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Extensions of the method of poles for code construction
Marie-Pierre Béal
To cite this version:
Marie-Pierre Béal. Extensions of the method of poles for code construction. IEEE Transactions on Information Theory, Institute of Electrical and Electronics Engineers, 2003, 49 (6), pp.1516-1523.
�hal-00619207�
of poles for ode onstrution
Marie-Pierre B
eal
Abstrat
The method of poles is a method introdued by P. A. Franaszek
for onstruting a rate 1 : 1 nite state ode from k-ary data into
aonstrainedhannel of nitetype whoseapaity isstritly greater
than log(k). The method is based on the omputation of a set of
states alled poles. To eah pole is assoiated a set of paths going
from thispole to others. Eah setveriesanentropyondition. The
ode produed by themethod of poles has asliding blok deoder if
eah set of paths satises moreover an optimization ondition based
on the sum of the path lengths of the set. In this paper we give a
newoptimizationonditionwhihguaranteestheslidingblokwindow
deodingpropertyandhasaloweromputationalomplexitythanthe
previousone. Wealsoextendthemethodofpolestothemoregeneral
aseofso onstrainedhannels.
Index Terms: methodofpoles, sliding-blokdeoder,enoder,shift
ofnitetype,sosystem,stronglysynhronizingstate.
1 Introdution
Weareinterestedintheproblemofenodingdigitaldataintoaonstrained
set of sequenes. Typial appliations are oding for storage systems or
transmissionsystems. Fordigitalmagneti storageforinstane,onstraints
are run-length onstraints on the sequenes of bits stored. They are due
to physiallimitationsof the storagesystems. The onstraints that an be
modeled by a nite state mahine are alled rational onstraints or so
onstraints. The problem is then to enode a free soure of data, usually
sequenes of bits 0 and 1, into an available sequene, that is a sequene
InstitutGaspardMonge,UniversitedeMarne-la-Vallee,77454Marne-la-ValleeCedex
2,Frane. http://www-igm.univ-mlv.fr/~ beal .
onstraints. With a xed rate p :q oding strategy, eah blok of p bits is
enodedinablokofqbits,wherepandqaresmallintegers. Afterhanging
the alphabets, that is after reoding eah soure blok of length p by one
letterand eah onstrainedblokoflengthq byoneletter, theoding(seen
with these new alphabets), is a 1 : 1 rate oding from an unonstrained
soureinto a onstrainedhannel.
In thispaperwe willonsiderrst thelassof onstraints ofnitetype
that we dene as the onstraints that an be reognized or representedby
a deterministi loal (or denite) automaton. An automaton is omposed
of a nite set of states and a nite set of edges. Eah edge is dened by
its origin state, end state, and a letter alled its label. The automaton is
deterministi if there is at most one edge going out of a given state and
with a given label (the end state is then determined). The automaton is
deterministi and loal if it has in addition the following property : there
is an integer n suh that all equally labeled paths of length n end at the
same state. The nalstate dependsthenonlyon thelabel. The method of
poles is one method of hanneloding for hannels modeled by nitetype
onstraints. The method onstruts a rate 1 :1 enoder from arbitrary k-
arydatainto agiven nitetypehannelS witha topologialentropyh(S),
alsoalledtheShannonapaityofthehannel,stritlygreaterthanlog (k).
The log is usually taken base 2. The algorithm starts with a nite loal
automaton A representing the hannel, and builds a transduer, that is a
nite automaton labeled by pairs of letters, whose input labeling is right
losing (or deterministi with a nite delay), and whose output labeling is
loal. Sinetheinputlabelingisrightlosing,theenodingissequentialand
sinetheoutputlabelingisloalthe deoderisa sliding-blokdeoderand
therefore propagates a symbolerrorin an enodedS-stringbyat mostthe
deoderwindowlength. Thedeodingissometimesalledstateindependent
beausethenetworkofthedeoderisthenapurelyombinatorialone. Final
automataredutionstend to minimizethesizeof thetransduerand hene
the size of the deoding window. The onstrution of the output of the
transduer is based on the omputation of a set of states alled prinipal
states. Itispossibleto assoiate toeah prinipalstate asetofpathsofthe
automatonA. Itonsistsinallpathsgoingoutofagivenprinipalstateand
endingat another one. All possible sets of paths satisfy a Kraft ondition
onthelengthsofthepaths. Foreahprinipalstate,thehosensetofpaths
isone that minimizestheset length(that isthesum of thepathlengths of
the set) among all possible sets. Finally the prinipal states that are not
reahed by anypath of any set are removed and theremainders are alled
featureofstate independentdeoding.
The method of poles and theomputation of theset of prinipal states
has been introdued by P. A. Franaszek in [14 ℄. The above optimization
onditionandtheproofofits inueneonthesliding-windowpropertywere
givenin[8 ℄,(see also[9℄). Thismethoddoesnotwork ingeneralinthease
ofequalityofapaities ofthe soureand oftheonstrained hannel[4 ℄. A
dierentmethod,whihallows alsointheaseof equalityofapaities,has
beenobtainedin[1 ℄(seealso[24 ℄,[25 ℄). Itisalledthestatesplittingmethod
or the ACH algorithm and involves state splittings of some states of the
representationof thehannel. Manyothermethodsbased onstatesplitting
aredesribed in[27 ℄ (see also[3 ℄, [6 ℄,[20 ℄). Thesehannelodingproblems
areloselyrelatedwiththemathematialtheoryofsymbolidynamis[25℄.
