Union

types

for

object-oriented

programming

We

propose

union

types

for

statically

typed

class-based

objectoriented

languages

as

a

means

to

enhance

the

flexibility

of

subtyping.

As

its

name

suggests,

a

union

type

can

be

considered

a

set

union

of

instances

of

several

types

and

behaves

as

their

least

common

supertype.

It

also

plays

the

role

of

an

interface

that

'factors

out'

commonality

of

given

typesfields

of

the

same

name

and

methods

with

similar

signatures.

Union

types

can

be

useful

for

implementing

heterogeneous

collections

and

for

grouping

independently

developed

classes

with

similar

interfaces,

which

has

been

considered

difficult

in

languages

like

Java.

To

rigorously

show

the

safety

of

union

types,

we

formalize

them

on

top

of

Featherweight

Java

and

prove

that

the

type

system

is

sound.

Background.

The

design

of

good,

reusable

class

libraries

is

known

to

be

a

very

hard

problem

and,

in

mainstream

objectoriented

languages

like

Java

and

C++,

inheritance

and

subPermission

to

make

digital

or

hard

copies

of

all

or

part

of

this

work

for

personal

or

classroom

use

is

granted

without

fee

provided

that

copies

are

not

made

or

distributed

for

profit

or

commercial

advantage

and

that

copies

bear

this

notice

and

the

full

citation

on

the

first

page.

To

copy

otherwise,

to

republish,

to

post

on

servers

or

to

redistribute

to

lists,

requires

prior

specific

permission

and/or

a

fee.

typing

(and,

more

recently,

generics)

have

been

used

as

main

mechanisms

to

promote

code

reuse.

While

inheritance

enables

one

class

to

reuse

implementation

(declarations

of

instance

variables

and

methods)

of

another

class,

subtyping

is

for

substitutability-the

property

that

if

an

object

of

one

type

can

be

used

at

a

certain

place,

then

another

object

of

a

subtype

can

be

used

at

the

same

place,

too.

(Substitutability

may

be

rephrased

as

reusability

of

contexts,

in

the

sense

that

if

some

context

is

applicable

to

an

object

of

one

type

then

the

same

context

is

also

applicable

to

any

object

of

its

subtype.)

Thus,

design

concerns

about

inheritance

and

subtyping

relations

are

somewhat

different:

it

has

to

be

taken

into

account,

for

inheritance,

how

new

classes

may

reuse

existing

implementation

and,

for

subtyping,

how

objects

may

be

used

in

client

code..

In

mainstream

languages,

however,

subtyping

relation

is

mostly

based

on

inheritance

relation

It

can

happen

that

two

classes

used

in

similar

contexts

but

with

rather

different

implementations

are

placed

apart

in

the

class

(inheritance)

hierarchy,

resulting

in

no

useful

supertype

of

those

classes.

Interfaces

(as

a

programming

construct)

in

Java

are

a

solution

to

this

problem:

one

can

define

a

super-interface

of

classes

of

similar

use,

regardless

of

a

given

inheritance

hierarchy,

and

enjoy

benefits

of

subtyping.

However,

interfaces

cannot

be

added

once

a

class

is

defined,

so

library

designers

still

have

to

do

a

lot

of

planning

of

their

interface

hierarchies

before

the

library

is

shipped.

This

problem

has

been

considered

a

significant

limitation

of

type

systems

with

declaration-based

subtyping,

as

in

Java..

Our

Proposal-Union

Types.

In

this

paper,

we

propose

union

types

to

partially

address

the

problem

of

the

inability

of

adding

supertypes

to

existing

types

(classes

and

interfaces).

As

its

name

suggests,

a

union

type

denotes

a

set

union

of

given

some

types

(viewed

as

sets

of

instances

that

belong

to

those

types)

and

behave

as

a

least

common

supertype

of

them.

Since

union

types

are

composed

from

existing

types,

they

give

an

ability

to

define

a

supertype

even

after

a

class

hierarchy

is

fixed.

Union

types

can

be

used

not

only

by

case

analysis

as

in

ML

datatypes,

but

also

by

direct

member

access

as

ordinary

types.

In

fact,

given

some

types,

their

union

type

can

be

viewed

as

an

interface

that

'factors

out'

their

common

features,

that

is,

the

fields

of

the

same

name

and

methods

with

similar

signatures..

We

expect

that

union

types

can

be

useful

for

grouping

independently

developed

classes

with

similar

interfaces,

by

giving

their

supertype,

and

for

implementing

heterogeneous

collections

like

lists

where,

say,

strings

and

integers

are

mixed

as

elements..

Our

contributions

in

this

paper

can

be

summarized

as

follows:.

The

proposal

of

union

types

for

class-based

objectoriented

languages

with

a

name-based

type

system;.

A

formalization

of

a

core

object-oriented

language

FJ∨

with

union

types

on

top

of

Featherweight

Java

and.

A

proof

of

type

soundness

of

the

core

language..

Organization

of

The

Paper.

The

rest

of

the

paper

is

organized

as

follows.

We

first

give

an

overview

of

union

types

and

related

constructs

in

Section

and

then

formalize

FJ∨

and

prove

its

type

soundness

in

Section

We

discuss

the

interactions

of

union

types

with

other

common

features

such

as

generics

and

method

overloading

in

Section

then

discuss

related

work

in

Section

and

finally

give

concluding

remarks

in

Section

For

brevity,

we

omit

proofs

of

theorems;

they

will

appear

in

a

full

version,

available

at

http:

We

have

discussed

a

possible

introduction

of

union

types

for

class-based

object-oriented

languages.

Union

types

can

be

used

to

represent

a

group

of

classes

by

forming

their

supertype

after

those

classes

are

defined.

A

union

type

allows

direct

member

access

on

it

by

playing

a

role

of

an

interface

consisting

of

common

features

of

classes.

Also,

case

expressions

provide

exhaustive

case

analysis

on

the

run-time

types

of

objects;

we

believe

that

exhaustive

case

analysis

would

be

useful

even

for

a

language

without

union

types.

We

have

also

formalized

the

core

of

the

type

system

on

top

of

Featherweight

Java

and

proved

that

the

type

system

is

sound..

Although

we

expect

it

is

useful

as

it

is,

the

mechanism

of

direct

member

access

may

be

criticized

that

it

heavily

depends

on

member

name

equality,

which

can

be

purely

coincidental.

To

remedy

the

situation,

member

renaming

operations

as

found

in

the

recent

proposal

of

traits

may

be

combined..

We

do

not

discuss

implementation

issues

in

this

paper

and

leave

them

for

future

work.

Straightforward

implementation

would

be

by

erasure

a

union

type

C∨D

can

be

translated

to

a

common

superclass

of

C

and

D

(or

simply

to

Object);

case

and

transparent

access

can

be

expressed

in

terms

of

instanceOf

and

downcasts.

Efficient

implementation

of

direct

member

access

is

an

interesting

research

topic.

The

notion

of

union

types

in

programming

languages

can

be

classified

into

two

categories:

tagged

(or

disjoint)

union

types

and

untagged

union

types.

