An omega-power of a context-free language which is Borel above Delta^0_omega

0.15 MB·English

by  Jacques Duparc

StefanBold,BenediktLo¨we,ThoralfRa¨sch,JohanvanBenthem(eds.)
InfiniteGames
Papersoftheconference“FoundationsoftheFormalSciencesV”
heldinBonn,November26-29,2004
8
0 An ω-power of a context-free language
0
2 which is Borel above ∆0
ω
n
a
Jacques Duparc and OlivierFinkel
J

1 Universite´deLausanne,
1 InformationSystemsInstitute,and
WesternSwissCenterforLogic,HistoryandPhilosophyofSciences

] BaˆtimentProvence
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CH-1015Lausanne
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[ EquipeMode`lesdeCalculetComplexite´

Laboratoiredel’InformatiqueduParalle´lisme⋆⋆
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CNRSetEcoleNormaleSupe´rieuredeLyon
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3 46,Alle´ed’Italie69364LyonCedex07,France.
8 E-mail:[email protected] and [email protected]
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.
1
Abstract. Weuseerasers-likebasicoperationsonwordstoconstructasetthatis
80 bothBorelandabove∆0ω,builtasasetVωwhereV isalanguageoffinitewords
acceptedbyapushdownautomaton.Inparticular,thisgivesafirstexampleofan
0
ω-powerofacontextfreelanguagewhichisaBorelsetofinfiniterank.
:
v
i
X
1 Preliminaries
r
a
Given a set A (called the alphabet) we write A∗, and Aω, for the sets
of finite, and infinite words over A. We denote the empty word by ǫ. In
⋆⋆UMR5668-CNRS-ENSLyon-UCBLyon-INRIA
LIPResearchReportRR2007-17
Received:...;
Inrevisedversion:...;
Acceptedbytheeditors:....
2000MathematicsSubjectClassification. PRIMARYSECONDARY.
(cid:13)c 2006KluwerAcademicPublishers.PrintedinTheNetherlands,pp.1–14.
2 JACQUESDUPARCANDOLIVIERFINKEL
order to facilitate thereading, we use u,v,w for finitewords, and x,y,z
for infinite words. Given two words u and v (respectively, u and y), we
writeuv (respectively,uy)for theconcatenationofu andv (respectively,
ofuandy).LetU ⊆ A∗ andY ⊆ A∗∪Aω,weset:UY = {uv,uy : u ∈
X ∧v,y ∈ Y}.
We recall that, given a language V ⊆ A∗, the ω-power of this lan-
guageis
Vω = {x = u u ...u ... ∈ Aω : ∀n < ω u ∈ V \{ε}}
1 2 n n
Aω isequippedwiththeusualtopology.i.e.theproductofthediscrete
topology on the alphabet A. So that every open set is of the form WAω
for any W ⊆ A∗. Or, to say it differently, every closed set is defined as
thesetofallinfinitebranchesofatreeoverA.WeworkwithintheBorel
hierarchyofsetswhichisthestrictlyincreasing(forinclusion)sequence
of classes of sets (Σ0) - together with the dual classes (Π0) and
ξ ξ<ω1 ξ ξ<ω1
the ambiguous ones (∆0) - which reports how many operations of
ξ ξ<ω1
countable unions and intersections are necessary to produce a Borel set
on thebasisoftheopenones.
A reduction relation between sets X, Y is a partial ordering X ≤ Y
whichexpressesthattheproblemofknowingwhetheranyelementxbe-
longs to X is at most as complicated as deciding whether f(x) belongs
toY,forsomegivensimplefunctionf.Averynaturalreductionrelation
between sets of infinite words (closely related to reals), has been thor-
oughly studiedby Wadge in the seventies.From thetopologicalpoint of
view, simple means continuous, therefore the Wadge ordering compares
sets of infinite sequences with respect to their fine topological complex-
ity. Associated with determinacy, this partial ordering becomes a pre-
wellorderingwithanti-chainsoflengthatmosttwo.ThesocalledWadge
HierarchyitinducesincrediblyrefinestheoldBorelHierarchy.Determi-
nacy makes it way through a representation of continuous functions in
termsofstrategiesforplayerIIinasuitabletwo-playergame:theWadge
game W(X,Y). In this game, players I and II, take turn playing letters
of the alphabet corresponding to X for I, and letters of the alphabet cor-
responding to Y for II. In order to get the right correspondence between
a strategy for player II and a continuousfunction, player II is allowedto
skip,whereas Iis not.However,II mustplay infinitelymanyletters.