In the ase of a hannel of nite type whose Shannon apaity is stritly
greater than the apaity of the soure, both methods lead in pratie to
transduers that an have about the same omplexity. By omplexity, we
meanthesizeofthelengthof thewindowofthedeoderand thesizeofthe
enoder. In thease of equality of apaities, thestate splittingmethod is
morepowerfulsinethemethodofpolesdoesnotapplyingeneralforthese
hannels.
The aim of the paperis to improve the method of poles for onstraints
of nite type by hoosing a better optimization ondition. We also show
that themethod an be extended to more general so onstraints. These
two points an be merged and one an make the improved optimization
onditionwork inthesoase.
The rst improvement is a omplexity improvement on the optimiza-
tion ondition that ensures the state independent deoding property. The
methodofpolesomputessetsofpathsoflengthatmostanintegerMwhih
depends on the entropy of the onstrained hanneland of the geometry of
theautomaton. Letusonsiderann-stateloalautomatonwhihrepresents
theonstraints. LetusalsoassumethattheShannonapaityofthehannel
is stritlygreater thanlog (k). In mostappliations, the integer n is small.
The boundM of thelength of the paths an asymptotially be k n
(see [4 ℄
forinstane)and is nevertheless nottoo bigin pratie. Foreah prinipal
statep,asetofpathsisomputedasanitetreeT
p
ofheightM andwhose
arity is the maximal outdegree of the graph representing the hannel. An
algorithmfor omputingthesetof prinipalstates and of one possibletree
assoiated to eah prinipal state, is given by Franaszek in [14 ℄. Its time
omplexity is exponential in M. A omputational problemappears when
wehave toapplytheoptimizationonditiongivenin[8℄. Indeed,ift
p isthe
p
onditionrequires to hek all treesrepresenting prex sets of paths being
prexes of paths of T
p
. This omputation is exponentialin the sum of the
sizes t
p
. We present here a new optimization ondition whih allows us to
omputeoptimaltreesina lineartimeinthesum of thesizest
p
. We prove
that the optimal trees obtained with this new optimization ondition still
leadto a sliding-blokwindowdeoder.
In the seond part of thepaper, we show how to extendthe method of
poles to so onstraints that are no more of nite type. This is possible
after a preliminarytransformation of the nite state mahine that models
the onstraints, in order to make the representation ontain some speial
statesalledstronglysynhronizingstates. Wedesribethistransformation
thanan be donewitheasy roundsof statesplittings oralsowitha bered
produt of two automata. This preproessing is very dierent and muh
simpler than the ACH algorithm [1 ℄. We then show that the method of
poles an be performed on this so representation and still builds a 1 : 1
enoderwith slidingblokdeoder.
Anothermethodto solvetheaseofsoonstraintshasbeendesribed
by R. Karabed and B. Marus in [21 ℄. It is based on the state splitting
proessof [1 ℄. Comparedto theirmethod,ourmethoddoesnotallowus,in
general, to treat thease of equalityof apaities forthe lass ofalmost of
nitetypeonstraints(see[21 ℄). Butthemethodofpolesforsoshiftsgives
apratialalternativetostatesplittingmethodsthatareratherompliated
inthesoase.
In Setion 2 we give the primary denitions and we briey reall the
method of poles. As it is used in Setion 3, we desribe the omputation
of the set of prinipal states. The algorithm of onstrution of the trees
obtained with the new optimization ondition is given in Setion 3. We
prove here that we do not lose the state independent deoding property.
The extension of the method of poles to the ase of so onstraints is
desribedinSetion4. Inthispaper,we willsometimesdesribealgorithms
asprogramswrittenina pseudoode. We adoptsome onventionsgiven in
[12 , p.4℄.
2 Denitions and bakground
LetA bean alphabet,thatis, aniteset ofsymbols alledletters.
A nite state automaton A=(Q;E) on thealphabetA is omposed of
twonitesets: Q,thesetofstates,andE,thesetofedges. Thesetofedges
edges((q
i
;a
i+1
;q
i+1 ))
0i<n
,theworda
1 a
2 :::a
n
beingthelabelofthepath.
A nite automaton is deterministi if and only if, whenever there are
two edges(p;a;q) and (p;a;r) thenq =r.
Aniteautomatonisloal iftherearethreenonnegativeintegersn;m;a
withm + a=nsuhthatwhenevertwonitepaths((q
i
;a
i+1
;q
i+1 ))
0i<n and
((q 0
i
;a
i+1
;q 0
i+1 ))
0i<n
have the same label w = a
1 a
2 :::a
n
, then q
m
= q 0
m .