The

former,

found

for

instance

in

ML's

datatypes

or

Pascal's

variant

records,

usually

requires

an

explicit

operation

of

tagging

(or

constructor

applications)

to

form

an

expression

of

a

union

type,

while

the

latter

does

not

(and

uses

subtyping

and

subsumption,

instead).

A

language

with

tagged

unions

is

equipped

with

a

construct

for

case

analysis

to

use

tagged

values,

while

an

untagged

union

can

usually

be

used

with

only

operations

that

are

valid

for

both

summands.

Our

union

types

can

be

considered

a

hybrid

of

the

two

kinds;

thanks

to

the

fact

that

every

object

is

inherently

tagged

by

the

name

of

the

class

from

which

it

was

instantiated,

explicit

tagging

is

not

needed

to

construct

an

expression

of

a

union

type

and,

furthermore,

as

we

have

shown,

both

case

analysis

and

direct

member

access

are

supported.

However,

in

our

language,

unlike

ML

datatypes,

forming

a

union

of

the

same

type

results

in

a

compatible

type

of

the

original

type,

so

it

is

not

meaningful

to

perform

case

analysis

on

such

a

type

(the

first

branch

will

always

be

taken)..

More

recently,

untagged

union

types

are

studied

in

the

context

of

programming

languages

for

processing

semi-structured

data

such

as

XML

Subtyping

supporting

"distributivity"

of

unions

over

record

field

types,

exemplified

as

{a:S,b:T}∨{a:U,c:V}

{a:S∨U},

({a:S,b:T}

is

a

type

for

records

that

have

a

field

a

of

type

S

and

b

of

T),

has

inspired

us

in

the

development

of

our

direct

member

access

mechanism.

The

type

system

of

Xtatic

an

extension

of

C#

with

mechanisms

for

native

XML

processing,

is

equipped

with

regular

expression

types

which

include

the

union

type

constructor.

In

Xtatic,

however,

types

for

XML

documents

and

those

for

objects

are

separated

and

so

there

are

not

union

types

for

object

types..

The

mechanism

called

intertype

declarations

to

add

supertypes

to

existing

classes

can

be

found

in

the

AspectJ

language

an

aspect-oriented

extension

of

Java.

However,

our

direct

access

mechanism,

which

allows

to

access

members

of

the

same

name

but

different

types,

provides

more

than

just

the

ability

to

add

supertypes.

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


Input feature selection by mutual information based on Parzen window Mutual information is a good indicator of relevance between variables, and have been used as a measure in several feature selection algorithms. However, calculating the mutual information is difficult, and the performance of a feature selection algorithm depends on the accuracy of the mutual information. In this paper, we propose a new method of calculating mutual information between input and class variables based on the Parzen window, and we apply this to a feature selection algorithm for classification problems. Mutual information is considered as a good indicator of relevance between two random variables [1] . Recently, efforts to adopt mutual information in feature selection problem resulted a series of researches [2] - [4] . Because the computation of mutual information between continuous variables is a very difficult job requiring probability density functions (pdf ) and involving integration of those functions, mutual information feature selector (MIFS) [2] and its variants [3] [4] used histograms in approximating the pdf s to avoid these complexities. Thus, the performance can be degraded as a result of large errors in estimating the mutual information. In addition, MIFS methods have another limitation in that these methods do not provide a direct measure to judge whether to add additional features or not. More direct calculation of mutual information is attempted using the quadratic mutual information in the feature transformation field [5] - [7] , but the relationship between Shannon's mutual information and the quadratic mutual information is not clear so far.. In this paper, a new feature selection method with the mutual information maximization scheme is proposed for classification problems. In calculating the mutual information between the input features and the output class, instead of dividing the input space into several partitions, we use the Parzen window method to estimate the input distribution.. With this method, more accurate mutual information is calculated giving better performance than other methods.. In the following section, the basics of information theory and the Parzen window method are briefly presented. In Section III, we propose a new feature selection method and in Section IV, the proposed algorithms are applied to several classification problems to show their effectiveness. And finally, conclusions follow in Section V. In this paper, we have proposed a method for calculating mutual information between continuous input features and discrete output class and applied this to a greedy input feature selection algorithm for classification problems. Although the mutual information is a very good indicator of the relevance between variables, the reasons why it is not widely used is its computational difficulties, especially for continuous multiple variables.. The proposed method make use of the Parzen window in getting the conditional density in a feature space. With this method, we can compute the mutual information between output class and multiple input features without requiring a large amount of memory.. The computational complexity of the proposed method is proportional to the square of the given sample size. This might be a limiting factor for huge data sets, but with a simple modification that confines each influence field in a finite area, we can greatly reduce the computational efforts. Furthermore, it is expected that a clustering or sample selection method can be used to overcome this limitation.. We applied the method for several classification problems and obtained better performances than those of the conventional methods such as MIFS, MIFS-U, and the stepwise regression.

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


On directed local chromatic number, shift graphs, and Borsuk-like graphs We investigate the local chromatic number of shift graphs and prove that it is close to their chromatic number. This implies that the gap between the directed local chromatic number of an oriented graph and the local chromatic number of the underlying undirected graph can be arbitrarily large. We also investigate the minimum possible directed local chromatic number of oriented versions of "topologically t-chromatic" graphs. We show that this minimum for large enough t-chromatic Schrijver graphs and t-chromatic generalized Mycielski graphs of appropriate parameters is ⌈t/4⌉ + 1. The local chromatic number of a graph G, defined by Erdős, Füredi, Hajnal, Komjáth, Rödl, and Seress [6] is a coloring parameter that was further investigated recently in the papers [13, 22, 23] . (See also [3] for some related results.) Denoting the set of neighbors of a vertex v by N(v), it is defined as follows. Thus ψ(G) is the minimum number of colors that must appear in the most colorful closed neighborhood of a vertex in any proper coloring that may involve an arbitrary number of colors. It was shown in [6] that there exist graphs G with ψ(G) = 3 and χ(G) > k for any positive integer k, where χ(G) denotes the chromatic number of G.. Changing "neighborhood" to "outneighborhood" in the previous definition we arrive at the directed local chromatic number (of a digraph) introduced in [13] . For a directed graph F let the set of outneighbors of a vertex v be N + (v) = {u ∈ V (F ) : (v, u) ∈ E(F )}. By a proper vertex-coloring of a directed graph we mean a proper vertex-coloring of the underlying undirected graph. The directed local chromatic number of a digraph is always less than or equal to the local chromatic number of the underlying undirected graph and we obviously have equality if our digraph is symmetrically directed, i.e., for every ordered pair (u, v) of the vertices (u, v) is an edge if and only if (v, u) is an edge. A digraph F = (V, E) is called oriented if the contrary is true: (u, v) ∈ E implies (v, u) / ∈ E. An orientation of an undirected graph G is an oriented graphĜ that has G as its underlying undirected graph.. It is a natural question whether every undirected graph G has an orientation the directed local chromatic number of which achieves the local chromatic number of G. Currently we know very little about this question. (See [20] for a problem of similar flavor: the relation of Shannon capacity and the maximum possible Sperner capacity of its orientations.). In this paper we explore the other extreme: what is the minimum possible directed local chromatic number that an orientation of a graph can attain.. In the following section we give some more definitions and summarize some facts about the investigated parameters. In Section 3 we investigate shift graphs. We observe that they have an orientation with directed local chromatic number 2 and prove that their local chromatic number can be arbitrarily large, in particular, it differs at most 1 from their chromatic number. We also consider the behavior of a symmetrized variant of shift graphs.. In section 4 we concentrate on Borsuk-like graphs: these are graphs the chromatic number of which can be determined by applying Lovász's topological method (cf. [15] ), while, at the same time they admit optimal colorings where no short odd length walks exist that start and end in the same color class. Several graphs have this property. In [22] we have shown that the local chromatic number of these graphs is around one half of their chromatic number. Here we show that the minimum directed local chromatic number of a Borsuk-like graph of appropriate parameters is about one quarter of its chromatic number.