Asusual,reductionrelationsinducethenotionofacompleteset:aset
that both belongs to some class, whose members it also reduces. In the
ABORELω-POWEROFACONTEXT-FREELANGUAGEABOVE∆0ω 3
contextofWadgereducibility,asetiscompleteifitbelongstosomeclass
closed by inverse image of continuous functions, and reduces everyone
of its members. A class which admits a complete set is called a Wadge
Class.Asamatteroffact,allΣ0,andΠ0,areWadgeclasses,whereas∆0
ξ ξ ξ
(ξ > 1)arenot.
For instance, the set of all infinite sequences that contains a 1 is Σ0-
1
complete, the one that contains infinitely many 1s is Π0-complete. As
2
a matter of fact, reaching complete sets for upper levels of the Borel
hierarchy,requires othermeanswhichweintroduceinnextsections.
2 Erasers
For climbing up along the finite levels of the Borel hierarchy, we use
erasers-likemoves,see[Dup01]. Forsimplicity,imagineaplayer(either
IorII)playingaWadgegame,inchargeofasetX ⊆ Aω,withtheextra
possibilityto deleteany terminalpartofherlast moves.
WerecallthedefinitionoftheoperationX 7→ X≈oversetsofinfinite
words. It was first introduced in [Fin01] by the second author, and is a
simplevariantofthefirst author’soperationofexponentiationX 7→ X∼
whichfirst appeared in[Dup01].
We denote |v| the length of any finite word v. If |v| = 0, v is the
empty word. If v = v v ...v where k ≥ 1 and each v is in A, then
1 2 k i
|v| = k and we write v(i) = v and v[i] = v(1)...v(i) for i ≤ k ; so
i
v[0] = ǫ. The prefix relation is denoted ⊑: the finite word u is a prefix
of the finite word v (denoted u ⊑ v) if and only if there exists a (finite)
word w such that v = uw. the finite word u is a prefix of the ω-word x
(denoted u ⊑ x)iffthereexistsan ω-wordy such thatx = uy.
GivenafinitealphabetA, wewriteA≤ω forA∗ ∪Aω.
Definition 2.1. Let A be any finite alphabet, և ∈/ A, B = A ∪ {և},
andx ∈ B≤ω, then
և
x is inductivelydefinedby:
և
ǫ = ǫ, andfora finiteword u ∈ (A∪{և})∗:
և և
(ua) = u a, ifa ∈ A,
և և և
(uև) = u with itslastletterremoved if|u | > 0,
և և
(uև) isundefinedif |u | = 0,
andfor uinfinite:
4 JACQUESDUPARCANDOLIVIERFINKEL
և և
(u) = lim (u[n]) , where, given β andv inA∗,
n∈ω n
v ⊑ lim β ↔ ∃n∀p ≥ n β [|v|] = v.
n∈ω n p
We now make easy this definition to understand by describing it in-
և
formally.Forx ∈ B≤ω,x denotesthestringx,onceeveryևoccurring
in x has been “evaluated” to the back space operation (the one familiar
toyourcomputer!),proceedingfromlefttorightinsidex.Inotherwords
և
x = x from which every interval of the form “aև” (a ∈ A) is re-
և և
moved.Byconvention,weassume(uև) isundefinedwhenu isthe
empty sequence. i.e. when the last letter և cannot be used as an eraser
(because every letter of A in u has already been erased by some eraser
և
և placed in u). We remark that the resulting word x may be finite or
infinite.
Forinstance,
և
– ifu = (aև)n, forn ≥ 1, oru = (aև)ω then(u) = ǫ,
և
– ifu = (abև)ω then (u) = aω,
և
– ifu = bb(ևa)ω then(u) = b,
և
– if u = և(aև)ω or u = aևևaω or u = (aևև)ω then (u) is
undefined.
Definition 2.2. ForX ⊆ Aω,
և
X≈ = {x ∈ (A∪{և})ω : x ∈ X}.
The following result easily follows from [Dup01] and was applied in
[Fin01,Fin04]to studytheω-powersoffinitarycontextfree languages.
Theorem 2.3. Let n be an integer ≥ 2 and X ⊆ Aω be a Π0-complete
n
set. Then X≈ is a Π0 -completesubsetof(A∪{և})ω.
n+1
Nextremarks willbeessentiallater.
Remark 2.4. Considerthefollowingfunction:
f : x ∈ (A∪{և})ω 7→ y ∈ Aω
defined by:
և
– y = 0ω ifx isfiniteorundefined,
և
– y = x otherwise.