Theinteger misformemoryand aforantiipation. IfAismoreoverdeter-
ministi, it is possible to have a nullantiipation, or, equivalently, to take
m=n. Loalautomata arealso alled denite automata [28 ℄. An automa-
tonissaid to be irreduible ifits graphis stronglyonneted. A onstraint
hannel S is said to be reognizedor representedbythe automaton Aif it
istheset oflabelsofbi-innitepaths oftheautomaton. Suh aonstrained
hannelisalledasohannel. Ifitanberepresentedbyaloalautoma-
ton,theonstraintorthehannelissaidof nitetype. Itisharaterizedby
a niteset of nitebloksavoided by any bi-innitesequene of the han-
nel. Thissetisalledasetofforbiddenwordsforthehannel. Itispossible
to represent the hannelby an automaton whih is both deterministiand
loal.
A word w of length m +a, where m and a are nonnegative integers,
is said to be (m;a)-synhronizing if there is a state p suh that eah path
(q
i
;a
i+1
;q
i+1 )
0i<m+a
labeled by w satises q
m
= p. We say in this ase
thatw synhronizesonto the state p. Awordwis said to be synhronizing
ifitis (m;a)-synhronizingforsome m and a.
We introdue the notion of strongly synhronizing states that will be
usefulinthelastsetion to extendthemethodof poles to soonstraints.
Foreahstatepandeahnonnegativeintegersm;a,wedenethesetE (m;a)
p
ofnitewordsuv,whereuisthelabelofapathendingatp,andvthelabel
of a path starting at p. A state p of an automaton A is said to be (m;a)-
strongly synhronizing, where m and a are nonnegative integers, if forany
state q distintfrom p,
E (m;a)
p
\E (m;a)
q
=;:
A state p is said to be strongly synhronizing if it is (m;a)-strongly syn-
hronizingforsome mand a. A state pof anautomatonA isthusstrongly
synhronizingifandonlytherearenottwodistintequallylabeledbi-innite
pathssuhthat therst onegoesthrougha state p at some index,and the
seond one goesthrougha state q 6=pat thesame index. This property is
omputableinapolynomialtime inthenumberofstates ofA (see[9 ℄ page
72).
We are going to enode a free k-ary soure into the onstraint han-
nelS representedbyanirreduible,deterministiand loalautomaton. We
assume that the topologial entropy, or the Shannon apaity, h(S) of S
satisesh(S)>log(k). The entropy is omputed asthe logof the spetral
radiusof the adjaeny matrixof thegraph of theautomaton A. It is de-
nedasthelimitof1=n log (ard (A n
\S
n
),whereS
n
is theset ofbloks of
lengthnthatan appearas asubblokof abi-innitesequene ofS.
The rst step in the method of poles onsists in ndinga subset P of
statesofQalledprinipal states,suhthatthereexistsapositiveintegerM
(as smallas possible) suh that one an assoiate to eah prinipal state p
aniteprexset Z
p
of nitepathsthat satisfythefollowingproperties
1. eahpath ofZ
p
isa pathof A thatbeginsat state p;
2. eahpath ofZ
p
endsat some state of P;
3. thelengthofeah pathof Z
p
islessthan orequalto M;
4. thesetZ
p
satisesthe Kraftinequality
X
z2Zp 1
k l (z)
1;
wherel(z) denotesthelengthof thepathz.
We reallthatasetofpathisa prexset 1
ifnopathisthestritbeginning
ofanotherone. A maximalsubsetof Qsatisfyingthese above onditions is
uniqueand isalled thesetof prinipal states of theautomaton.
It is shown in [8 ℄ that if h(S) > log(k), then, for a suÆiently large
integer M, suh a nonempty set P of prinipal states always exists. The
searhbeginswithM =1;2; ,(M isinrementedby1at eahstepwhen
theprevious one has failed). One a nonemptyset P of prinipal states of
Ais found, themaximalsizeM of thelengths ofthe paths of any possible
setZ
p
is xed.
Franaszek'salgorithmgivesthen,foreahprinipalstatepofP,aprex
set of paths Z
p
satisfying the above onditions, whih rst maximizes the
sum
X
z2Z
p 1
k l (z)
;
1
alsoalledaprexfreeset.
lengthl(Z
p )of Z
p
denedby
l(Z
p )=
X
z2Z
p
l(z): (1)
We an here remark that the last minimization ondition is a loal mini-
mizationonditioninthesensethat itdoesnotimplythat ahosen setZ
p
hasaminimallengthamong all sets thatsatisfyonditions1 to 4.
We desribebelowFranaszek's algorithminapseudoode. Theompu-
tationof thesetof prinipalstates an be performedasfollows
Computation of the set of prinipalstates
begin
P Q //whereQ istheset of statesof A
while (P 6=;and there isa state p2P withS
P
(p)<1)
doP P fpg
end,
whereS
P
(p)is themaximumof thesums
X
z2Zp 1
k l (z)
;
forallpossiblehoiesofprexsets Z
p
ofpathssatisfyingonditions1to 4.