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Supporting young children's communication with adult relatives across time zones Regular contact between children and their adult relatives can be a problem if they live in different time zones. In this situation, finding an agreed time to contact each other can be both confusing and complicated. This paper presents a study of the effect of time zone differences on communication between grandparents and grandchildren living in different time zones. We deployed a system between time zone distributed families to study this effect and analysed its use based on four parameters of time and events based theory: rigid sequential structures (that some events cannot occur before others), fixed durations (that most events always last the same time), standard temporal locations (that events have a standard time when they occur during the day) and uniform rates of recurrence (that some events always reoccur at a uniform rate). Our findings highlight the importance of: the need to consider the parents' role in facilitating contact and making the technology easy to use by children independently; the advantage of concurrent synchronous and asynchronous interaction forms; and the need to respect people's private time. These findings can inform the design of technology for supporting young children's communications with adult relatives across time zones. Communication over distance between children and their adult relatives is complicated when a family has to adapt to being in different time zones. There are a variety of reasons why families become distributed across time zones. One reason is migration. An adult son or daughter might move to a new country, eventually settling permanently with a husband/wife and children. This results in children growing up without easy regular physical contact with his/her grandparents, if at all. If this move includes significant geographical distance in an east-west direction, different circadian rhythms are introduced between the two families; for example, one might be eight hours ahead of the other. Another reason that time zone differences might be introduced between children and other family members is if the parents get divorced, and one parent moves to another time zone without the child. Similarly, time zone differences can occur on a temporary basis whenever one parent travels, for example, on a business trip. Whatever the reason, it might be hard to find an appropriate time of day that suits both households, as their daily schedules can be misaligned due to time zone differences. Even though children can generally tell time by the age of six (Freidman & Laycock, 1989) , it is unclear how they cope with understanding time zone differences. This paper explores such challenges of communication between children and adult relatives across time zones by deploying a system for families with children in one time zone and adult relatives in another. We study how this system supports communication between family members separated by both distance and time zones.. In this paper we start by motivating our study with related work, followed by findings from interviews with time zone distributed families in respect to their use of our prototype system. We then describe modifications made to the system to ease the impact of time zone differences as a direct result of those findings. Outcomes from studying use of the modified system then inform a discussion on how time zone differences influenced communication between the children and their adult relatives. We conclude with consequences of these findings on designing technology for communication between children and adult relatives across time zones. This paper has explored the important issue of the role of technologies for supporting communication between young children and their grandparents across time zones. A system was deployed and studied in two families, each of which had grandparents living in Denmark and grandchildren living in Australia. By analysing interviews with the families and logged usage patterns, we derived the following understandings that can be used to inform the design technology for this context. We found the following unique problems in communication and maintaining relationships across time zones:. • The two households have misaligned daily routines.. There are two main implications of this. Firstly, it is difficult to find a mutual time for the grandchildren and grandparents to communicate. The consequence of this is that parents are needed to negotiate the next meeting time with the grandparents to account for the time difference. Secondly, when talking together the two parties sometimes miss the shared conversational context of the current state of their day. The time indicator showing day and night in the other household solved the need for some of this negotiation, and gave a contextual talking point about where the others where in their day.. • Children have activity patterns related to circadian rhythms that don't necessarily fit with the grandparent's rhythm, and hence desire to respond. This is partly solved by the availability indicator, which meant children understood that their grandparents were available to play or read a story "now", and children could then decide if that suited them. • Children's daily rhythms might more closely match distant grandparents than their parents. Children generally go to sleep earlier than, and rise before, their parents. This might be a time when the grandparents are able and keen to communicate, so the child needs to be able to use the system, and initiate adhoc interactions without the help of a parent. The simple interaction design of the system using touch screens, continual mirroring of content across both sites and a single button to launch the storytelling made this possible.. We understand that these findings on the "stubbornness" of the time zone problem presented are not exhaustive, due to our small sample. As future work, we would like to confront this issue in a more theoretical and philosophical way by expanding this study to include more families using modified technology probes to understand more deeply what it is that makes consistent cross-time zone communication difficult, if not impossible. There have been some interesting investigations into the relationship between time and people. Sociologist Eviatar Zerubavel (1982) describes the emergence of the international standardization of time, and thus the concept of time zones so that people could have fixed points in time to refer to when the world was connected by railway transportation and telegraphic communication. In his book "Hidden rhythms", Zerubavel (1985) explores and dissects the concept of time. Of interest to our study, he defines four parameters concerning the concepts of time and events: rigid sequential structures (that some events cannot occur before others), fixed durations (that most events always last the same time), standard temporal locations (that events have a standard time when they occur on the day) and uniform rate of recurrence (that some events always reoccur at a uniform rate). After a family agreed to participate in the field study, set up dates were scheduled with both the grandparents in Denmark and the parents (and children) in Melbourne.. At the set up, in their homes, a semi-structured interview was conducted giving insight into the current challenges of staying in contact with their remote family members.. Following this, the system computer and the touch screen monitor were installed concurrently in the homes in both Denmark and Melbourne. To facilitate use, it was placed in a centrally located, high traffic area in the home, such as on a kitchen table. The computer and the keyboard were hidden away as much as possible. The mouse was kept visible and usable. The computer was wirelessly connected to the household's existing ADSLbased Internet connection (all households had that already). The family was informed that they could use the system in whatever way they found suitable or exciting and the different activities available in the system were demonstrated.. Every time a family used the Storytelling component, a screen capture program (CamStudio) recorded both what was shown on the touch screen and their speech. This happened automatically in the background without interfering with the use of the system. The screen shot was supplemented by a log file with time stamped entries including: every time a household tried to initiate a storytelling session; what the response was from the other household ("accept" or "deny"); and how long initiated sessions lasted. When a photo was sent to the system and therefore to both components, the sender and the time of day were logged.. When the system was collected from the households, a second semi-structured interview was conducted investigating how they used the system, what they liked and disliked about it and if, and how, the system supported communication with their remote family members.. To harvest enough data and ensure that system usage routines developed, the system was deployed in each household for three weeks. The system had to be deployed sequentially between Family 1 and Family 2 for technical reasons. The time zone difference during the deployment with Family 1 was 10 hours, because of DST in Australia, where as it was between 8 and 10 hours during deployment with Family 2, because DST came into effect in Denmark and ended in Australia.. After the system had been deployed with Family 2 for three weeks, their use informed modifications to the system. Family 2 subsequently agreed to have this modified system installed for another three weeks. The modified system was deployed remotely to the households using remote desktop software (TeamViewer). After three weeks, the complete system was collected from them and another semi-structured interview was conducted exploring if, and how, the modifications impacted how family members used the system.

