claims

that

John

wasn't

over

the

speeding

limit

when

the

accident

happened.

Then

we

have

an

inconsistent

knowledge

base

from

the

two

witnesses,

i.e.

K

={Break(John,speed_limit),¬Break(John,speed_limit)}.

If

we

know

that

Alice

has

a

good

reputation,

but

Bob

is

a

drunken

witness,

we

may

consider

that

is

much

more

reliable

than

Intuitively,

the

degree

of

responsibility

of

for

the

inconsistency

in

K

should

be

more

than

that

of

However,

the

Shapley

inconsistency

value

cannot

make

a

distinction

between

the

blames

of

Break

John,

and

since

it

assigns

half

of

the

amount

of

inconsistency

in

K

to

and

respectively..

As

mentioned

earlier,

the

fundamental

viewpoint

of

the

second

category

is

that

minimal

inconsistent

subsets

of

a

(classical

or

prioritized)

knowledge

base

can

be

considered

as

the

purest

form

of

inconsistency

in

that

base.

Following

this

viewpoint

in

the

case

of

prioritized

knowledge

bases,

it

is

also

natural

to

explore

connections

between

the

measures

for

the

blame

of

each

prioritized

formula

for

the

inconsistency

of

the

knowledge

base

the

formula

belongs

to

and

the

minimal

inconsistent

subsets

of

that

base.

Our

previous

article

can

be

considered

as

an

attempt

to

follow

this

viewpoint

for

prioritized

knowledge

bases,

in

which

we

explored

a

possible

combination

of

the

scoring

function

presented

in

and

the

priority

levels

of

formulas.

However,

as

pointed

out

in

the

scoring

function

of

a

singleton

set

for

a

given

knowledge

base

is

very

sketchy

for

measuring

the

degree

of

blame

of

the

corresponding

formula

for

the

inconsistency

of

the

knowledge

base.

With

regard

to

another

representative

proposal

of

the

second

category,

one

of

the

fundamental

characteristics

of

the

MinInc

inconsistency

value

MIV

C

and

some

revised

MinInc

inconsistency

values

presented

in

is

that

each

formula

of

a

given

minimal

inconsistent

knowledge

base

M

takes

the

same

degree

of

responsibility

(i.e.

|M|

for

the

inconsistency

of

M.

This

is

not

always

satisfiable

for

prioritized

knowledge

bases

such

as

the

example

of

two

conflicting

witnesses

mentioned

above..

In

this

article,

we

argue

that

it

is

desirable

to

have

proportional

distributions

of

the

overall

inconsistency

of

a

minimal

inconsistent

prioritized

knowledge

base

among

formulas

belonging

to

that

base,

according

to

severity

of

contribution

of

each

formula

to

cause

the

inconsistency.

That

is,

a

reasonable

measurement

of

the

blame

of

each

formula

of

a

prioritized

knowledge

base

for

the

inconsistency

of

that

base

should

obey

the

so-called

principle

of

proportionality.

In

order

to

rationally

justify

any

measures

based

on

this

principle,

we

first

propose

a

set

of

properties

to

characterize

measures

of

the

blames

of

formulas

in

the

inconsistency

of

a

prioritized

knowledge

base,

which

accords

with

the

principle

as

well

as

the

view

of

minimal

inconsistent

subsets

being

the

purest

form

of

inconsistency.

In

particular,

these

properties

take

the

special

characteristics

of

minimal

inconsistent

subsets

as

well

as

the

priorities

of

formulas

into

account.

This

makes

the

set

of

properties

more

appropriate

to

capturing

the

most

intuitive

constraints

in

general

such

as

most

of

properties

presented

in

as

well

as

special

constraints

presented

in

this

article

for

inconsistency

measures

based

on

minimal

inconsistent

subsets.

We

then

define

a

family

of

measures

of

the

blame

of

each

formula

for

the

inconsistency

of

a

prioritized

knowledge

base

under

the

guidance

of

the

principle

of

proportionality.

We

finally

show

these

measures

satisfy

these

intuitive

properties.

Compared

to

the

current

proposals,

the

measures

guided

by

these

properties

are

more

succinct

and

intuitive

to

capture

the

degree

of

blame

of

each

formula

for

inconsistency

in

prioritized

knowledge

bases

as

well

as

in

classical

knowledge

bases.

Especially,

the

corresponding

measures

in

the

case

of

classical

knowledge

bases

provide

support

for

the

intuition

about

the

inconsistency

of

minimal

inconsistent

subsets

illustrated

by

the

well

known

Lottery

Paradox

The

rest

of

this

article

is

organized

as

follows.

Some

preliminaries

are

given

in

the

next

Section.

In

Section

we

provide

some

rational

properties

to

characterize

an

intuitive

and

natural

measure

of

the

blames

for

formulas.

In

Section

we

define

a

family

of

natural

measures

of

the

blames

of

formulas

from

minimal

inconsistent

subsets.

In

Section

we

present

a

case

study

to

illustrate

the

application

of

our

approach

to

requirements

engineering.

In

Section

we

compare

our

work

with

related

works.

Finally,

we

conclude

this

article

in

Section

We

have

presented

a

family

of

measures

for

the

blame

of

each

formula

in

a

prioritized

knowledge

base

in

terms

of

minimal

inconsistent

subsets

of

that

prioritized

knowledge

base.

This

article

presented

the

following

contributions

to

measuring

inconsistency

for

a

knowledge

base:.

We

presented

a

set

of

properties

to

characterize

a

desirable

measure

for

the

amount

of

inconsistency

of

a

prioritized

knowledge

base

defined

from

the

minimal

inconsistent

subsets

of

that

base,

including

MinInc

Additivity,

Contradiction,

Monotony

with

significance,

Attenuation

with

the

size

and

Almost

consistency..

