Regular sets over extended tree structures

HAL Id: hal-00379256

https://hal.archives-ouvertes.fr/hal-00379256

Preprint submitted on 28 Apr 2009

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Regular sets over extended tree structures

Severine Fratani

To cite this version:

Severine Fratani. Regular sets over extended tree structures. 2009. �hal-00379256�

SéverineFratani

Laboratoired'InformatiqueFondamentaledeMarseille(LIF)CNRS:UMR6166

UniversitédelaMéditerranée-Aix-MarseilleII

Universitéde Provene-Aix-MarseilleI

Abstrat

WeinvestigatenotionsofdeidabilityanddenabilityfortheMonadiSeond-

Order Logi of labeled tree strutures, and links with nite automata using

oralestotestinputprexes.

A generalframework is dened allowing to transfersome MSO-properties

from agraph-strutureto a labeledtree struture. Transferredpropertiesare

deidabilityofsentenesandexisteneofadenablemodelforeverysatisable

formula. Alassofniteautomatawithprex-oralesisalsodened,reogniz-

ingexatlylanguagesdened byMSO-formulasinanylabeledtree-struture.

Applying these results, the well-known equality between languages reog-

nized by nite automata, sets of verties MSO denable in a tree-struture

andsetsofpushdownontextsgeneratedbypushdown-automataisextendedto

k

^{-iteratedpushdownautomata.}

Keywords: Labeledtreestrutures; MSOdenablesets; Automatawith

orale;Iteratedpushdownstrutures.

Introdution

InitiatedbytheworkofBühionwords,thestudyoflinksbetweenautomata

andlogihaspermittoidentifystrutureshavingadeidableMonadiSeond-

Ordertheory. Inpartiular,Rabinprovedin[28℄deidabilityoftheMSO-theory

ofinnitetreestruturesinwhihnumerouspropertiesaredenableandtheories

areinterpretable. Theseworkshavealsoledtoalogiharaterisationofregular

languages: languagesreognisedbyniteautomata areexatlysets denedby

MSO-formulasinatreestruture.

The goal of this paper is to extend these works to the study of labelled

tree strutures: identify labellings for whih tree struture have a deidable

MSO-theory, for whih every formula admits a denable model and give an

automata-haraterizationofthedenablesets.

Toahievethisgoal,weintroduenewinterestingobjetsandresults. First,

wedene alassof word/tree automatawith prex-orales(i.e.,sets ofwords

overthe inputalphabet) used to testthe alreadyproessed prexesofinputs.

Languagesandforestreognizedbyprex-oralesautomataenjoynieproperty,

ofMSO-formulasoverinnitetreesanbeextendedtotheselanguages: forests

reognized by automata with orales

O 1 , . . . , O m

^{are forests}MSO-denable in treestruturesextendedbyunaryrelations

O 1 , . . . , O m

^{. Remarkthat this ap-}

proahhasalreadybeendevisedin [32℄to haraterisesomepropersublasses

ofregularlanguagesbyusingregularprex-oralesandtostudytheirdenabil-

ityinFirst-OrderLogioverextendedwordstrutures. However,thedenition

of automata with prex-oralesdoes not expliitlyappear in this paper sine

regularprex-oralesanbesimulatedbythediretprodutofniteautomata.

Seond,weestablishtransfertheorems, allowingfrom astruture,to onstrut

atreestruturehavingsomesimilarMSOproperties. Thisapproahisommon

for the transferof deidability (for example the transfer of deidability from

astruture to its tree-like struture,(see [31℄ or[35℄), or from a graphto it's

unfolding(see[10℄)),buthere,inadditiontodeidability,transferredproperties

alsoapplied to sets MSO denable in suhstrutures and lassesof automata

reognisingthem. Inaddition,ourtransferofdeidabilityallowstoobtainnew

deidabilityresultswhiharenotoverbytheonesitedabove. Propertiesare

transferred to a labelled tree struture from its image struture by any mor-

phism. If

µ : D → D

^{′ isamappingand}

S

isarelationalstruture over

D

^{, the}

imagestruture

µ( S )

^of

S

has

D

^{′ asdomainanditsrelationsaretheimagesby}

µ

^{oftherelationsof}

S

.