ABORELω-POWEROFACONTEXT-FREELANGUAGEABOVE∆0ω 5
It isclearly Borel. Infact aquickcomputationshowsthattheinverse
imageofany basicclopen setis Borel oflowfiniterank.
Remark 2.5. Let X beany subset oftheCantor space{0,1}ω, and f as
inremark 2.4. If0ω 6∈ X, thenforanyx ∈ {0,1,և}ω
x ∈ X≈ ⇐⇒ f(x) ∈ X
In otherwords,X≈ = f−1X. Inparticular, ifX isBorel, so isX≈
3 Increasing sequences of erasers
The followingconstruction has been partly used by the second authorin
[Fin04]to construct a Borel set ofinfiniterank which is an ω-power,i.e.
in the form Vω, where V is a set of finite words over a finite alphabet
Σ. We iterate the operation X 7→ X≈ finitely many times, and take the
limit.Moreprecisely,
Definition 3.1. Given anyset X ⊆ Aω:
– X≈0 = X,
k
– X≈1 = X≈,
k
– X≈2 = (X≈1)≈,
k k
– X≈(k) = (X≈(k−1))≈, where we apply k times the operation X 7→
k k
X≈ withdifferent newlettersև , և , ...,և , և ,
k k−1 2 1
in suchaway thatwehavesuccessively:
• X≈0 = X ⊆ Aω,
k
• X≈1 ⊆ (A∪{և })ω,
k k
• X≈2 ⊆ (A∪{և ,և })ω,
k k k−1
• X≈(k) ⊆ (A∪{և ,և ,...,և })ω.
k 1 2 k
– Weset X≈(k) = X≈(k).
k
X≈∞ ⊆ (A∪{և : 0 < n < ω})ω isdefined by
n
x ∈ X≈∞ ⇐⇒
def
և
– foreach integern, x = xև1...ևn−1 n isdefined, infinite,and
n
– x = lim x isdefined, infinite,andbelongstoX.
∞ n<ω n
Remark 3.2. Considerthefollowingsequenceoffunctions:
6 JACQUESDUPARCANDOLIVIERFINKEL
– f (x) = x (f is theidentity),
0 0
– f : (A∪{և : k < n < ω})ω 7−→ (A∪{և : k+1 < n < ω})ω
k+1 n n
defined by:
և և
• f (x) = x k+1 ifx k+1 isinfinite,
k+1
և
• f (x) = 0ω ifx k+1 isfiniteorundefined,
k+1
By induction on k, one shows that every function f is Borel - and
k
evenBorel offiniterank.
Moreover, since Borel functions are closed under taking the limits
[Kur61], thefollowingfunctionisBorel.
f : (A∪{և : 0 < n < ω})ω 7−→ Aω
∞ n
defined by:
– f (x) = lim f (x) iflim f (x) is defined, andinfinite,
∞ n<ω n n<ω n
– f (x) = 0ω otherwise.
∞
Remark 3.3. LetX ⊂ {0,1}ω with0ω 6∈ X, thenforany
x ∈ ({0,1}∪{և : 0 < n < ω})ω
n
x ∈ X≈∞ ⇐⇒ f (x) ∈ X
∞
In other words, X≈∞ = f −1(X), which shows that whenever X is
∞
Borel, X≈∞ isBorel too.
In fact, with tools described in [Dup01], and [Dup0?], it is possible
toshowthatgivenanyΠ0-completesetY,thesetY≈∞ belongstoΠ0 .
1 ω+2
IfX isthesetofinfinitewordsoverthealphabet{0,1}whichcontainsan
infinite number of 1s, then it is also possible to show that X≈∞ is Borel
by completely different methods involving decompositions of ω-powers
[FS03,Fin04].
Proposition3.4. Let X be the set of infinitewords over {0,1}that con-
taininfinitelymany1s,
X≈∞ ∈ ∆1 \∆0
1 ω
ABORELω-POWEROFACONTEXT-FREELANGUAGEABOVE∆0ω 7
Proof. The fact X≈∞ is Borel is Remark 3.3. As for X≈∞ ∈/ ∆0, it
ω
is a consequence of the fact that the operation Y 7−→ Y≈ is strictly
increasing(fortheWadgeordering)inside∆0 (see[Dup01][Dup0?]).In
ω
other words, for any Y ∈ ∆0 the relation Y < Y≈ holds (< stands
ω W W
for the strict Wadge ordering). But, as a matter of fact, (X≈∞)≈ ≤
W
X≈∞ holdswhich forbidsX≈∞ tobelongto ∆0.