Motivated

by

the

principle

of

proportionality,

we

presented

a

set

of

properties

to

characterize

a

desirable

measure

for

the

blame

of

each

formula

in

inconsistency

of

a

minimal

inconsistent

prioritized

knowledge

base,

including

Accumulation,

Innocence,

Necessity,

MinInc

Distribution

and

Proportionality..

We

presented

a

family

of

measures

for

the

blame

of

each

formula

in

inconsistency

of

a

prioritized

knowledge

base

under

guidance

of

the

principle

of

proportionality..

We

showed

that

the

measures

defined

in

this

article

satisfy

all

the

expected

properties..

We

used

a

simple

but

explanatory

example

in

requirements

engineering

to

illustrate

the

potential

application

of

the

measures

for

the

blames

of

formulas

in

inconsistency

presented

in

this

article..

As

pointed

out

earlier,

the

Shapley

inconsistency

value

is

inappropriate

to

model

the

blame

of

formulas

in

inconsistency

of

a

strictly

prioritized

knowledge

base.

How

to

characterize

the

blame

of

each

formula

for

the

inconsistency

of

a

prioritized

knowledge

base

by

using

some

other

appropriate

social

models

will

be

the

main

direction

for

our

future

work.

Measuring

inconsistency

in

knowledge

bases

has

received

considerable

attention

in

computer

science

as

well

as

in

artificial

intelligence

recently.

Many

approaches

have

been

proposed

accordingly.

In

this

article,

we

concentrated

on

prioritized

knowledge

bases

and

presented

a

family

of

measures

for

the

blames

of

formulas

in

inconsistency

of

a

prioritized

knowledge

base

by

using

the

minimal

inconsistent

subsets

of

the

base.

In

the

following,

we

compare

our

measures

with

some

of

closely

related

approaches..

Most

of

the

approaches

proposed

so

far

are

mainly

concerned

with

measuring

inconsistencies

in

knowledge

bases

There

are

relative

few

techniques

for

identifying

the

degree

of

blame

or

responsibility

of

each

formula

for

the

inconsistency

of

a

knowledge

base.

Hunter

and

Konieczny

presented

two

approaches

to

measuring

the

degree

of

blame

of

individual

formulas

in

inconsistency

of

a

classical

knowledge

base

in

and

respectively.

The

first

approach

focuses

on

the

distribution

of

the

measures

of

inconsistency

for

a

whole

knowledge

base

among

formulas

by

using

a

cooperative

game

theory

model,

i.e.

Shapley

inconsistency

value

However,

we

have

argued

that

Shapley

value

is

inappropriate

for

modeling

the

blame

of

each

formula

for

the

inconsistency

of

a

prioritized

knowledge

base.

Moreover,

the

symmetry

property,

one

of

the

four

properties

completely

characterizing

the

Shapley

value,

does

not

hold

in

the

case

of

prioritized

knowledge

bases..

The

second

approach

presented

in

emphasizes

the

importance

of

defining

the

measure

of

the

blames

of

formulas

of

a

knowledge

base

from

the

minimal

inconsistent

subsets

of

that

base

directly.

The

MinInc

inconsistency

values

and

the

scoring

function

can

be

considered

as

the

most

representative

work

of

the

second

approach.

However,

as

pointed

out

in

the

scoring

function

is

very

sketchy

for

measuring

the

blame

of

each

formula,

since

it

does

not

consider

the

size

of

each

minimal

inconsistent

subset..

The

MinInc

inconsistency

value

presented

in

can

be

considered

as

one

of

the

most

representative

proposals

of

the

second

approach.

Our

measures

presented

in

this

article

comply

with

the

second

approach

in

upholding

the

viewpoint

of

minimal

inconsistent

subsets

of

a

knowledge

base

as

the

purest

form

of

inconsistency

in

that

base.

In

this

sense,

these

approaches

presented

in

this

article

may

be

considered

as

an

extension

of

the

second

approach..

However,

the

following

aspects

distinguish

our

measures

from

the

MinInc

inconsistency

value.

First

of

all,

the

MinInc

inconsistency

value

focuses

on

classical

knowledge

bases.

In

contrast,

the

family

of

measures

presented

in

this

article

focuses

on

prioritized

knowledge

bases

as

well

as

classical

knowledge

bases.

Second,

the

MinInc

inconsistency

value

MIV

C

insists

that

the

amount

of

inconsistency

in

a

minimal

inconsistent

knowledge

base

is

shared

equally

among

all

the

formulas

of

that

base.

It

distributes

|M|

to

each

formula

of

a

minimal

inconsistent

subset

M.

In

contrast,

we

combined

the

principle

of

proportionality

and

the

view

of

minimal

inconsistent

subsets

of

a

knowledge

base

as

the

purest

form

of

inconsistency

of

that

base

The

principle

of

proportional

is

central

to

characterize

the

blame

of

each

formula

for

the

inconsistency

of

a

prioritized

knowledge

base.

Roughly,

the

blame

of

each

individual

formula

for

the

inconsistency

is

determined

by

the

opposed

formulas

to

the

formula

in

causing

the

inconsistency.

Third,

we

present

some

more

intuitive

properties

to

characterize

the

measure

of

inconsistency

for

a

prioritized

knowledge

bases

in

terms

of

minimal

inconsistent

subset.

Moreover,

in

the

case

of

classical

knowledge

bases,

a

combination

of

some

(not

all)

new

properties

can

derive

all

the

properties

presented

in

to

capture

a

basic

inconsistency

measure

except

the

property

of

Dominance.

However,

as

illustrated

earlier,

the

property

of

Dominance

does

not

hold

for

the

MI

inconsistency

measure

I

MI

which

was

considered

as

a

basic

inconsistency

measure

in

Also,

it

is

inappropriate

to

describe

the

measures

based

on

minimal

inconsistent

subsets

if

we

consider

the

amount

of

inconsistency

of

a

knowledge

base

as

the

sum

of

the

amounts

of

inconsistency

of

minimal

inconsistent

subsets..