Let

t

^{be a labelled tree, and}

t

^{be the struture assoiated to}

t

^{. For any}

morphismofmonoid

µ

^{,andundersomesimplehypothesisonthelabellingof}

t

^,

weobtainthefollowingmainresults:

•

^Transferofdeidability: (Theorem55)if

µ(t)

^{hasadeidableMSO-theory,}

then

t

^hasadeidableMSO-theory,

•

^{Transfer of the property of Denable Model: (Theorem57) under aon-}

dition on

µ

^{, if}

µ(t)

^{satisesthe propertyof Denable Model (DM),then}

t

^{satises DM. This propertyensures for a struture}

S

^{that any satis-}

ableformulaadmitsatleastonemodelwhih isMSO-denablein

S

^(see

Denition15),

•

^{Theorem of struture: (Theorem 58) under the same ondition on}

µ

^{, if}

µ(t)

^{satises DM, then anyset is} MSO-denablein

t

^{i it is reognised}

byanite automatonusing onlyorales oftheform

µ

⁻¹

(D)

^where

D

^is

MSO-denablein

µ(t)

^{. (Theneahoraletestsaproperty}MSO-denable in

µ(t)

^{, ontheimageby}

µ

^{ofinputwordprexes).}

Applying these results, weobtaintree strutures having adeidable MSO

theoryandlassesoflanguageshavingtwoequivalentharaterizations:aslan-

guagesreognizedbyautomatawithorales,andassetsMSO-denableinsome

labelled tree strutures. We thus extend the two haraterizations of regular

languagesmentionedabove.

But regularlanguagesadmit athird haraterization,assets of pushdown

ontextsgeneratedbyapushdownsystemoftransitions[21℄. Somereentworks

roughlyastakofstak... ofstak(seeforexamples[5,7,24,19℄),itisthennat-

uralto attemptto deneanotionofregular setsof

k

^{-pushdowns(i.e.,staks}

with

k

^{level of embedded pushdowns) whih generalize the previousequality.}

Wegiveequalities betweenlanguagesof

k

^{-pushdownsreognized byautomata}

withp-orales,languagesMSO-denableinapartiulartreestrutureandsets

of

k

^{-pushdownontextsgenerated bya}

k

^{-pushdownsystemof} transitions. We iteratively use the three transfer theorems on a family of strutures

( P

k

) _k≥1

havingaprexwordslanguage

P k

^{fordomain. Thelanguage}

P k

^denesanen-

odingoftheset

k

^-pdsofall

k

-pushdowns. Thestruture

P

k^isMSO-equivalent withthestruture PDS

k

^{whose domain is}

k

^{-pdsand whoserelations arethose}

induedby thelassial instrutionson

k

-pushdowns. This allows to dene a lassoflanguagesin

P k

^{thatanbeexpressedinfourequivalentways(Theorem}

85):

•

^{aslanguagesreognisedbyniteautomatawith}prex-orale,

•

^{aslanguagesdenedby}MSO-formulasinthetreestrutureofdomain

P _k

^,

•

^{asenodingsofsetsdened by}MSO-formulasinPDS

k

^,

•

^{asenodingofsets of}

k

^{-pushdownsgenerated bya}store-ontrolled

k

^-pds

systemoftransitions.

Weshowin additionthatPDS

k

^{satisesthepropertyofDenableModel.}

Thispaperisorganizedasfollows. Setion 1isdevoted tobasidenitions

onwords,logi,automataand

k

^{-pushdownsstrutures. Itisalsointroduedthe}

notionofwordautomatawithorales. InSetion2,weextendtotreeautomata

the use of orales. The Rabin's orrespondene between regular forests and

modelsofMSO-formulasovertreesisadaptedtotheselanguages.InSetion3,

we developa game-theoretial approah to provethe three transfertheorems.