ω
Indeed,toseethat(X≈∞)≈ ≤ X≈∞ holds,itisenoughtodescribe
W
awinningstrategyforplayerIIintheWadgegameW (X≈∞)≈,X≈∞ .
(cid:0) (cid:1)
In this game, player II uses ω many different erasers: և ,և ,և ,...
1 2 3
whose strength is oppositeto their indices (և erases all erasers և for
k j
any j > k but no և for i ≤ k). While player I uses the same erasers as
i
player IIdoes, plus an extraone(և) which is strongerthan all theother
ones.
The winningstrategyforII derivesfrom ordinal arithmetic:1+ω =
ω.It consistsin copyingI’srun withashifton theindicesoferasers:
– ifIplaysaletter0 or1,then IIplaysthesameletter,
– ifIplaysan eraser և , IIplays theeraser և .
n n+1
– if I plays the eraser և (the first one that will be taken into account
when theerasingprocess starts),then IIplaysև .
0
Thisstrategyisclearly winning.
4 Simulating X≈∞ by the ω-power of a context-free
language
It was already known that there exists an ω-power of a finitary language
which is Borel of infinite rank [Fin04]. But the question was left open
whethersuchafinitary languagecouldbecontext free.
This article provides effectively a context free language V such that
Vω isaBorelsetofinfiniterank,andusesinfiniteWadgegamestoshow
thatthisω-powerVω islocated above∆0 intheBorel hierarchy.
ω
The idea is to have X≈∞, where X stands for the set of all infinite
words over {0,1} that contain infinitely many 1s to be of the form Vω
forsomelanguageV recognizedbya(nondeterministic)PushdownAu-
tomaton.Wefirstrecallthenotionofpushdownautomaton[Ber79,ABB96].
Definition 4.1. A pushdownautomaton(PDA)is a7-tuple
M = (Q,A,Γ,δ,q ,Z ,F)
0 0
8 JACQUESDUPARCANDOLIVIERFINKEL
where
– Qis afinitesetof states,
– Ais afiniteinputalphabet,
– Γ isa finitepushdownalphabet,
– q ∈ Qistheinitialstate,Z ∈ Γ isthestartsymbol,
0 0
– δ isa mappingfromQ×(A∪{ε})×Γ tofinitesubsetsof Q×Γ∗.
– F ⊆ Qisthesetof finalstates.
If γ ∈ Γ+ describes the pushdown storecontent, the leftmost symbol
of γ will be assumed to be on “top” of the store. A configuration of a
PDAisa pair(q,γ) whereq ∈ Qandγ ∈ Γ∗.
Fora ∈ A∪{ε},γ,β ∈ Γ∗ and Z ∈ Γ,if (p,β)isinδ(q,a,Z),then
we writea : (q,Zγ) 7→ (p,βγ).
M
7→∗ isthetransitiveandreflexive closureof 7→ .
M M
Let u = a a ...a be a finite word over A. A finite sequence of
1 2 n
configurations r = (q ,γ ) is called a run of M on u, starting in
i i 1≤i≤p
configuration(p,γ), iff:
1. (q ,γ ) = (p,γ)
1 1
2. for each i, 1 ≤ i ≤ p − 1, there exists b ∈ A ∪ {ε} satisfying
i
b : (q ,γ ) 7→ (q ,γ ) suchthat a a ...a = b b ...b .
i i i M i+1 i+1 1 2 n 1 2 p−1
This run is simply called a run of M on u if it starts from configuration
(q ,Z ).
0 0
The languageaccepted byM is
L(M) = {u ∈ A∗:thereisa runr ofM onu endingina finalstate}.
Forinstance,theset 0∗1 ⊂ {0,1}∗ istriviallycontext-free.
Proposition4.2 (Finkel). Let L bethemaximalsubsetof
n և
և և ... n
{0,1,և ,և ,...,և }∗ suchthatL 1 2 = 0∗1,
1 2 n n
L is context-free
n
Thiswas first noticedby thesecond authorin [Fin01].