On

the

other

hand,

in

the

case

of

classical

knowledge

bases,

we

use

the

inconsistency

measure

Inc

c

presented

in

to

derive

the

measures

for

the

blame

of

each

formula

guided

by

the

five

properties

The

measure

Inc

c

states

that.

complies

with

the

maximal

η-consistency

presented

by

Knight

in

upholding

the

intuition

that

bigger

size

means

a

smaller

amount

of

the

inconsistency.

According

to

the

inconsistency

measure

of

maximal

η-consistency,

a

minimal

inconsistent

classical

knowledge

base

of

n

(n

size

is

maximally

n

-consistent.

Note

that

the

inconsistency

measures

for

prioritized

knowledge

bases

presented

in

this

article

take

the

priority

level

of

formulas

into

account.

In

some

sense,

these

measures

consider

the

significance

or

strength

of

inconsistency

as

well

as

the

degree

of

inconsistency.

In

particular,

in

the

case

of

Type-I

prioritized

knowledge

bases,

Inc

max

and

Inc

mean

agree

with

the

significance

functions

for

quasipossibilistic

logic

presented

in

in

taking

the

strength

of

conflicts

on

literals

into

account.

However,

the

significance

functions

defined

in

are

model-based

measures,

i.e.

these

were

built

upon

a

mass

assignment

over

the

set

of

(quasi-possibilistic)

models.

In

contrast,

Inc

max

and

Inc

mean

were

defined

from

the

priority

level

of

formulas

as

well

as

the

minimal

inconsistent

subsets

directly,

that

is,

there

are

syntax-based

measures.

Moreover,

these

measures

are

characterized

by

properties

in

terms

of

minimal

inconsistent

subsets.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O B I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B I B I I B I I B I I I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O B I I O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I I O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O B I I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I I I O O O O B I I O O O O O O O O O O O O O O O O O B I I I I I O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O B I I


A study on workload-aware wavelet synopses for point and range-sum queries In this paper we perform an extensive theoretical and experimental study on common synopsis construction algorithms, with emphasis on wavelet based techniques, that take under consideration query workload statistics. Our goal is to compare, "expensive" quadratic time algorithms with "cheap" near-linear time algorithms, particularly when the latter are not optimal and/or not workload-aware for the problem at hand. Further, we present the first known algorithm for constructing wavelet synopses for a special class of range-sum query workloads. Our experimental results, clearly justify the necessity for designing workload-aware algorithms, especially in the case of range-sum queries. Compact synopses have proven very popular, recently, as an effective means of dealing with massive multi-dimensional data-sets, such as those typically encountered in modern data intensive scenarios, e.g. OLAP queries. Such synopses serve two goals. First, synopses can assist in the query optimization process by providing highly accurate selectivity estimates, as they efficiently summarize the joint data distribution. Second, synopses can be used instead of the actual data providing with approximate answers to large costly range aggregate queries. Indeed, it can be argued that users, such as data analysts, exhibit exploratory behavior: they pose complex queries over large parts of the data, which would require considerable resources to process, and yet, they can tolerate some imprecision in the answers as they try to identify/extract interesting patterns, as long as these approximate answers are fast and are accompanied with some error guarantees.. In particular, summarization techniques that take user behavior under consideration are very common. User behavior is expressed in terms of workload statistics collected by logging the queries posed to the system. The argument here is that these statistics accurately predict future behavior and thus, can be used to identify "hot" areas of the data. The synopsis construction algorithm is then designed so as to pay closer attention to these hot areas by providing increased accuracy, at the expense of less interesting areas. We refer to such algorithms as being workload-aware.. Among the most popular data summarization techniques proposed in literature are space partitioning approaches, such as histograms and wavelet synopses. Briefly, for a domain of size N , a histogram construction algorithm searches for a partition of the domain space in B arbitrarily sized buckets and approximates all data inside a bucket by a single value; the optimal partition is chosen so as to minimize some error metric, while not exceeding the available budget B. On the other hand, the wavelet decomposition employs a fixed, data-independent hierarchical partitioning of the domain space (better known as a linear transformation) into N intervals, each having a single value, a coefficient, associated with; the wavelet synopsis algorithm optimizes for some error metric by choosing to maintain only B of the resulting coefficients, in accordance to a space budget. The emphasis of this study is on space partitioning approaches and, in particular, on workload-aware wavelet synopsis construction algorithms.. Various algorithms for constructing wavelet synopses can be found in the database research literature; some of them can also be tuned to become workload-aware. These algorithms try to minimize the aggregate error incurred when all possible point, or equality, queries are considered, given a space budget. Interestingly enough, their algorithmic requirements range from (pseudo-)linear in domain size 1 and budget up to quadratic in domain size. A natural question that arises is how "cheap", linear algorithms fare against "expensive", quadratic algorithms in terms of the accuracy achieved by the resulting synopsis.. Surprisingly, the problem of minimizing the aggregate error when range queries are involved has attracted little attention. Clearly, workload-aware algorithms should take into consideration such queries, in order to be more applicable to real-life scenarios. In this work, we propose a novel workload-aware algorithm, RangeWave, that optimizes for a special class of hierarchical range queries. To the best of our knowledge, this is the first treatment for range queries given workload statistics. Our Contribution Our study is focused primarily on wavelet synopsis construction algorithms and presents the following contributions:. 1. We provide a detailed theoretical presentation of the time and space complexities associated with the most common algorithms found in literature, with particular interest in the algorithms that can become workload aware. In the extensive experimental study, our goal is to seek for the algorithm that offers the optimal compromise in terms of achieved accuracy compared to its running time overhead. For the sake of comparison, we also include in our study the optimal histogram construction algorithm. 2. We present a novel workload-aware synopsis construction algorithm, RangeWave, for a special class of rangesum queries. We experimentally validate its increased accuracy and further investigate the performance overhead it introduces, against previous non workload-aware wavelet techniques. Histograms have a long tradition as an effective synopsis technique and are, thus, used widely in commercial systems. The bulk of early research focused on heuristically minimizing approximation errors in some estimation problem with the validity of the heuristics shown only experimentally. The interested reader can refer to the work of Ioannidis [9] for detailed discussion. A special class of heuristic algorithms is those where the histogram is constructed solely by query answer feedback; these include self-tuning, or workload-aware histograms [1, 2] .. The first work to tackle the problem of constructing an optimal histogram from an algorithmic standpoint is [10] . The authors make the observation that once the optimal bucket boundaries have been found the value to assign to each bucket can be independently derived. Their optimization is for the sum-squared-error of point queries, yet extensions to other metrics such as weighted variations and maximum absolute/relative error are possible, albeit only for point queries. In the case of aggregating range-sum errors, the pioneering work of [12] dealt with workloads of hierarchical range queries, minimizing the (weighted) sumsquared-error. For arbitrary range queries approximation algorithms were provided in [16] , assuming, however, uniform workload distribution.. The wavelet decomposition has been applied successfully as a data reduction mechanism in a wide variety of applications. To name a few, wavelets have been used in answering range-sum aggregate queries over data cubes [19, 18] and in selectivity estimation [14] . The effectiveness of Haar wavelets as a general-purpose approximate query processing tool was demonstrated in [3] . For the case of data sets with multiple measures, the authors in [4] introduce the notion of extended wavelets, while I/O efficient algorithms for common maintenance tasks were presented in [11] .. A lot of recent work focuses on developing optimal algorithms for constructing wavelet synopses that minimize more meaningful than the sum-squared-error metrics, taking into consideration workload statistics. The work in [5] constructs wavelet synopses that probabilistically minimize the maximum relative or absolute error incurred for reconstructing any data value. The work in [6] showed that this is also possible for deterministically constructed synopses and provided a novel dynamic programming recurrence, applicable [7] to incorporating workload statistics. Similar ideas were employed in [15] to construct optimal synopses in nearquadratic time for a particular class of workload statistics. Further, the work in [8] provides with improvements to the space requirements of the aforementioned dynamic programming algorithms. For the same problem, that of constructing an optimal workload-aware wavelet synopsis, the work in [17] constructs a novel workload-dependant Haar-like basis by changing the decomposition process so that the conventional greedy thresholding technique applies. Assuming a uniform workload, the authors in [13] proved that the heuristics employed in [14] were in fact optimal -however, their approach requires the construction of the prefix-sum array/cube.. Let us note that various approximation schemes exist for fast construction of histograms and wavelet synopses, better suited for low-resource settings such as data stream environments. Such schemes, however, are beyond the scope of this study, where we primarily focus on optimal workload-aware wavelet synopses.