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


RESOURCE BOUNDED SYMMETRY OF INFORMATION REVISITED Abstract. The information contained in a string x about a string y is the difference between the Kolmogorov complexity of y and the conditional Kolmogorov complexity of y given x, i.e., I(x : y) = C(y) − C(y | x). The Kolmogorov-Levin Theorem says that I(x : y) is symmetric up to a small additive term. We investigate if this property also holds for several versions of polynomial time bounded Kolmogorov complexity. We study symmetry of information for some variants of distinguishing complexity CD where CD(x) is the length of a shortest program which accepts x and only x. We show relativized worlds where symmetry of information does not hold in a strong way for deterministic and nondeterministic polynomial time distinguishing complexities CD poly and CND poly . On the other hand, for nondeterministic polynomial time distinguishing complexity with randomness, CAMD poly , we show that symmetry of information holds for most pairs of strings in any set in NP. Our techniques extend work of Buhrman et al. (CCC 2004) on language compression by AM algorithms, and have the following application to the compression of samplable sources, introduced in Trevisan et al. (CCC 2004): any element x in the support of a polynomial time samplable source X can be given a description of size − log Pr[X = x] + O(log 3 n), from which x can be recovered by an AM algorithm. One of the most beautiful theorems in Kolmogorov Complexity is the principle of "Symmetry of Information", independently proven by Kolmogorov and Levin (ZL70) . Roughly speaking, symmetry of information states that for any two strings x and y, the information contained in x about y is equal to the information contained in y about x, up to logarithmic factors. More formally, letting C(x) be the length of a shortest program which prints x, and C(y | x) be the length of a shortest program which prints y when given input x, symmetry of information can be stated as C(y) − C(y | x) ≈ C(x) − C(x | y). Besides its inherent attractiveness, this principle has also seen applications in diverse areas of theoretical computer science, for example in (ABK + 02; JSV97; VV02). In this paper, we investigate the principal of symmetry of information when resource bounds are placed on the program to describe y given x. While the argument of Kolmogorov-Levin (ZL70) can be used without modification to show that symmetry of information holds for programs using exponential time or polynomial space, things become trickier with polynomial time bounds. Though this question has been around for some time, indeed as early as 1967 Kolmogorov suggested time-bounded versions of symmetry of information as an interesting avenue of research (Lev04), still few definite answers are known. See section 7.1 of (LV97) for a survey and open problems.. The main contributions to the problem of polynomial time symmetry of information appear in the series of works (LM93; LW95) which show, in particular, the following:. • If P=NP then polynomial time symmetry of information holds (LW95).. • If cryptographic one-way functions exist, then polynomial time symmetry of information does not hold up to a O(log n) factor (LM93; LW95).. The intuition behind the second result is, if f is a polynomial time computable one-way function, and f (x) = y, then y is simple given x. On the other hand, if x is simple in polynomial time given y then this would provide a way to invert the function, by cycling through all small programs. Revisiting these works, several interesting questions arise:. • Can polynomial time symmetry of information hold up to a factor larger than O(log n)? The same argument sketched above shows that if symmetry of information holds up to a factor of δ(n) then there do not exist polynomial time computable cryptographical functions which cannot be inverted in time 2 δ(n) . However, as, for example, factoring n-bit integers can be done in 2 O( √ n) time (BJP93), it is not implausible that symmetry of information could hold up to a factor of δ(n) = n or even δ(n) = n 1/2+ . It is the case that 2C(x, y) ≥ C(x) + C(y | x), could we show (2 − )C(x, y) ≥ C(x) + C(y | x) for some ?. • Can symmetry of information hold for complexity measures other than polynomial time printing complexity? En route to showing that BPP is in the polynomial hierarchy, Sipser (Sip83) introduced a relaxation of printing complexity called distinguishing complexity, denoted CD. For a string x, CD(x) is the length of a shortest program which accepts x and only x.. The arguments of (LM93; LW95) leave open the question if symmetry of information can hold for distinguishing complexity. Now if f is a polynomial time computable one-way permutation and f (x) = y, then CD poly (x | y) is constant, as with a description of f , on input z we accept if and only if f (z) = y. More recently, distinguishing complexity measures using nondeterminism, denoted CND, and nondeterminism and randomness (based on the complexity class AM), denoted CAMD, have been introduced (BFL02; BLvM04). Does symmetry of information hold for these measures?. • Is there an assumption weaker than P=NP which implies polynomial time symmetry of information?. Addressing the first two questions, we show relativized worlds where symmetry of information fails in a strong way for CD poly and CND poly (the existence of such worlds was claimed in (BF95), though without a complete proof). On the other hand, we show that for any set A ∈ NP symmetry of information holds for most pairs of strings x, y ∈ A with respect to the measure CAMD poly . We also unconditionally show that C poly (x, y) ≥ CAMD poly (x) + CAMD poly (y | x). This implies that symmetry of information holds under the condition C poly (x | y) ≤ CAMD poly (x | y). We show that this statement, however, is equivalent to P=NP.. The main tool of our positve results is an extension of the language compression technique of (BLvM04). This extension itself has an interesting implication for the compression of samplable sources, the study of which is introduced in (TVZ04). We show that for any polynomial time samplable source X, any element x in the support of X can be given a description of size − log Pr[X = x] + log 3 n, such that x can be recovered from this description by an AM algorithm. Note that this means the source can be compressed to expected length H(X) + O(log 3 n), differing from optimal by just a O(log 3 n) additive factor.. Another interesting approach to the definition of time-bounded Kolmogorov complexity is L. Levin's Kt complexity introduced in (Lev73) . Recently D. Ronneburger proved that symmetry of information does not hold for Kt complexity in a very strong sense (Ron04).