Wegivealsoasimpleappliation ofthetransfertheorems. Finally,wegivein

Setion4adenitionof

k

^{-regularsets ofpushdowns.}

1. Preliminaries

1.1. Basi denitions

1.1.1. Somenotations andonventions

Givenaset

A

^,wedenoteby

|A|

^theardinalof

A

^{. If}

s

^{isamapfromaset}

A

^{, then}

s(A) = {s(a) | a ∈ A}

^{. If}

V ~ = (V 1 , . . . , V n )

^{isavetorofsubsets of}

A

then

s( A) = (s(V ~ 1 ), . . . , s(V n ))

^{. The}harateristifuntion of

V ~

ⁱⁿ

A

^isamap

χ ^V _A ^~ : A → {0, 1} ⁿ

^denedforall

x ∈ A

^,by

χ ^V _A ^~ (x) = (b 1 , . . . , b n )

^where

∀i

^,

b i = 1

i

u ∈ S i

^.

If

A

^isaset,

A

^{∗ denotesthesetofwords(nitesequenes)over}

A

^,and

ε

^the

emptyword. For

u, v ∈ A

^{∗,thelengthof}

u

^isdenoted

|u|

^andwewrite

v 4 u

^if

v

^isprexof

u

^{. Aset}

P ⊆ A

^{∗ isaprexlosedlanguageif}

∀u ∈ P

^,

∀v ∈ A

^{∗, if}

v 4 u

^then

v ∈ P

^.

1.1.3. Freegroup

Givenanitealphabet

A

^{,letusassoiatetoeah}

a ∈ A

^{theinversesymbol}

¯

a

^{whihdoesnotbelongto}

A

^{. Wedenote by}

A

^{theset of inverselettersof}

A

anddene

A b = A ∪ A

^{. Forevery}

u = a 1 · · · a n ∈ A b

^{∗,theinverse wordof}

u

^is

u = b n · · · b 1

^where

∀i ∈ [1, n]

^:

if

a i ∈ A

^then

b i = ¯ a i

^,andif

a i = ¯ a ∈ A

^then

b i = a

^.

Letusthenonsidertheredutionsystem

S = {(a¯ a, ε), (¯ aa, ε)}

^{. Awordin}

A b

^∗

issaidto bereduedifitis

S

^{-redued,i.e.,itdoesnotontainourrenesof}

a¯ a

^or

¯ aa

^,for

a ∈ A

^{. Wedenote by}

Irr(A)

^{the setof reduedwordsin}

A b

^{∗. As}

S

^{isonuent,eahword}

w

^{is equivalent(mod}

↔

^∗

_S

^{)to auniquereduedword}

denoted

ρ(w)

^.

Wedenethefreegroup

(Irr(A), ε, •)

^{, where}

∀u, v ∈ Irr(A)

^,

u • v = ρ(u · v)

^.

1.1.4. Projetions

Foranyintegers

0 < i ≤ j ≤ n

^{, forany vetorofelements}

(a 1 , . . . , a n )

^{, we}

dene the projetion

π i (a 1 , . . . , a n ) = a i

^and

π i,...,j (a 1 , . . . , a n ) = (a i , . . . , a j )

^.

For any alphabets

B

^and

A

^with

B ⊆ A

^{, the projetion}

π B : A

^∗

→ B

^{∗ is a}

morphismdened

∀a ∈ A

^by

π B (a) = a

^if

a ∈ B

^and

π B (a) = ε

^else.

1.1.5. Treesand forests

Given nite alphabets

Σ

^and

A

^{and a prex losed language}

P ⊆ A

^{∗, a}

P

^-tree

(Σ)

^{(treeofdomain}

P

^labelledby

Σ

^{)is atotalfuntion}

t : P → Σ

^{. The}

setofall

P

^-tree

(Σ)

^isdenoted

P

^-Tree

(Σ)

^{. Inordertodealwithunlabelledtrees}

in an uniform way, we introdue the speial symbol

⊤

^{. Unlabelled trees are}

then funtions

t : P → {⊤}

^{. We will often onsider trees in}

P

^-Tree

({0, 1} ⁿ )

^,

for

n ≥ 0

^{(withthe onvention that}

{0, 1} ⁰ = {⊤}

^{), we will denote this lass}

P

^-Tree

n

^{. Remarkthatatreein}

P

^-Tree

n

^{analwaysbeseenasthe}harateristi funtion

χ ^S _P ^~

^ofavetor

S ~ = (S 1 , . . . , S n )

^,for

S i ⊆ P

^.