We now desribe the omputation of the prediate fS
P
(p) < 1g for a
statepinP. ReallthatM isaxedintegerthatboundsthelengthsofthe
pathsonsidered. WerstbuildatreeT ofheightM whosenodesrepresent
thepaths in A of length lessthan or equalto M starting at p. The nodes
ofT arelabeledbystatesof Aand wedenote byr theroot labeledbyp. If
nis a node, its height is the lengthof the path from the root to the node.
EahnodeatheightatmostM 1labeledbyastate qadmitsasonlabeled
by s for eah edge (q;a;s) in A. This ompletely denes a tree that is a
overingtree starting at p and of height M, of theautomaton A. The size
ofthetree is its numberof nodes.
WeassignabooleanmarktoeahnodeofthetreeT. Anodeismarkedor
unmarked. ItismarkedifitslabelbelongstosetP andunmarkedotherwise.
We then assoiate to eah node a rational value. The value of a node n is
denoted by v(n)inthe pseudo ode below. Theomputations neessaryto
alulatevarelinearinthesizetofthetree. Theyareperformedbottom-up
fromthe leavesto theroot ofT.
P
begin
if(n isa leaf)
thenif (nis marked)thenv(n) 1 elsev(n) 0
else//n is notaleaf
if (nis markedand distintfrom theroot)
thenv(n) max(1;
P
ssonsofn v(s)
k )
elsev(n) P
ssonsofn v(s)
k
end
It is easy to verify that fS
P
(p) 1g if and only if v(r) 1. We point
out that this omputation depends on the urrent set P and an thus be
performedseveraltimesforasamestate pduringthesearhoftheprinipal
states. One the set of prinipal states is obtained, the same algorithm
an be slightly modied to produe the Franaszek's sets of paths Z
p of
Equation (1). We more preiselyompute, from the overing tree T of A
starting at p and of height M, a tree whose set of paths from the root to
theleaves isZ
p .
Computation of the Franaszek treeof the state p
begin
if(n isa leaf)
thenif (nis marked)then v(n) 1 elsev(n) 0
elsebegin //nis notaleaf
let v= P
ssonsofn v(s)
k
if(nis markedand distintfrom theroot)
thenif (v1)
thenv(n) 1 and utall branhes underthenode n
elsev(n) vand ut thelinkbetweennand eah of
its sonsssuhthat v(s)=0
elsev(n) v and utthelinkbetweenn andeah of its sonss
suh that v(s)=0
end
end
In the sequel, we denote by T
p
the nal tree assoiated to eah prinipal
state p omputed by the above algorithm. We refer to it as the Franaszek
treeassoiatedtotheprinipalstatep,oralsoastheprinipaltreeassoiated
to p.
statesandofthetreesT
p
,wherepisaprinipalstate. Weonsidertheloal
automaton of Figure 1 whih represents a onstrained hannel of entropy
greaterthanlog (2). Wehoosek=2andomputethesetofprinipalstates
foramaximalpathlengthM equalto 1. The searh failsbyomputingan
empty set. We try again with M = 2. The set of states P is initialized
to f1;2;3g. The marked nodesare irledand the valuesof the funtionv
aregiven inthesquaresbesideeahnode of thetree.
2 3
a
1 b f
e c
d
Figure 1: Aloalautomaton
FirststepWehaveP =f1;2;3g. We
hek if fS
P
(3) 1g. The omputa-
tionof v isdoneonthetree ofFigure
2whose root is denoted by r. We get
v(r)=3=4andweremovestate3from
thesetP.
1 1 1
1
1 2 3
3 3/4
3/2
Figure 2: Firststep.
Seond step We have P = f1;2g.
WehekiffS
P
(2)1g. Theompu-
tationof v is doneon thetree of Fig-
ure 3. We again get v(r) = 3=4 and
we remove state 2 from theset P. In
thisomputation,somebranheshave
been ut during the proess. This is
symbolizedbyadashed line.
1 1
1 1/2
1 2 3
1
0 3
1
2 3/4 1
Figure 3: Seondstep.
Third step We have P = f1g. We
hek fS
P
(1)1g. The omputation
ofvisdoneonthetreeofFigure4. We
now get v(r) = 1 and then S
P (1) =
1. The set of prinipal states is f1g.
Some branhes have been ut during
the proess. This is symbolized by a
dashedline. WeobtaintheFranaszek
treeT
1 .
1
1 1
1
1/2 1/2
1 1 1
1 1
2
0 0 0 1
3
2 3 3
Figure4: Thirdstep.
To eah tree T
p
, where p is a prinipal state, is assoiated in a natural
waya set of paths Z
p
satisfying theonditions 1 to 4 dened as theset of
pathsof thetreegoing fromtheroot to aleaf. Eah suhpathorresponds
to apathin A. Sineondition4 issatised, thatis,
X
z2Zp 1
k l (z)
1;
it is possible to extrat from Z
p
a subset Z 0
p
suh that the above Kraft
inequalitybeomes aKraft equality
X
z2Zp 1
k l (z)
=1:
Inorder to minimize thenumber ofnotations, we stillallthisnew set Z
p .