O O O B B O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B B O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B B B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B B O


On expander codes Abstract-Sipser and Spielman have introduced a constructive family of asymptotically good linear error-correcting codes-expander codes-together with a simple parallel algorithm that will always remove a constant fraction of errors. We introduce a variation on their decoding algorithm that, with no extra cost in complexity, provably corrects up to 12 times more errors. In [1] , Sipser and Spielman introduced a constructive family of asymptotically good linear error-correcting codes together with a simple parallel algorithm that will always remove a constant fraction of errors.. More precisely, they prove [1, Theorem 19] .. Theorem 1 (Sipser-Spielman) : For all 0 such that 102H( 0 )>0;. where H (1) is the binary entropy function, there exists a polynomial-time constructible family of expander codes of rate 1 0 2H ( 0 ) and minimum relative distance arbitrarily close to 2 0 in which any < 2 0 =48 fraction of error can be corrected by a circuit of size O(N log N) and depth O(log N) where N is the length of the code.. In one of the two open questions of [1] , Sipser and Spielman ask whether one can obtain better constants from the construction of the above theorem.. To obtain such an improvement we shall introduce a variation on their decoding algorithm. It will enable us to replace in Theorem 1 the correction of "any < 2 0 =48" fraction of error by "any < 2 0 =4.". Formally, Theorem 1 is improved to the following.. Theorem 1A: For all 0 such that 102H(0) > 0,where H(1) is the binary entropy function, there exists a polynomial-time constructible family of expander codes of rate 1 0 2H( 0 ) and minimum relative distance arbitrarily close to 2 0 in which any < 2 0 =4 fraction of error can be corrected by a circuit of size O(N log N) and depth O(log N).