To be more precise, by u ∈ L we mean: we start with some u,
n
then weevaluateև as an eraser, and obtainu (providingthat wemust
1 1
neveruseև toerasetheemptysequence,i.e.everyoccurrenceofaև
1 1
symbol does erase a letter 0 or 1 or an eraser և for i > 1). Then we
i
ABORELω-POWEROFACONTEXT-FREELANGUAGEABOVE∆0ω 9
start again with u , this time we evaluate և as an eraser, which yields
1 2
u ,andsoon.Whenthereisnomoresymbolև tobeevaluated,weare
2 i
left withu ∈ {0,1}∗. Wedefine u ∈ L iffu ∈ 0∗1.
n n n
To makeaPDA recognizeL , theideaistohaveitguess(non deter-
n
ministically), for each single letter that it reads, whether this letter will
be erased later or not. Moreover, the PDA should also guess for each
eraser it encounters, whether this eraser should be used as an eraser or
whether it should not - for the only reason that it will be erased later
on by a stronger eraser. During the reading, the stack should be used to
accumulate all pendant guesses, in order to verify later on that they are
fulfilled.
We would very much like to prove that L∞ = [ Ln is context-
n<ω
free. Unfortunately, we cannot get such a result. However, we are able
to show that a slightly more complicated set (strictly containing L ) is
∞
indeed context-free.
Of course, the first problem that comes to mind when working with
L , is to handle ω many different erasers with a finite alphabet. This
∞
impliesthaterasers mustbecodedbyfinitewords.Thiswas donebythe
second author in [Fin03b]. Roughly speaking, the eraser և is coded
n
by the word αBnCnDnEnβ with new letters α,B,C,D,E,β. It is a
little bit tricky, but the PDA must really be able to read the number n
identifyingtheeraser fourtimes.
The very definition of the sets L , requires the erasing operations
n
to be executed in an increasing order: in a word that contains only the
erasersև ,...,և ,onemustconsiderfirsttheeraserև ,thenև ,and
1 n 1 2
so on...
Therefore thiserasingprocess satisfythefollowingproperties:
(a) An eraser և may only erase letters c ∈ {0,1} or erasers և with
j k
k > j.
(b) Assume that in a word u ∈ L , there is a sequence cvw where c is
n
either in {0,1} or in the set {և ,...,և }, and w is (the code of)
1 n−1
an eraser և which erases c once the erasing process is achieved.
k
If there is in v (the code of) an eraser և which erases e, where
j
e ∈ {0,1} or e is (the code of) another eraser, then e must belong to
v (itisbetweencandw inthewordu);moreovertheerasing-bythe
eraser և - has been achieved before the other one with the eraser
j
10 JACQUESDUPARCANDOLIVIERFINKEL
և . Thisimpliesj ≤ k. Thustheintegerk mustsatisfy:
k
k ≥ max{j : an eraser և was usedinsidev}
j
Theessentialdifferencewiththecasestudiedin[Fin03b]isthathere
aneraserև mayonlyeraseletters0or1orerasersև fork > j,while
j k
in [Fin03b] an eraser և was assumed to be only able to erase letters 0
j
or1 orerasers և fork < j.So theaboveinequalitywas replaced by:
k
k ≤ min{j : an eraser և was usedinsidev}
j
However,withaslightmodification,wecanconstructaPDABwhich,
amongwordswherelettersα,β,B,C,D,Eareonlyusedtocodeerasers
ofthe form և , accepts exactly thewords which belongto thelanguage
j
L .WenowexplainthebehaviorofthisPDA.(Forsimplicity,wesome-
∞
timestalkabouttheeraser և insteadofitscodeαBjCjDjEjβ.)
j
Assume that A is a finite automaton accepting (by final state) the
finitary language0∗1overthealphabetA = {0,1}.
We can informally describe the behavior of the PDA B when reading a
word u such that the letters α,B,C,D,E,β are only used in u to code
theerasers և for1 ≤ j.
j
B simulates the automaton A until it guesses (non deterministically)
that it begins to read a segment w which contains erasers which really
eraseandsomelettersofAorsomeothereraserswhichareerasedwhen
theoperationsoferasingare achievedinu.
Then, still non deterministically, when B reads a letter c ∈ A it may
guess that this letter will be erased and push it in the pushdown store,
keepingin memorythecurrent stateoftheautomatonA.
In a similar manner when B reads the code և = αBjCjDjEjβ,
j
it may guess that this eraser will be erased (by another eraser և with
k
k < j)andthenmaypushinthestorethefinitewordγEjν, whereγ,E,
ν are inthepushdownalphabetofB.
ButB mayalsoguessthattheeraserև = αBjCjDjEjβ willreally
j
be used as an eraser. If it guesses that the code of և will be used as an
j
eraser, B has to pop from the top of the pushdown store either a letter
c ∈ A or the code γEi.ν of another eraser և , with i > j, which is
i
erased by և .
j