We moreoverassume,bypossiblyremovingsomeunusefulstatesinP,that
the following ondition is satised: for eah pair of prinipal states p;p,
there isa onatenationof pathsof S
p2P Z
p
goingfrom p to p 0
.
Nowa transduerT,used to enode and deode,an be onstrutedas
follows. Foreahprinipalstate p,wehooseaprexodeX
p
on ak-letter
alphabet,thathasthesame lengthdistributionasZ
p
. Wehoosea length-
preservingbijetion
p
from Z
p to X
p
. We dene a state p,^ alled a pole,
foreah prinipalstate p. Foreah pathz inZ
p
,thetransduerhas apath
^
zoflengthl(z) fromstate p^tostate
^
p 0
,wherep 0
istheterminalstate ofthe
path inA. One an imaginel(z) 1 dummystates of T strung along the
pathz.^ Tothepathz^is assigned asinputlabel
p
(z),and asoutput label,
thelabelof thepathz inA.
As shown in [9 , p. 172℄, this proess does not ensure that the output
automaton of the transduer is a loal automaton, and therefore that the
deodingis slidingblok. Thisis baseduponthefatthat thesets Z
p have
to satisfy a length optimization ondition dierent from the ondition of
Equation(1), inorder to getthe state independent deodingproperty.
3 A new optimization ondition in the method of
poles
In [8 ℄ is given a length optimization onditionthat ensures that the trans-
duer has a loal output, and then a sliding blok window deoder. The
ondition is to hoose, among all possible sets Z
p
of paths satisfying on-
ditions 1 to 4, a set that minimizes the sum of the path lengths. This
optimization ondition is a global optimization ondition ompared to the
loal one desribed inthe denitionof Franaszek's sets Z
p
(see theremark
below the denition of the sets Z
p
in the seond setion). This requires a
searhofallprexsets ofpathssatisfyingonditions1to 4,thepathsbeing
prexes of pathsof theFranaszek tree T
p
obtained intheprevious setion.
Ifwedenote byt
p
thesizeofthetreeT
p
,thatisthenumberof nodesofT
p ,
thisexhaustivesearh hasan exponentialtime ostint
p .
In thissetion, we give a new optimizationondition that an be om-
putedfrom theFranaszek treesina linear time.
We allsubtree ofa treeT thepart ofT formedbya node ofT withall
itsdesendants. We allprinipal subtreeofaFranaszek treeT asubtree of
T whoseroot is notaleaf of T and islabeledbyaprinipal state.
We nowgive a family of new optimization onditions. We assume that
theprinipalsubtreesarepreorderedwithapreorder,denotedbyord,whih
satisesthefollowingondition:
A isa strit subtreeof B =)ord(A)<ord(B): (2)
We hoose, for eah prinipal state p, a prinipal subtree of any Franaszek
tree,whihisrootedbyanodelabeledbyp,andminimizesthepreorderord.
We denote itbyB
p
. We pointout thatthe trees(B
p )
p2P
satisfy theKraft
inequalityonditionsinetheyareprinipalsubtreesofsometreeT
q
. Thisis
duetothefatthateahnodelabeledbyaprinipalstateofaFranaszektree
hasanalvaluev,omputedduringtheomputationoftheFranaszektrees,
whih is greaterthan orequal to 1. Thispropertyis equivalent to the fat
that the prinipal subtree satises the Kraft inequality. A nal extration
onsistsinremovingsomebranhesoftheprinipalsubtrees(B
p )
p2P to get
nally sets of paths whih satisfy theKraft equality. The transduer used
to enode and deode is builtfrom thenew sets Z
p
asexplained inSetion
2. Ithas astrongly onnetedgraph.
The following proposition states that the important property of state
independentdeodingisguaranteed.
Proposition 1 The output labeling of the transduer onstruted from the
trees (B
p )
p2P
isa loal automaton.
Proof : We prove that the transduer onstrutedfrom thetrees (B
p )
p2P
that we get before the nal extration has a loal output. Sine this au-
tomatonisobtainedfrom thenalpruned trees(B
p )
p2P
byremovingsome
pathsfrom it, itwillprovethe result.
Reall that an irreduible automaton is loal ifand only if it does not
admittwo distintequallylabeledyles. Letusthusonsidertwo distint
equallylabeledylesoftheoutputautomatonofthetransduerT,denoted
by ((p
i
;a
i+1
;p
i+1 )
0i<n
and ((p 0
i
;a
i+1
;p 0
i+1 )
0i<n
. The addition on theset
of indies f0;1;:::;n 1g of the yles, has to be understood modulusn.
We rst onsiderthease where thereis an indexisuhthat p
i and p
0
i are
bothpolesofthetransduer. ThetwoylesoftheoutputofT projetonto
twoylesoftheautomatonAand thetwo poles p
i andp
0
i
projetontotwo
prinipalstates. Sine theprinipal statesof A arestrongly synhronizing,
the two projeted prinipal states are equal, and then the poles p
i and p
0
i
also. It follows that the two yles of T are also equal, by onstrution of
thetransduer.