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


PARTIC: Power-Aware Response Time Control for Virtualized Web Servers Abstract-Both power and performance are important concerns for enterprise data centers. While various management strategies have been developed to effectively reduce server power consumption by transitioning hardware components to lower power states, they cannot be directly applied to today's data centers that rely on virtualization technologies. Virtual machines running on the same physical server are correlated because the state transition of any hardware component will affect the application performance of all the virtual machines. As a result, reducing power solely based on the performance level of one virtual machine may cause another to violate its performance specification. This paper proposes PARTIC, a two-layer control architecture designed based on well-established control theory. The primary control loop adopts a multi-input multi-output control approach to maintain load balancing among all virtual machines so that they can have approximately the same performance level relative to their allowed peak values. The secondary performance control loop then manipulates CPU frequency for power efficiency based on the uniform performance level achieved by the primary loop. Empirical results demonstrate that PARTIC can effectively reduce server power consumption while achieving required application-level performance for virtualized enterprise servers. In this paper, we have presented PARTIC, a two-layer control architecture based on well-established control theory. The primary control loop adopts an MIMO control strategy to maintain load balancing among all virtual machines so that they can have approximately the same relative response time. The secondary performance control loop then manipulates CPU frequency for power efficiency based on the uniform relative response time achieved by the primary loop. Empirical results demonstrate that our control solution can effectively reduce server power consumption while achieving required response times for all virtual machines. Xiaoyun Zhu received the dual BS degree in automation and applied mathematics from Tsinghua University in 1994 and the PhD degree in electrical engineering from California Institute of Technology in 2000. She is a staff engineer at VMware, Inc., specializing in resource management for virtualized data centers and cloud computing environments. Her general interests are in applying control theory, optimization, algorithms, statistical analysis and simulation to IT systems and services management. She has coauthored more than 40 refereed papers in journals and conference proceedings. She holds five patents and has more than 20 pending patent applications. She has been a program committee member for a number of leading management conferences, including the IM, NOMS, DSOM, and ICAC, and an associated editor for the American Control Conference.. . For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib. Control theory has been applied to manage applicationlevel performance metrics for virtualized computer servers. Padala et al. [15] propose a two-layered architecture to control resource utilization and response time. Xu et al. [16] use predictive control for dynamic resource allocation in enterprise data centers. Liu et al. [17] control the mean response time of individual applications by adjusting the percentage of allocated CPU entitlements. Zhang et al. [18] adjust the resource demands of virtual machines for fairness and efficiency. Wang et al. [19] present an adaptive controller for more robust performance. However, all the aforementioned works are not designed to reduce power consumption for virtualized servers.. Some prior work has been proposed to use power as a tool for guaranteeing required application-level SLOs in a variety of heuristic-based ways, such as using approximation algorithms [20] . More closely related to this paper, Horvath et al. [21] use dynamic voltage scaling (DVS) to control end-to-end delay in multitier web servers. Zhu and Mueller [7] develop a feedback control scheduling algorithm using DVS. Sharma et al. [5] effectively apply control theory to control application-level quality of service requirements. Chen et al. [6] present a feedback controller to manage the response time in a server cluster. Bertini et al. [22] also use SISO control to achieve energy efficiency. Although they all use control theory to manage system performance while reducing power consumption, they cannot be directly applied to virtualized computing environments because multiple application servers are correlated due to shared hardware resource. A recent work [23] discusses a heuristic solution to power control in virtualized environments. In contrast to their work, we rely on rigorous control theory for systematically developing control strategies with analytic assurance of control accuracy and system stability.. Several research projects [24] , [25] , [26] have successfully applied control theory to explicitly control power or cooling of computing servers. Lefurgy et al. [14] have shown that control-theoretic solution outperforms a commonly used heuristic-based solution by having more accurate power control and less overhead. Ranganathan et al. [27] propose a negotiation-based algorithm to allocate power budget to different servers in a blade enclosure for better performance. Wang and Chen [28] develop an MIMO control algorithm for cluster-level power control. Those projects are different from our work because they control power consumption only, and thus, cannot provide guarantees for application-level performance. They also are not designed for virtualized enterprise servers.. Various admission control algorithms have been presented in related work to address server overload. Some of them adopt control theory as the foundation of their algorithm design, such as PI control [29] , fuzzy control [30] , and nonlinear control [31] . In addition, Elnikety et al. [11] have introduced an advanced scheduling algorithm for admission control in multiple-tier e-commerce websites. These algorithms are complementary to PARTIC and can be easily integrated into our control architecture.. Control-theoretic approaches have also been applied to other computing systems. A survey of feedback performance control in various computing systems is presented in [32] . Feedback control scheduling algorithms have been developed for real-time systems and applications [33] , [34] . Control techniques have also been applied to digital control applications [35] , data service and storage systems [36] , and caching services [37] . While most existing works are based on simpler SISO control strategies, Diao et al. develop MIMO control algorithms to control the processor and memory utilizations for Apache web servers [38] and to achieve load balancing for database servers [39] . In our work, we develop a two-layer control architecture to provide real-time guarantees for virtualized web servers and reduce power consumption to the maximum degree.

O O B I O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O


Advance Bandwidth Scheduling Algorithms in Dedicated Networks An increasing number of high performance research network testbeds and production networks have the capability of provisioning dedicated channels for high speed data transfer in support of large scale scientific applications. Each dedicated channel in these networks typically consists of one or more physical links that are shared by multiple applications in both time and bandwidth through in advance reservation. The design of efficient bandwidth scheduling algorithms is critical to maximizing the utilization of dedicated network resources and meeting diverse end-to-end transport performance requirements.The topology of a dedicated network is represented by a graph, where each link maintains a list of residual bandwidths specified as segmented constant functions of time. Given a graph with an aggregated time-bandwidth table combining the reservation information on all links, designated source, and destination vertices, and a specified data size, we formulate and investigate five bandwidth scheduling problems to minimize the data transfer end time under different path and bandwidth constraints:(i) variable path with variable bandwidth (VPVB), which computes the widest (highest bandwidth) path in each time slot; (ii) fixed path with variable bandwidth (FPVB), which computes a fixed path with varying bandwidths across different time slots; (iii) variable path with fixed bandwidth (VPFB), which computes one path in each time slot with the same bandwidth; (iv) fixed path with fixed bandwidth (FPFB), which computes a fixed path with a constant bandwidth; and (v) multiple fixed paths with fixed bandwidths (MFPFB), which computes multiple concurrent fixed paths with constant bandwidths.Among these problems, VPVB represents the most flexible scenario where the network resources are fully utilized and the minimal transfer end time is achieved, while FPFB imposes the most stringent transport conditions by fixing both path and bandwidth.We design an optimal algorithm for each of these scheduling problems. VPVB is solved using an extension of the classical Dijkstra's algorithm, and FPVB is solved using Maximum Permutation Algorithm, which tries all possible permutations of throughputs at different time slots to obtain the minimal transfer end time. The algorithm for VPFB, FPFB, and MFPFB are based on Adjacent Time Slots Search, which finds the minimum transfer end time by examining the possibility of data transfer in any number of contiguous time slots. All these algorithms are of polynomial or pseudo polynomial time complexity with respect to the network size and the total number of time slots in a bandwidth reservation table.