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


Syntactic control of concurrency We consider a finitary procedural programming language (finite data-types, no recursion) extended with parallel composition and binary semaphores. Having first shown that may-equivalence of second-order open terms is undecidable we set out to find a framework in which decidability can be regained with minimum loss of expressivity. To that end we define an annotated type system that controls the number of concurrent threads created by terms and give a fully abstract game semantics for the notion of equivalence induced by typable terms and contexts. Finally, we show that the semantics of all typable terms, at any order and in the presence of iteration, has a regular-language representation and thus the restricted observational equivalence is decidable. Game semantics has emerged as a powerful paradigm for giving semantics to a spectrum of programming languages ranging from purely functional languages to those with non-functional features such as control operators and references [1] [2] [3] [4] [5] [6] . Recently it has been developing in a new, algorithmic direction. Hankin and Malacaria have applied it to program analysis [7, 8] . Ghica and McCusker [9] found that the game semantics of a second-order fragment of a procedural language can be captured by regular languages, demonstrating a new, semantics-directed, approach to software model-checking [10] . The approach has subsequently been extended in various directions: to third order [11] , call-by-value [12, 13] , Hoare-style assertions [14] and specification [15] .. In this paper we propose a game-based framework for compositional model checking of concurrent programs. Although a fully abstract game model for a Work funded by British EPSRC, Canadian NSERC and St John's College, Oxford.. concurrent programming language exists [16] , it seems unsuitable as a model of computation for model-checking applications. We can show that observational equivalence, even at second order in the absence of recursion, is not decidable. The sources of non-finitary behaviour are the free identifiers of first or higher-order types, which correspond to procedures using an argument in an unbounded number of concurrent threads of computation.. In the game model, active threads at any moment correspond to pending questions in a play. Hence, we constrain plays by placing bounds on the allowable number of pending questions and enforce these restrictions syntactically using a type system augmented with resource bounds. The key differences between this type system and the standard type system, are the "linearization" of application and parallel composition, i.e. requiring the environments of the two sub-terms to be disjoint. We also revise the contraction rule to count the number of contracted occurrences of a variable. We call this type system Syntactic Control of Concurrency (SCC); it is a generalization of Serially Reentrant Algol (SRA), a type system introduced by Abramsky to identify higher-order terms of a sequential language denotable by "pointer-free" finitary strategies [17] .. The bounds imposed on the number of pending questions by SCC can be seen as a kind of assume-guarantee reasoning (see e.g. [18] ): bounds on the behaviour of the Opponent represent assumptions on the behaviour of the environment, while bounds on the behaviour of the Proponent represent guarantees on the behaviour of the system. Typability can be seen as composition, made possible by the fact that the guarantees and the assumptions match. Unsurprisingly, not all terms of the original language admit a resource-bounding typing.. Resource-sensitive type systems are an area of research with numerous applications; the examples mentioned below are only entry points to a vast literature. The nature of the controlled resource is usually duration [19] or space [20] ; applications of such systems are as diverse as execution in embedded systems [21] , memory management [22] , compilation to hardware [23] or proof-carrying code [24] . Type systems have been also used to control more abstract resources, such as variable usage for improved compilation [25] or interference effects for specification and verification [26] .. The motivation behind SCC is to isolate (open) terms with finitary models for the purpose of automated verification. The notion of resource in SCC, which we may call active threads of computation, has a computational meaning, but it is primarily motivated by the game-semantic analysis of the language [16] . The main thrust of the paper is thus semantic; we plan to investigate the type-theoretic issues of SCC separately.

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


LMI-based adaptive fuzzy integral sliding mode control of mismatched uncertain systems Integral sliding mode design is considered for a class of uncertain systems in the presence of mismatched uncertainties in both state and input matrices, as well as norm-bounded nonlinearities and external disturbances. A sufficient condition for the robust stability of the sliding manifold is derived by means of linear matrix inequalities. The initial existence of the sliding mode is guaranteed by the proposed control law. The improvement of the proposed control scheme performances, such as chattering elimination and estimation of norm bounds of uncertainties, is then considered with the application of an adaptive fuzzy integral sliding mode control law. The validity and efficiency of the proposed approaches are investigated through a sixth order uncertain mechanical system. The relationship between models and the reality they represent is subtle and complex. The differences or errors between them define the term uncertainty. Thus, it is necessary to analyze dynamic systems subject to uncertainties. In other words, for control design purposes we need to handle simple models. However, the obtained controller must work when connected to a real system. Control strategies based on this philosophy have attracted the attention of researchers and have been widely considered. Variable Structure Control (VSC) with Sliding Modes (SMs) has been regarded as a robust technique for its insensitivity to external disturbances and model uncertainties satisfying the matching condition, i.e., perturbations that affect the system model through the input channel (Decarlo et al., 1988; Hung et al., 1993) . In addition, the use of SMC can offer fast response, good transient performance and order reduction. These advantages make the SM technique very widespread in robust control design (Ha et al., 1999; Utkin, 1992) .. The existence of the reaching phase in the resulting closed loop system yields the sensitivity of the dynamics to perturbations in an initial period of time in which the system has not yet reached the sliding manifold (Ackermann and Utkin, 1998) . Hence, the new approach known as Integral SMC (ISMC) has as the main contribution the elimination of the reaching phase. The basic idea of this concept is the addition of an integral term in the sliding surface allowing immediate sliding mode emergence (Utkin and Shi, 1996) . ISMC is used in the works of Mnasri and Gasmi (2007a; 2007b) and good results are obtained for large scale systems with matched uncertainties.. Application of the same method in the control of matched uncertain MIMO systems produced good performance as for robustness and tracking (Mnasri and Gasmi, 2008) . However, in many cases, the uncertainties and external disturbances do not always satisfy the matching conditions. Contrary to the matched case, any mismatched uncertainty affects the behavior of the sliding mode directly, even when the ISMC approach is used. To solve this problem, the main idea is the combination of SMC with other robust techniques.. A majority of the existing methods are based on Classic SMC (CSMC) (Kim et al., 2000; Choi, 2001; Xia and Jia, 2003) . These methods are affected by the aforementioned insufficiency of SMC with the reaching phase. Recently, much research has been focused on the advantages of ISMC in the control of systems with mismatched uncertainties. This method is considered by Cao and Xu (2004) for the case of systems with mismatched uncer-tainties in the state matrix, but it is limited to matched uncertainties in the input matrix and to external disturbances. A few recent studies have included the case of mismatched uncertainties in the input matrix (Shaocheng and Yongji, 2006; Choi, 2007) . However, neither of them is applicable in the presence of mismatched disturbances not related to the input channel.. In this paper, we propose a new approach to design ISMC for a class of uncertain systems. This class regroups mismatched uncertainties in both state and input matrices, as well as mismatched norm-bounded nonlinearities and external disturbances. Our approach, based on Linear Matrix Inequalities (LMIs), combines the advantages of ISMC with H ∞ control. This gives a sufficient condition for the robust stability of the system in the sliding mode. The chattering phenomenon, the main drawback of SMC, is also considered in this paper. An adaptive fuzzy ISMC law is proposed to improve the performance of the control scheme, by allowing elimination of chattering and the estimation of the norm bounds of uncertainties. The efficiency of the proposed control laws is investigated through a sixth-order mechanical system example. A robust ISMC design for mismatched uncertain systems has been studied. A sufficient condition for quadratic stability of sliding motion has been established in terms of LMIs. The immediate sliding mode existence has been guaranteed by the proposed ISMC law. The induced chattering phenomenon has been eliminated by the introduction of an AFISMC law. Finally, the effectiveness of the proposed methods has been proved through a sixth order uncertain mechanical system example.. Boyd, S., Ghaoui, L.E., Feron, E. and Balakrishnan, V. (1994) . Linear Matrix Inequalities in System and Control Theory,

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


On the Construction of Prefix-Free and Fix-Free Codes with Specified Codeword Compositions We investigate the construction of prefix-free and fix-free codes with specified codeword compositions. We present a polynomial time algorithm which constructs a fix-free code with the same codeword compositions as a given code for a special class of codes called distinct codes. We consider the construction of optimal fix-free codes which minimize the average codeword cost for general letter costs with uniform distribution of the codewords and present an approximation algorithm to find a near optimal fix-free code with a given constant cost.