Weannowassumethatforeahindexi,p
i andp
0
i
arenotsimultaneously
poles. Sineeah yleofT goesthroughapole, thereisat leastone index
isuh that p
i
is apole and p
i
isnot one. Thetwo yles of thetransduer
projet onto two yles in A, one going through the projetion of p
i , the
other one going through the projetion of p 0
i
at the same time. Sine the
projetion of p
i
is a prinipal state whih is strongly synhronizing, the
projetions of p
i and p
0
i
are equal. We all it the projetion of the pair
(p
i
;p 0
i
). Sine we are onlyinterested inthepairs (p
j
;p 0
j
) suh that p
j orp
0
j
isapole, we anassume,after renumbering,that allpairsarelikethis,two
states of onseutive indiesbeing linked bya path of length at least one.
We dividethese indiesinto two disjoint sets, one set A where p
j
is a pole
andonesetA 0
wherep 0
j
isapole. Werestritthenourattentiononlytothe
boundaryset I onsistingof all indiesj suh that
(j2A and(j+1)2A 0
) or(j 2A 0
and (j+1)2A):
LetusassumethatthesetI isf0;1;:::;r 1gand thattheadditionon I is
modulusr. We onsiderthen theirularsequene (e
j )
0j<r
of theproje-
tionsof pairsof states (p
i
;p 0
i
) indexed bysuessive points of I modulusr
(see Figure5 wherethepoles areirled).
p
p’ k p k+1
e w e z
e i
j j+1 j+1 j+2 j+1
w j z j
Figure 5: Pathsenroahing uponeah other
Let e
j
be one element of this sequene whih is the projetion of a
pair (p
i
;p 0
i
). Let us assume that p
i
is a pole (the ase where p 0
i
is a pole
is symmetri). Then e
j+1
is the projetion of a pair (p
k
;p 0
k
) where p 0
k and
p
k+1
are poles. Let usdenote byw
j
(resp. z
j
the pathfrom p
i to p
k (resp.
from p
k to p
k+1
) in the rst yle. The projeted path onto A of w
j z
j
belongs to the set of paths assoiated to the tree B
e
j
. Sine the states p
k
and p 0
k
projet onto the same prinipal state of A, the state e
j+1
,the tree
B
e
j
has a proper prinipalsubtree B rooted by e
j+1
. By denition of the
optimizedtrees(B
p )
p2P
,we then have
ord(B
e
j+1
)ord(B)<ord(B
e
j ):
ej+1 ej
proofbya ontraditionsine thesequene(e
j )
0j<r
isnite.
We now give some examples of possiblepreorders on prinipalsubtrees
satisfyingondition(2):
theheightof a tree.
thelength of a tree, that is thesum of thelengths of all paths going
fromtheroot to eah leaf.
For pratial appliations, we adopt this seond preorder as optimization
ondition. This new optimization ondition an be stated as follows. We
ompute and assoiate to eah prinipal state p a prinipal subtree B
p ,
rootedbya node labeledbyp,whih hasa minimallength amongall prin-
ipal subtrees of all Franaszek trees. This hoie is due to two reasons.
First,thesizeoftheenodingtransduerdependsonthelengthofthetrees
(B
p )
p2P
. The deodingwindowlengthisalso bounded above bya funtion
ofthesizeofthistransduer. Seond,theomputationofthetrees(B
p )
p2P
islinear inthesum ofthesizes ofthe Franaszek trees.
Wepreisebelowtheomputationofthetrees(B
p )
p2P
fromtheFranaszek
trees. A bottom up omputation of these trees is possible as follows. We
rstomputethelengthsofallsubtreesoftheprinipaltrees. Thisiseasily
donebyomputingtogetherforanodenthepair(l
n
;x
n
) ofthelengthl
n of
thesubtreerootedbythisnode,andthenumberx
n
ofleavesofthissubtree.
Ifnis itselfaleaf,wehave (l
n
;x
n
)=(0;1). Ifnis anode whihhasssons
denoted by1;2;:::;s, we have thefollowingtrivialequalities:
x
n
= s
X
i=1 x
i
;
l
n
= s
X
i=1 (l
i +x
i )=
s
X
i=1 l
i +x
n
Let us now denote by N the set of roots of all prinipal subtrees of all
Franaszek trees and by (n) the label of a node n in N. We dene B
p as
thesubtreerooted byn wherenis anode of theforest(B
p )
p2P
suh that
l
n
=minfl
p
jp2N and (n)=pg:
A bottom-up explorationof theforest (T
p )
p2P
allows simultaneous ompu-
tationsofpairs(l
n
;x
n
)andoftheforest(B
p )
p2P
. Sineeahtreeisexplored
one, thetimeomplexityof theomputationis O(
P
p2P t
p ).
log(2) pitured in Figure 6. The trees (T
p )
p2P
are given in Figure 7. The
set of prinipal states with M = 2 is P = f1;2;3;4g. The trees (B
p )
p2P
obtainedattheendoftheomputationhavetheirrootpointedinthegure.