O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


A sparse preconditioned iterative method for vibration analysis of geometrically mistuned bladed disks Vladislav Ganine, Mathias Legrand, Hannah Michalska, Christophe Pierre. A sparse preconditioned iterative method for vibration analysis of geometrically mistuned bladed disks. This paper addresses the problem of quantifying and predicting forced response of geometrically mistuned rotors by building compact ROM in a computationally efficient way. First we have analyzed the behavior of SMC method revealing that it is closely related to the generalized Davidson algorithm. There are all indications that from the memory efficiency and accuracy point of view it is a good choice for the moderate order FEM models under low rank localized perturbation if narrow clustered areas of spectrum are analyzed. For very large scale industrial models as well as for the areas of spectrum where multiple mode families interact, a new method is proposed. It stems from the Jacobi-Davidson algorithm implementing a number of simple heuristic strategies based on the block-circulant structure of the nominal system and assumptions on perturbation. In particular, a number of typical industrial applications of ROM are considered such as manufacturing imprecision, erosion or foreign objects damage event that constitute rather local low rank high amplitude perturbation destroying symmetry. A set of numerical experiments have been conducted on an industrial bladed disk model. In its current simple form the implementation of the method demonstrates promising numerical results. Our experience indicates that the algorithm combined with proposed preconditioning scheme routinely converges to the perturbed interior eigenspace within reasonable time, provided that the perturbation is localized to a few sectors. However this work is preliminary, and more research is needed, both to further develop the method and to study the extension of these results to the case of high rank low magnitude perturbation that cannot be effectively handled by existing "small mistuning" ROM. On the other hand, the new method may replace or complement the SMC in the algorithm proposed in [10] to study the combined effect of "large" deterministic mistuning and "small" random mistuning on bladed disks.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O


A non-topological view of dcpos as convergence spaces The category TOP of topological spaces is not cartesian closed, but can be embedded into the cartesian closed category CONV of convergence spaces. It is well known that the category DCPO of dcpos and Scott continuous functions can be embedded into TOP, and so into CONV, by considering the Scott topology. We propose a di erent, "cotopological" embedding of DCPO into CONV, which, in contrast to the topological embedding, preserves products. If X is a cotopological dcpo, i.e. a dcpo with the cotopological CONV-structure, and Y is a topological space, then [X → Y ] is again topological, and conversely, if X is a topological space, and Y a cotopological complete lattice, then [X → Y ] is again a cotopological complete lattice. For a dcpo D, the topological and the cotopological convergence structures coincide if and only if D is a continuous dcpo. Moreover, cotopological dcpos still enjoy some of the properties which characterise continuous dcpos. For instance, all cotopological complete lattices are injective spaces (in CONV) w.r.t. topological subspace embeddings. It is well known that the category DCPO of dcpos and Scott continuous functions can be embedded into TOP, the category of topological spaces and continuous functions, by endowing each dcpo D with its Scott topology, leading to the topological space D s . This embedding hinges on the fact that a function between dcpos is Scott continuous This embedding provides a useful way to look at dcpos as topological spaces, yet it has its drawbacks. For instance, it does not preserve products, i.e., the Scott topology of a product dcpo is not necessarily the same as the product topology derived from the two Scott topologies (in short, (D × E) s = D s × E s does not generally hold); see the discussion in [5, p. 106 ]. There are even complete lattices. Connected with this product problem is a problem about binary joins in complete lattices. Binary join ∨ : L × L → L is obviously Scott continuous, and therefore continuous in the sense (L × L) s → L s . Yet, it is not always continuous in the proper topological sense, i.e., as a function. There is a similar problem with pointwise join of functions. While the pointwise join of a directed set of continuous functions is continuous again, this does not hold for the pointwise join of two functions: there are continuous functions f; g : X → L s such that their pointwise join f ∨ g : X → L s is not continuous (in [8, 9] , we had to work around this problem by restricting attention to those X where f ∨ g is continuous again).. A concrete example where all these problems occur is the complete lattice L constructed in [13] as an example of a complete lattice which is not sober in its Scott topology. The problems listed above are not very well known because they do not occur for continuous dcpos (cf. II-4.12 and II-4.13 in the Compendium [5] ). Yet they can be avoided altogether by considering a di erent embedding of DCPO into a topological category-not quite TOP itself, but the larger category CONV of convergence spaces [18] (also known as ÿlter spaces [12] ), whose objects are characterised by the convergence properties of ÿlters.. Every topological space carries a notion of ÿlter convergence which leads to an embedding of TOP as a re ective full subcategory into CONV. Moreover, CONV is cartesian closed in contrast to TOP, i.e., it provides a function space construction such that [X × Y → Z] and [X → [Y → Z]] are naturally isomorphic, and -calculus can be interpreted in the category.. In this paper, we propose a new embedding (−) c of DCPO into CONV, which, in contrast to the topological embedding (−) s , preserves products and avoids all the prob- The convergence spaces D c , which we call cotopological dcpos, exhibit an interesting behaviour in the function space construction:. • If X = D c is a cotopological dcpo and Y is topological, then [X → Y ] is topological.. • If X is topological and Y = L c is a cotopological complete lattice, then [X → Y ] is a cotopological complete lattice again. These properties were the reason for choosing the name "cotopological".. As indicated above, a dcpo D is continuous i D c is topological, or shortly, continuous = topological+cotopological. Indeed, the cotopological dcpos (lattices) still enjoy many properties familiar from continuous dcpos (lattices). For instance, it is well known that continuous lattices are injective spaces w.r.t. topological embeddings [5, . Here, we show that L c is injective w.r.t. topological embeddings for any complete lattice L whatsoever.. We start out by a quick recap of ÿlters (Section 2) and convergence spaces (Section 3). There is not much new in there, and most proofs are omitted. In Section 4, we rule out some ugly convergence spaces by imposing certain "niceness conditions" which are obeyed by topological spaces and preserved by product, subspace, and exponentiation. Then we consider d-spaces in Section 5, which are spaces whose structure is similar to that of dcpos. In Section 6, D s is identiÿed as the strongest topological d-space structure on D, while D c is introduced as the strongest d-space structure of all. The ÿnal, quite large Section 7 is devoted to prove the main properties of cotopological dcpos (or lattices), i.e., the properties that have been presented in this introduction, and a few more.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O