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Bounds on the performance of belief propagation decoding Abstract -We consider Gallager's soft decoding (belief propagation) algorithm for decoding low density parity check (LDPC) codes, when applied to an arbitrary binary-input symmetric-output channel. By considering the expected values of the messages, we derive both lower and upper bounds on the performance of the algorithm. We also derive various properties of the decoding algorithm, such as robustness to the initial values of the messages. Our results apply both to regular and irregular LDPC codes. We assume a binary-input ( ( 0 , l ) ), symmetric-output, memoryless channel and consider ensembles of regular and irregular LDPC codes [2] . Let N be the block length of the code, and let sv v = 1 , . . . , N be the channel output for the w-th input bit. qv = P ( l 1 s v ) denotes the conditional probability of the w-th transmitted bit given s v . The algorithm is initialized by assigning rightbound message values to each edge in the bipartite graph representation of the code, such that for an edge e with left vertex w this value is 7. . The algorithm then proceeds as described in [l] .. Consider the subgraph of depth two spanned from an edge e , with left vertex w. We assume that this subgraph is tree like. This assumption holds with probability arbitrarily close to one for a sufficiently large block length. Denote the edges of this tree by e = (v,w), ei = ( V , T I~) and ei,j. and 2 of the tree respectively. Denote by Xt+' the random variable (r.v.) corresponding to the rightbound value passed in the decoding algorithm along e at time t + 1. Similarly, denote by Kt the leftbound message along the edge ei at time t , and by Xt,j the rightbound message along the edge ei3j at time t. Finally let q = qv. Then the following relations hold:. , where c = deg(w) (2) 11. BOUNDS ON THE PERFORMANCE In order to analyze the algorithm we assume without loss of generality that the all-zero codeword was transmitted. We then obtain bounds on the conditional expectation of the messages given this assumption.