The value l
n
for a node n is given besidethe node. The nodesare labeled
bytheirprojetionstate onto theautomatonofFigure6thatrepresentsthe
hannel. Analoperation,madeinorderto gettreesthatsatisfytheKraft
equality, onsists in removing some branhes of the trees obtained at the
previousstep. Thisis showninFigure 8.
q 1
5
4 6
2
3 a b c
u
t
d e
y z
f
p x r
Figure6: Channel of entropygreater thanlog(2)
B4
4
1
3 T3 T4
T2
5 1 6 13 3
3 4 3 1
13 2
2 3 2 3 1
T1 B3
B2 B1
5
Figure 7: Computationofthe forest(B
p )
p2P
Here we an remark that we have B
p
= T
p
for at least one prinipal
state p, sineotherwise, we ould hoose a smaller integer M that bounds
thepathlengths duringtheomputation of theprinipalstates.
3 4
a c d b e f x
yz tu
p q r
1 2
Figure8: Final extration
4 Extension of the method of poles to so on-
straints
In thissetion,we extendthe methodof poles to the more general lass of
onstraints representedby a notneessarily loal automaton. We onsider
atransitivesohannelS,thatis,ahannelthatanberepresentedbyan
irreduibleautomaton. Ifitisnotthease, itisalwayspossibletoonsider
a subset of the hannel that has thisproperty and thesame apaity. We
alsoassumethattheentropyofSisstritlygreaterthanlog(k),wherekisa
positiveinteger. Inorder to extendthemethod,we willusestate splittings
ofstatesof therepresentationof thehannel. Werst givethedenitionof
thenotionofstate splitting,whihomes from symbolidynamis.
We dene the operation of output state splitting in an automaton A =
(Q;E). Let q be a vertex of Q and let I (resp. O) be the set of edges
oming in q (resp. going out of q). Let O = O 0
+O 00
be a partition of
O. Theoperation of (output) state splitting relative to (O 0
;O 00
) transforms
A into the automaton B = (Q 0
;E 0
) where Q 0
= (Qnfqg)[fq 0
g[fq 00
g is
obtainedfromQbysplittingstate q intotwostatesq 0
andq 00
,andwhereE 0
isdened asfollows(see Figure 9and 10)
1. Alledges of E that arenotinidentto q areleft unhanged.
2. Thestates q 0
and q 00
have thesame inputedgesas q.
3. Theoutput edges of q aredistributedbetween q 0
and q 00
aording to
thepartition of O into O 0
and O 00
. We denote U 0
and U 00
the sets of
outputedges of q 0
and q 00
respetively
U 0
=f(q 0
;x;p)j(q;x;p)2O 0
g and U 00
=f(q 00
;x;p)j(q;x;p)2O 00
g.
c q
O’
O’’
d a
e
b
Figure9. Automaton A
q’’
e
q’
U’’
U’
c b a
a e
d d
Figure 10. Automaton B
The notionofinput state splitting isdenedsimilarly.
We now transform the automaton that represents the onstraints into
anotherone thathasat least one stronglysynhronizingstate.
Proposition 2 A transitive so hannel admits a representation that has
atleast onestrongly synhronizing state.
Proof : It is known that a transitive sohannel hasa unique minimal
deterministi representation. This representation admits a synhronizing
word. ItisalledtheminimalautomatoninautomatatheoryandtheFisher
over in the symboli dynamis theory (see for instane [10 , p. 478℄, [23 ℄
or[9℄).
LetAbesuharepresentationandletwbea(m;a)-synhronizingword
onto a state p. We assoiate to eah state q the set E (m;a)
q
of pairs (u;v)
of nitewords, whereu is the label of a path ending at q, and v the label
of a path starting at q. The pairs (u;v) are also denoted by u v. One
an remarkthat ifuv and u 0
v 0
belongs to a set E (m;a)
q
, then uv 0
and
u 0
v also. We now onsiderforeah state q the longestprex z
q
(possibly
equal to the empty word) of all words v suh that there is a word u with
uv2E (m;a)
q
. Wehooseastater suhthatz
r
hasaminimallengthamong
all(z
q )
q2Q
. Ifthelengthofz
r
isstritlylessthantheantiipationa,theset
E (m;a)
r
ontains two pairsuz
r
bv and u 0
z
r v
0
,where u;u 0
;v;v 0
are words,
andb;distintletters. Ifz
r
isnot theempty word, letd beits rst letter.
Wedenethewordx
r byz
r
=dx
r
. Wedo anoutputstatesplittingofstate
rbypartitioningtheoutgoingedgesofrintheonesendingatastatessuh
thatx
r
bis aprexofz
s
andtheother ones. Ifz
r
istheemptyword,wedo
an output state splittingof state r by partitioningthe outgoing edges of r
inthe oneslabeled by b and theother ones. This state splittingproess of
theautomaton is iterated from the new automaton obtained. This proess
alwaysstopssineifastate qissplitinq
1 andq
2
,theardinalitiesofE (m;a)
q1
andofE
q
2
arestritlylessthantheardinalityofE
q
. Theautomaton
omputedatthelaststep issuhthatallwordsz
q
have alengthequalto a.