Enhanced Qualitative Probabilistic Networks for Resolving Trade-offs Qualitative probabilistic networks were designed to overcome, to at least some extent, the quantification problem known to probabilistic networks. Qualitative networks abstract from the numerical probabilities of their quantitative counterparts by using signs to summarise the probabilistic influences between their variables. One of the major drawbacks of these qualitative abstractions, however, is the coarse level of representation detail that does not provide for indicating strengths of influences. As a result, the trade-offs modelled in a network often remain unresolved upon inference. We present an enhanced formalism of qualitative probabilistic networks to provide for a finer level of representation detail. An enhanced qualitative probabilistic network differs from a basic qualitative network in that it distinguishes between strong and weak influences. Now, if a strong influence is combined, upon inference, with a conflicting weak influence, the sign of the net influence may be readily determined. Enhanced qualitative networks are purely qualitative in nature, as basic qualitative networks are, yet allow for resolving more trade-offs upon inference. The formalism of probabilistic networks introduced in the 1980s [20] , is an intuitively appealing formalism for capturing knowledge of complex problem domains along with the uncertainties involved. Associated with the formalism are powerful algorithms for reasoning with uncertainty in a mathematically correct way; these algorithms allow for causal reasoning, diagnostic reasoning as well as case-specific reasoning. Applications of probabilistic networks can be found in areas such as (medical) diagnosis and prognosis, planning, monitoring, vision, and information retrieval (see, for example, [1, 2, 3, 4, 15, 25] ).. A probabilistic network basically is a concise representation of a joint probability distribution on a set of statistical variables. It consists of an acyclic directed graph encoding the relevant variables from a domain of application along with their probabilistic interrelationships. Associated with each variable is a set of conditional probability distributions describing the relationship of the variable with its predecessors in the graph. The first task in constructing a probabilistic network is to identify the important domain variables, their values, and their interdependencies. This knowledge is then modelled in a directed graph, referred to as the network's qualitative part. The final task is to obtain the probabilities that constitute the network's quantitative part. As (conditional) probabilities are required for each variable in the graph, their numbers can be quite large, even for small applications. While the construction of the qualitative part of a probabilistic network is generally considered feasible, its quantification is a far harder task. Probabilistic information available from literature or data is often insufficient or unusable, and domain experts have to be relied upon to assess the required 1 INTRODUCTION 2 probabilities [12] . Unfortunately, experts are often uncomfortable with having to provide probabilities. Moreover, the problems of bias encountered when directly eliciting probabilities from experts are widely known [14] . The usually large number of probabilities required for a probabilistic network, as a consequence, tends to pose a major obstacle to their application [12, 13] .. To mitigate the quantification bottleneck to at least some extent, qualitative probabilistic networks have been introduced [26] . Qualitative networks in essence are qualitative abstractions of probabilistic networks. Like a probabilistic network, a qualitative network encodes variables and the probabilistic relationships between them in a directed graph. However, while the relationships between the represented variables are quantified by conditional probabilities in a probabilistic network, these relationships are summarised in its qualitative abstraction by qualitative signs capturing stochastic dominance. For reasoning with a qualitative probabilistic network, an efficient algorithm is available, based on the idea of propagating and combining these signs [9] .. Qualitative probabilistic networks, by their nature, have a coarse level of representation detail. Influential relationships between variables can be modelled as positive, negative, zero or ambiguous, but no indication of their strengths can be provided as in a quantified network. One of the major drawbacks of this coarse level of representation detail is the ease with which the ambiguous '?'-sign arises upon inference. Ambiguous signs typically arise from trade-offs. A qualitative network models a trade-off if two nodes in the network's digraph are connected by multiple parallel reasoning chains with conflicting signs. In the absence of a notion of strength of influences, qualitative networks do not provide for resolving such trade-offs. Inference with a qualitative network for a real-life domain of application, as a consequence, often introduces ambiguous signs. Moreover, once an ambiguous sign has been generated, it will spread throughout major parts of the network. Although not incorrect, ambiguous signs provide no information whatsoever about the influence of one variable on another and are therefore not very useful in practice.. Ambiguous results from inference can be averted by enhancing the formalism of qualitative probabilistic networks to provide for a finer level of representation detail. Roughly speaking, the finer the level of detail, the more trade-offs can be resolved during inference. The finer levels of detail, however, typically come at the price of a higher computational complexity of inference. The problem of trade-off resolution for qualitative networks has been addressed by various researchers. S. Parsons has introduced, for example, the concept of categorical influence, which is either an influence that serves to increase a probability to 1, or an influence that decreases a probability to 0, and thus serves to resolve any trade-off in which it is involved [19] . Parsons has also studied the use of the less well-defined order-of-magnitude reasoning [19] . C.-L. Liu and M.P. Wellman have designed two methods for resolving trade-offs based upon the idea of reverting to numerical probabilities whenever necessary [17] . While only some trade-offs can be resolved by the use of categorical influences, the methods of Liu and Wellman provide for resolving any trade-off, but require a fully quantified probabilistic network.. To provide for trade-off resolution without resorting to numerical probabilities, we have designed an intuitively appealing formalism of enhanced qualitative networks. An enhanced qualitative probabilistic network differs from a basic qualitative network in that it introduces a notion of relative strength by distinguishing between strong and weak influences. If a trade-off is modelled in an enhanced network and the positive influence, for example, is known to be stronger than the conflicting negative one, we may upon inference conclude the net influence to be positive. Trade-off resolution during inference thus builds upon the idea that strong influences dominate over conflicting weak influences. To provide for inference with an enhanced network, we have generalised the sign-propagation algorithm for basic