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Fast Mixing of Parallel Glauber Dynamics and Low-Delay CSMA Scheduling Abstract-Glauber dynamics is a powerful tool to generate randomized, approximate solutions to combinatorially difficult problems. It has been used to analyze and design distributed CSMA (Carrier Sense Multiple Access) scheduling algorithms for multi-hop wireless networks. In this paper we derive bounds on the mixing time of a generalization of Glauber dynamics where multiple links are allowed to update their states in parallel and the fugacity of each link can be different. The results can be used to prove that the average queue length (and hence, the delay) under the parallel Glauber dynamics based CSMA grows polynomially in the number of links for wireless networks with bounded-degree interference graphs when the arrival rate lies in a fraction of the capacity region. We also show that in specific network topologies, the low-delay capacity region can be further improved. In wireless networks, the links (transmitter-receiver pairs) may not be able to transmit simultaneously due to interference. A scheduling algorithm, or MAC (Medium Access Control) protocol, determines which links can access the medium in each time instant so that no active links interfere with each other. Since many wireless network applications today have stringent bandwidth and delay requirements but the resources (e.g., spectrum, power) are often quite limited in a wireless setting, designing low-complexity scheduling algorithms to achieve high throughput and low delay is of great importance.. It is well known that the queue-length based Maximum Weighted Scheduling (MWS) algorithm is throughput-optimal [21] , meaning that it can stabilize the network queues for all arrival rates in the capacity region of the network. However, MWS requires the network to select a max-weight independent set in the interference graph in every time slot, which is NPhard for general interference graphs [5] . There exist several low-complexity alternatives such as Maximal Scheduling and Greedy Maximal Scheduling, but in general these algorithms can only guarantee to achieve a fraction of the capacity region (see [12] and references therein).. Due to their simplicity, random access type scheduling algorithms such as Aloha and Carrier Sense Multiple Access (CSMA) are widely used in practice. Performance analysis of random access algorithms in single-hop wireless networks can Authors names appear in the alphabetical order of their last names. This work is supported by MURI grant BAA 07-036. 18 , NSF Grants 07-21286, 05-19691, 03-25673, AFOSR Grant FA-9550-08-1-0432 and DTRA Grant HDTRA1-08-1-0016. be found in [1] . In [2] the authors introduced a continuous-time Markov chain model to analyze the performance of a (fixedparameter) CSMA algorithm in multi-hop wireless networks, and it was shown that the stationary distribution over the schedules has a product form. The model was used in [23] to study throughput and fairness issues of the CSMA algorithm, and its insensitivity properties were studied in [13] . A discretetime version of the algorithm was studied in [16] , [17] .. In [10] the authors proposed an adaptive CSMA algorithm where the links adaptively adjust their parameters based on locally measured arrival and service rates. The algorithm was shown to be throughput-optimal under a time-scale separation assumption (the CSMA Markov chain converges to its steadystate distribution instantaneously compared to the time-scale of adaptation of the CSMA parameters) which can be justified using a stochastic-approximation argument [14] , [11] . In [18] the authors established throughput-optimality of their adaptive CSMA algorithm without the time-scale separation assumption by choosing the link parameters to be slowly varying functions of the queue lengths. The discrete-time equivalent of this analysis for the model in [16] , [17] appears in [6] .. Central to these CSMA algorithms is the so-called Glauber dynamics, which is a Markov Chain Monte Carlo method that can be used to sample the independent sets of a graph according to a product-form distribution [4] , [22] . Under traditional Glauber dynamics, in each time slot one link will be selected uniformly at random, and only that link can change its state while other links will keep their states unchanged. For the chosen link, if all of its neighboring links were in state 0 (inactive) in the previous slot, then the link will choose to be in state 1 (active) with probability λ 1+λ and in state 0 with probability 1 1+λ ; otherwise (i.e., if at least one of its neighboring links was in state 1 in the previous slot), the link will choose to be in state 0 definitely. In statistical physics, the parameter λ is called the fugacity since it indicates how likely a selected site will change its state. 1 Glauber dynamics has many applications in statistical physics, graph coloring, approximate counting, and combinatorial optimization (e.g., [4] , [15] , [22] ). In these applications, the performance of the Glauber dynamics is often determined by how fast the Markov chain converges to the stationary distribution. The Glauber dynamics is said to be fast mixing if the mixing time (will be defined formally later) grows polynomially in the size of the graph. The algorithm in [16] , [17] is a generalization of Glauber dynamics where multiple links are allowed to update their states in parallel. The CSMA algorithm in [2] can be viewed as continuoustime Glauber dynamics with fixed parameters, and the CSMA algorithms in [10] , [18] can be viewed as continuous-time Glauber dynamics with adaptive parameters. In this paper, we focus on the discrete-time CSMA algorithm suggested in [16] , [17] . An important feature of this algorithm is that the overhead for signaling is constant (independent of the size of the network) even taking into account the collisions during the signaling phase. The results in this paper can be extended to other versions of CSMA algorithms as well.. The recently proposed Glauber dynamics based CSMA algorithms have made an important progress to the design of low-complexity distributed scheduling algorithms to achieve maximum throughput in wireless networks. On the other hand, the delay performance of these CSMA algorithms has not been well understood. As shown in a recent work [20] , for general networks, it may not be possible to design lowcomplexity scheduling algorithms which can achieve both low delay (i.e., grows polynomially in the number of links) and even a diminishingly small fraction (i.e., approaches zero when the number of links increases) of the capacity region unless NP ⊆ BPP or P = NP. In this paper we prove a positive result. We show that our parallel Glauber dynamics based CSMA scheduling algorithm can achieve both low delay and a fraction (independent of the size of the network) of the capacity region when the interference graphs satisfy certain properties.. The main contributions of this paper include:. • We analyze the mixing time of Glauber dynamics with parallel updates and heterogenous fugacities. We derive various conditions on the system parameters such as fugacities, vertex degrees and update probabilities, under which the mixing time grows logarithmically in the number of vertices.. • Based on the above mixing time results, we show that, for wireless networks with bounded-degree interference graphs, the parallel Glauber dynamics based CSMA algorithm can achieve a small queue length (O(log n) where n is the number of links in the network) at each link if the arrival rate lies within a fraction (independent of n) of the capacity region and the fugacities of the links are appropriately chosen and fixed. Moreover, we consider an adaptive version of the CSMA algorithm where the fugacities of the links are adjusted based on local queue length information. We show that the total queue length in the network grows polynomially (O(n 3 log n)) under the adaptive CSMA algorithm.. • Unlike prior analysis of Glauber dynamics based CSMA algorithms which uses the conductance method to obtain exponential bounds on the mixing time and the queue lengths, here we use the coupling method which allows us to obtain polynomial bounds on the queue lengths for a fraction of the capacity region.. • For a special but important network topology (Wireless LANs), we show that Glauber dynamics based CSMA can support the full capacity region with linear mixing time. In a related work [19] , the authors proposed a CSMA algorithm which can achieve order-optimal delay performance (i.e., the per-node delay is bounded by a constant) for networks with polynomial growth structure (the number of r-hop neighbors of any node is bounded by a polynomial of r). The key idea is to (periodically) partition the network into sub-networks with bounded number of nodes in each sub-network such that the mixing time of the Markov chain of the schedule in each subnetwork is bounded by a constant. However, the partition is realized by a distributed coloring algorithm which requires the nodes to exchange messages with their multi-hop neighbors, and this overhead can be significant and may increase with the size of the network, especially when the discrete nature of signaling is considered. So a direct comparison of our results and the results in [19] appears to be difficult.. The paper is organized as follows. In Section II we introduce a CSMA scheduling algorithm based on a generalization of Glauber dynamics with parallel updates and heterogenous fugacities. In Section III we derive bounds on the mixing time of parallel Glauber dynamics, with the proof presented in Section IV. In Section V we analyze the delay performance of the CSMA scheduling algorithms using the mixing time results. Section VI is dedicated to the analysis of complete interference graphs. The paper is concluded in Section VII. Fig.1(a) shows a wireless network of 3 links, where link 2 interferes with links 1 and 3, and links 1 and 3 don't interfere with each other. So, the associated interference graph is shown in Fig.1(b) , and the conflict set of link 2 is N 2 = {1, 3}, etc.) An independent set of G is a subset of the vertices in V where no two vertices are neighbors of each other. Let I be the set of all independent sets of G.. A feasible schedule of the network is a set of wireless links that can be active at the same time according to the conflict set constraint, i.e., no two links in a feasible schedule conflict with each other. This corresponds to an independent set of vertices in the interference graph. (In Fig.1(b) , for example, the sets {1}, {1, 3}, {2} are all feasible schedules.) We will use links and vertices interchangeably throughout this paper. Suppose |V | = n. We can represent a feasible schedule by a vector σ of the form (σ i ) i∈V , with σ i ∈ X = {0, 1} for all i ∈ V . For a link i and a schedule σ, we say link i is included in the schedule (written as i ∈ σ) if σ i = 1. Note that σ is a feasible schedule if the set {i ∈ V : σ i = 1} is an independent set of G, i.e., if. Let Ω ⊆ X n be the set of all feasible schedules on G.. We consider a time-slotted system. A scheduling algorithm is a procedure to decide which schedule to be used in every time slot for data transmission. In [16] we proposed a scheduling algorithm based on a generalization of Glauber dynamics (called parallel Glauber dynamics) where multiple links are allowed to update their states in a single time slot. The key idea is that in every time slot, we select an independent set of links m ∈ I to update their states according to a randomized procedure, i.e., we select m ∈ I with probability q m , where m∈I q m = 1. We call m the decision schedule. 2 The parallel Glauber dynamics is formally described as follows. In this paper, we have shown that Glauber dynamics based CSMA can result in queue lengths that only grow polynomially in the network size in bounded degree graphs if the arrival rates lie within a certain fraction of the capacity region. This establishes a positive result in contrast to previous results which showed that polynomial queue lengths are not possible if the interference graph of the network can be arbitrary [20] . To establish our results, we use Markov chain coupling theory to estimate the mixing time of parallel Glauber dynamics. It is interesting that our upper bound on the queue lengths is larger (in an order sense) for the dynamic-parameter algorithm compared to the fixed-parameter algorithm. However, it is unclear whether this is an artifact of our bounding techniques or if it is an inherent penalty due to the time required for adaptation. It would be interesting to explore this issue in future work. We are also interested to study the throughput and delay performance of CSMA scheduling algorithms in other important classes of interference graphs, and design enhanced or new algorithms to further improve the performance. . then we have. Proof: Choose f (v) = 1+λv qv , ∀v ∈ V in Theorem 2.