We do the symmetrial operations with the longest suÆxes y
q of all
words u suh that there is a wordv with uv 2 E (m;a)
q
. We use this time
inputstate splitting. The nal automaton that we get is suh that, forall
of its states q, the word y
q
has length m, and the word z
q
has length a.
This means that eah set E (m;a)
q
of the nal automaton is redued to one
pair y
q z
q
. Sine w isa synhronizingwordof the initialautomaton, y
p is
theprex of lengthm of w,and z
p
is its suÆx of length a. This state is a
stronglysynhronizingstate of thenalautomaton.
Wementionthatanotherproofofthepreviousresultanbeobtainedby
doinga diret (orberedprodut)of theinitialautomaton that reognizes
thehanneland a (m;a)-loal universalDe Bruinautomaton,(we refer for
instaneto[9℄forthisnotion). Intheaboveproof,theorderhosentotreat
thepastand thefuture anbehanged. Thesequenesofinputandoutput
statesplittingsanbemerged. Theinterestofthestatesplittingwayversus
the produt of automata is that one an stops the proess as soon as we
have obtained enoughstronglysynhronizingstates.
Let S be a transitive so hannel reognizedbyan automaton A with
anentropyh(S)>log (k). By thepreviouspropositionwe an assume that
A has a nonempty set of strongly synhronizing states. A set of prinipal
states for an integer M is obtained like in Setion 3 by starting this time
theomputation witha set P reduedto the stronglysynhronizing states
only.
Computation of the set of prinipalstates
begin
P theset ofstronglysynhronizingstates
while (P 6=;and there isa state q with S
P
(q)<1)
doP P fqg.
end,
whereS
P
(q) isthemaximum ofthe sums:
X
z2Zq 1
k l (z)
;
forallpossiblehoiesofprexsetsZ
q
ofpathssatisfyingonditions1to 4.
Sine the set of strongly synhronizing states is not empty and sine
theShannonapaity of thehannelis stritlygreater thatlog (k),one an
prove likeforonstraintsofnitetype(see Setion2), thatanonemptyset
set.
The onstrution of a oding and deoding transduerin thendoneex-
atlylike intheprevioussetion. Theproofthatits output automatonis a
loalautomatonisthesame. Itisduetothestronglysynhronizingproperty
oftheprinipal states.
ExampleLetusonsiderthe transitiveso system reognizedby theau-
tomatonAofFigure11. Itsentropyisstritlygreaterthanlog (2). Thisau-
tomatonistheminimaldeterministirepresentationofthesystem. Itadmits
at leastone synhronizingword: theword bb,whihis (2;0)-synhronizing.
Italsohasastronglysynhronizingstate: thestate4,whihis(2;0)-strongly
synhronizing. Inordertogetasmanystronglysynhronizingstatesaspos-
sible,wedoa sequeneofinputstate splittingsandgettheautomatonB of
Figure12 wherethe setE (2;0)
p
isrepresentedinsideeah state p.
b
1 2 3
4
b
c c
b c
a a
Figure11: AutomatonA
1ab
2ba 2ab 3ba
3aa 1bb
4bc
2cb
3cc
b a 3ac b
a b c
c
c c
c c c
b b b
a
b
a b
a
b
Figure12: AutomatonB
S =f(1;bb);(4;b);(2;b);(3; );( 3;a);(3;aa)g ;
where a state p is denoted here by its projetion state onto Aand theleft
omponent ofthe uniquepair of E (2;0)
p
. A nonempty setof prinipal states
isobtainedfor M =5. The setof poles is then
P =f(1;bb);(2;b);(3; );(3;a) ;(3 ;aa) g:
Thelabelsof thepaths Z
p
assoiated to eah polep are:
Z
(3;aa)
= f;ba;bbb;bbaa;bb;babb;baba;bbabbg
Z
(2;b)
= fa;bb;abb;baa;b;aba;babbg
Z
(1;bb)
= fb;;aag
Z
(3;a)
= C
(3;)
=fb;g
Sine the optimized treesassoiated to the poles (3;a) and (3;) are the
same,one an mergethesetwopolesinthetransduer. Thetransduerob-
tainedhas4polesand 41statesfortheintegerM =5. Abettertransduer
an be obtained withan initial transformationof theautomaton B of Fig-
ure12. If thestate (1;ab) is removed forinstane, thehannelrepresented
has an entropy whih is still greater than log (2). The number of strongly
synhronizingstatesis thenadvantageouslyinreasedin
S=f(1;bb);(4;b);(2;b);(3; );( 3;a);(3;aa);(2 ;ab)g ;
and thefollowingsetof polesis obtainedwithM =2 only
P =f(1;bb);(2;b);(3; );(3;a) ;(3 ;aa) ;(2 ;ab) g:
With a nal automata redution (state merging), we get the very small
enodingtransduerofFigure 13. Itsslidingblokdeodingwindowlength
isonly2.
5 Aknowledgment
We thank anonymousreferees forhelpfulomments.
4
1/b 0/b 0/c 0/c
1/a 0/a
1/b 1/c
1/a 0/b
1 5 3
2
Figure13: EnodingTransduer
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