qualitative networks to deal with strong and weak influences. This generalisation is rather straightforward and derives from the observation that the properties upon which the basic sign-propagation algorithm is based are also provided for in an enhanced network. The new infer-3 ence algorithm takes into account that the effect of one variable on another diminishes as variables are further apart in the network's graph; it also takes into account that a variable may affect another variable along multiple pathways with differing strengths. To maintain the correct strengths of indirect influences, the algorithm has to do some bookkeeping, as a result of which it is intuitively less appealing than the inference algorithm for basic qualitative networks. The distinction between strong and weak influences in an enhanced network, however, is very intuitive and domain experts should have no problems providing and interpreting the associated signs.. The paper is organised as follows. In Section 2, we provide some preliminaries from the fields of probabilistic networks and qualitative networks to introduce our notational conventions. In Section 3, we present our new formalism of enhanced qualitative probabilistic networks. In Section 4, we detail various properties of enhanced networks, on which we build two alternative sign-propagation algorithms. Section 5 provides an example of inference with an enhanced qualitative probabilistic network and discusses some complexity issues concerning the different propagation algorithms. Related work is reviewed in Section 6. The paper is rounded off with our conclusions and directions for future research in Section 7. The problem of trade-off resolution within the framework of qualitative networks has been addressed before by different researchers. In this section we briefly review this related work.. S. Parsons introduced the concept of categorical influence [19] . A categorical influence is a qualitative influence that serves either to increase a probability to 1 or to decrease a probability to 0, disregarding all other influences. For example, a positive categorical influence S [++] (A, B) of a variable A on a variable B is defined as Pr(c | ax) = 1 for all relevant variables X. A categorical influence thus serves to resolve any trade-off in which it is involved, but can only capture deterministic relationships between nodes; in real-life applications few to none of such relationships will exist. In addition, this extension requires new signs for indicating the increase to 1 and decrease to 0, respectively, and the incorporation of these new signs in the tables for the ⊗-and ⊕-operators.. Parsons also studied the use of both relative and absolute order-of-magnitude reasoning in the context of qualitative probabilistic networks [19] . Relative orders of magnitude can be used to relate different qualitative influences to each other. Using the relative order-of-magnitude system ROM[K] [6] , one qualitative influence can be specified as being, respectively, negligible with respect to, distant from, comparable to, or close to another influence. The use of relative orders of magnitude thus serves to relate the strengths of different influences, but it requires the specification of a relation between all pairs of influences, instead of a notion of strength per influence. In addition, the relations used seem to be ill-defined, which makes reasoning with them anything but intuitive. For absolute order-of-magnitude reasoning, Parsons proposes a method that revolves around the propagation of abstract intervals between −1 and 1, that correspond to labels like 'Strongly Positive', 'Weakly Positive', etc. Two different sets of labels are required: one for modelling influences that are associated with the arcs in the network's digraph, and one for modelling changes that occur at the nodes in the graph (comparable to 'node-signs'). The intervals corresponding to a set of labels do not overlap and together span the interval [−1, 1]. The boundaries of the intervals, however, are not actually quantified, but set to be α, β, etc.; this approach is therefore comparable to our treatment of the cut-off value. Probabilistic inference is based on propagating and combining the abstract intervals; the interval comparisons required to this end are done using ≥ int , where [α, β] ≥ int [γ, δ] iff α ≥ γ and β ≥ δ. Note that if one interval is considered larger than another with this operator, then they may in fact overlap. To prevent considerable loss of information, assumptions about the actual values of the interval boundaries have to be made.. κ-calculus [24] can be considered another absolute order-of-magnitude system. Using a probabilistic interpretation of the κ-calculus, probabilities can be abstracted to κ-values, where a κ-value of n indicates that the associated probability has the same order of magnitude as n for some infinitesimal 38 number [7] . Drawbacks of the use of these κ-values are that they are not concerned with changes in probabilities, but rather with the probabilities themselves, and that the probabilistic interpretation is suitable for infinitesimal probabilities only. The former problem can be lifted as an absolute change in probability is a number between zero and one and can therefore be abstracted to a κ-value. This principle has more recently been used in another approach to enhance the expressiveness of qualitative probabilistic networks [23] . With this approach, an interval of κ-values is associated with the sign of an influence to capture its possible strengths. These κ-intervals are propagated along with the qualitative network's signs. Propagation results, however, are only guaranteed to be correct for infinitesimal probabilities. Finally, the definition of a κ-value is not very intuitive and such values are therefore hard for domain experts to specify and interpret.. Categorical influences, order-of-magnitude reasoning and κ-calculus are of a purely qualitative nature, yet serve for resolving some trade-offs. C.-L. Liu and M.P. Wellman designed methods for resolving trade-offs based upon the idea of reverting to numerical probabilities whenever necessary [17] . They propose to reason with a probabilistic network in a qualitative way, thereby exploiting the efficiency of sign-propagation, and only reverting to the full quantification whenever a trade-off leads to an ambiguous result. Two methods are described for resolving the trade-off. The first method provides for incrementally applying numeric inference to the point where qualitative reasoning can produce a decisive result. That is, a trade-off between two variables is resolved numerically and then abstracted into a net qualitative influence between the two variables. The second method amounts to estimating bounds on the net influence along the trails that give rise to a trade-off. These bounds are then again used to compute the qualitative sign of the net influence. The methods presented by Liu and Wellman resolve any trade-off present in the network, but require a fully specified, numerical probabilistic network.. We would like to mention that some other approaches to dealing with uncertainty in a qualitative way have been proposed in the literature. As these approaches are not tailored for use within the framework of qualitative probabilistic networks, we do not review them here.

O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O