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


The single publication H-index of papers in the Hirsch-core of a researcher and the indirect H-index The single publication H-index of Schubert is applied to the papers in the Hirsch-core of a researcher, journal or topic. Four practical examples are given and regularities are explained: the regression line of the single publication H-index of the ranked papers in the Hirsch-core is decreasing. We propose two measures of indirect citation impact: the average of the single publication H-indices of the papers in the Hirsch-core and the H-index of these single publication H-indices, defined as the indirect H-index. Formulae for these indirect citation impact measures are given in the Lotkaian context. The single publication H-index was introduced in Schubert (2009) , for assessing single publications. Its definition is as follows. For a fixed publication, consider all publications that cite this fixed publication. If we consider this set of citing publications, one can count the citations to theses publications and rank them in decreasing order of received citations. The h-index of this ranked list (i.e. the largest rank r such that all publications receive at least r citations) is called the single publication H-index of the fixed publication.. The main idea behind this definition is that not only direct impact of a publication is important (direct citations to this publication) but also indirect impact, i.e. the citations to papers that cite this single publication.. The main topic of this paper is the following. Consider a set of papers (e.g. of a researcher, journal or topic) of which we can calculate the classical Hirsch-index (h-index) based on received citations. For this (easy) calculation, papers are ranked in decreasing order of received citations. For each paper in this ranked list we can calculate the single publication H-index as defined above. So, for each paper (say in the h-core) we have a single publication h-index, denoted H 1 , …, H h (we distinguish between h, the h-index of the set of papers and H i , the single publication H-index of the ith paper in this h-core. Let us give a concrete example: the publications of this author and their citations on April 21st 2011 according to Thomson Reuters' Web of Science (WoS). In Table 1 we can see that the h-index of Egghe is h = 19.. The single publication H-indices of these 19 papers is calculated as follows. In the WoS we ask for the papers of Egghe to be ranked in decreasing order of received citations. When clicking on each paper we can ask for seeing all papers that cite this paper (e.g. for the paper on rank r = 1 we ask for retrieving all 172 citing papers). Then we ask for putting these citing papers in decreasing order of received citations from which it is easy to calculate the single publication H-index (e.g. for the first paper H 1 = 19, for the second paper H 2 = 13 and so on). This procedure is very simple and can be executed in a few minutes.. In Thor and Bornmann (2011) one presents a web application where the single publication H-index can be automatically calculated for any publication indexed in Google Scholar.. It is clear that we can expect higher H r -values for lower r ranks. This will be studied theoretically in the next section based on the Lotkaian framework (see e.g. Egghe 2005 ). Also a formula for H h (the single publication H-index of the paper on rank r = h) is presented. In Sect. 3 we study four examples: the example of Egghe above, the example of Jean Bourgain (Field medalist in mathematics), the example of the journal ''Scientometrics' ' (period 2006-2011) In Sect. 4 we present two new measures of indirect citations impact: H, the average of the H i -values in the h-core and IH, the indirect H-index of the researcher, journal or topic, being the Hirsch-index of the H 1 , H 2 , …, H h -values. We calculate H and IH for the given examples and present formulae for H and IH in the Lotkaian framework.. The paper ends with a conclusion and open problems section.

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


A model of the electrical behaviour of myelinated sensory nerve fibres based on human data Calculation of the response of human myelinated sensory nerve fibres to spinal cord stimulation initiated the development of a fibre model based on electrophysiological and morphometric data for human sensory nerve fibres. The model encompasses a mathematical description of the kinetics of the nodal membrane, and a non-linear fibre geometry. Fine tuning of only a few, not well-established parameters was performed by fitting the shape of a propagating action potential and its diameter-dependent propagation velocity. The quantitative behaviour of this model corresponds better to experimentally determined human fibre properties than other mammalian, nonhuman models do. Typical characteristics, such as the shape of the action potential, the propagation velocity and the strength-duration behaviour show a good fit with experimental data. The introduced diameter-dependent parameters did not result in a noticeable diameter dependency of action potential duration and refractory period. The presented model provides an improved tool to analyse the electrical behaviour of human myelinated sensory nerve fibres.

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


Complementation of rational sets on scattered linear orderings of finite rank In a preceding paper [6] , automata have been introduced for words indexed by linear orderings. These automata are a generalization of automata for finite, infinite, bi-infinite, and even transfinite words studied by Büchi. Kleene's theorem has been generalized to these words. We show that deterministic automata do not have the same expressive power. Despite this negative result, we prove that rational sets of words of finite ranks are closed under complementation. Automata were first introduced by Kleene who showed that they have the same expressive power as rational expressions [13] . Since then, many extensions of this deep result have been proved. Different kinds of structures have been considered such as infinite words [8, 14] , bi-infinite words [2, 15] and transfinite words [10, 11] , finite and infinite trees [18] , finite and infinite traces and pictures.. In [6, 7] , automata that accept linearly-ordered structures were introduced. These automata are a simple and natural generalization of usual automata with additional limit transitions of the form P → q and q → P where P is a subset of states. They allow finite words, infinite words, bi-infinite words and transfinite words to be treated in the same framework. These automata have been shown to be equivalent to certain rational expressions when the orderings are restricted to scattered orderings [6] . Recall that scattered orderings are those orderings which do not contain a dense sub-ordering like Q. Scattered orderings include the ordinals and their mirrors.. One main property of rational sets is the closure under complementation. This property means that for any automaton A, there exists another automaton B which accepts exactly the structures not accepted by A. This property holds for almost all structures: finite and infinite words, finite and infinite trees and even for transfinite words on ordinals.. This property is important both from the practical and the theoretical point of view. The effective closure under complementation means that the class of rational sets forms an effective boolean algebra. This closure property is used whenever some logic is translated into automata. For instance, in the proof of the decidability of the monadic second-order theory of the integers by Büchi [9] , and also in the proof of the decidability of the monadic second-order theory of the infinite binary tree by Rabin [18] , the closure under complementation of automata is the key property. It is well known that automata have the same expressive power as the monadic second order theory on many structures such as finite, infinite, and transfinite words and trees. A nice result would be to extend this equivalence to linear orderings. Proving closure under complementation is one step towards this result.. In [7] , the question of closure under complementation was left open. In this paper, we address and solve this problem for a subclass of scattered linear orderings. Namely, we prove that rational sets of words on scattered orderings of finite ranks are closed under complementation. Recall that Hausdorff's result [12] states that scattered orderings can be obtained from finite orderings by repetitive applications of ω-sums and −ω-sums (see Theorem 1) . The rank of a scattered linear ordering is the number of nested ω-sums and −ω-sums needed to obtain it. The ranks of all countable scattered linear orderings range over all countable ordinals. Therefore the rank of a scattered linear ordering can be seen as a measure of its complexity. For instance, ω and ζ are scattered orderings of rank 1. Our result for scattered linear orderings of finite rank generalizes both the complementation of infinite and bi-infinite words. The class of scattered orderings of finite rank includes all the ordinals smaller than ω ω . Therefore, our result holds for the sets of transfinite words studied by Choueka [11] .. The classical method for constructing an automaton for the complement of a set of finite words accepted by an automaton A is through determinization [1] . Another method uses algebraic objects such as semigroups [17] . The determinization method can still be used for infinite words but it becomes more involved [21, 3] . This method has been pushed further by Büchi for countable transfinite words but it is then very complex [10] . The algebraic method can also be extended to ordinals [4, 5] . In our case, this method can not be applied since automata can not be made deterministic. In Section 4, we give an example of a rational set of words that cannot be accepted by a deterministic automaton. Therefore, to prove the closure under complementation, we use another method which was introduced by Büchi for infinite words. This method is based on an equivalence relation on words whose classes are shown to be rational.. The paper is organized as follows. In Section 2, we introduce words indexed by linear orderings and recall the Hausdorff characterization of countable scattered linear orderings. Then rational sets of words are defined in terms of rational operators and automata in Section 3. We finally prove in Section 4 that rational sets of words indexed by countable scattered linear orderings of finite ranks are closed under complementation.

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