Wewillusetwokindsofoperationsontreesandtree-languages:

•

Restrition: let

t ∈ A

^∗-Tree

(Σ)

^,

t

|P ^isthe

P

^-tree

(Σ)

^{obtainedbyrestrit-}

ingthedomainof

t

^to

P

^{. If}

F ⊆ A

^∗-Tree

(Σ)

^,then

F

_|P

= {t

_|P

, t ∈ F }

^.

•

^{Produt: let}

t 1

^{be a}

P

^-tree

(Σ 1 )

^and

t 2

^a

P

^-tree

(Σ 2 )

^{, the produt of}

t 1

and

t 2

^{is thetree}

t 1 b t 2 ∈ P

^-Tree

(Σ 1 × Σ 2 )

^fullling

∀u ∈ P

^,

t 1 b t 2 (u) = (t 1 (u), t 2 (u))

^{. Thisdenitionanbeextendedto treelanguages:}

if

F 1 , F 2 ⊆ P

^-Tree

(Σ)

^,then

F 1 b F 2 = {t 1 b t 2 | t 1 ∈ F 1 , t 2 ∈ F 2 }

^.

Finite automata with prex-orale (or p-orale) extend the lass of nite

automatabyallowingsomemembershiptests onprexof theinputword. An

automaton

A

^{with p-orales, on the input alphabet}

A

^{is a nite automaton}

assoiatedtoavetor

O ~ = (O 1 , . . . , O m )

^ofsubsetsof

A

^{∗ andwhose}transitions ontainabooleanvetorofsize

m

^{alledtest. Duringtheomputationby}

A

^of

aninputword,thealreadyproessedpart

u

^{oftheinputiskeptinmemoryand}

atransition withtest

~o

^{anbeappliedif}

~o

^isequaltotheharateristivetor of

u

^inside

O ~

^(i.e.,if

~o = χ ^O _A ^~

^∗

(u)

^).

Denition1(Finite automatonwith p-orales). Given

m ≥ 1

^{, an au-}

tomaton with

m

^{p-orales is a tuple}

A = (Q, A, ~ O, ∆, q 0 , F )

^where

Q

^{is a}

nite set of states,

A

^{is the input alphabet,}

O ~ = (O 1 , . . . , O m )

^,

O i ⊆ A

^∗,

∆ ⊆ Q × A × {0, 1} ^m × Q

^{is theset of}transitions,

q 0 ∈ Q

^{isthe initialstate,}

and

F ⊆ Q

^{isthesetofnalstates.}

Aongurationof

A

^{is apair}

(q, u

_↑

v)

^where

uv ∈ A

^{∗ and}_↑ ^{isasymbolwhih}

doesnotbelongto

A

^{. Thebinaryrelationon}ongurationsis

→

_A^andonsists

inallpairs

(q, u

_↑

av) →

_A

(p, ua

_↑

v)

^suhthat

(q, a, χ ^O _A ^~

∗

(u), p) ∈ ∆

^{. Thelanguage}

reognisedby

A

^is

L(A) = {u ∈ A

^∗

| (q 0 ,

_↑

u) →

^∗_A

(q F , u

_↑

), q F ∈ F}

^.

Wewill usethefollowingnotations: FA

O ~ (A)

^{isthefamilyof automataover}

A

withp-orale

O ~

^,thelassof

O ~

^{-regularlanguages(i.e.,reognisedbyautomata}

inFA

O ~ (A)

^{) isREG}

^{O ~} (A)

^{. Remark that anautomatonwithorale}

∅

^{is simplya}

niteautomaton. WewritethenFA ratherthanFA

∅

andREGforREG

∅

.

Denition2. Anautomatonwith

m

^p-orales

A = (Q, A, ~ O, ∆, q 0 , F )

^issaid

to be deterministi if

∀p ∈ Q, a ∈ A, ~o ∈ {0, 1} ^m

^{, there is oneand only one}

q ∈ Q

^suhthat

(p, a, ~o, q) ∈ ∆

^.

Example3. Thefollowing automatonisdeterministi and reognizethelan-

guage

{a ⁿ b ⁿ c ⁿ } _n≥1

^.

A = ({q 0 , q 1 , q 2 , q 3 , q F }, {a, b, c}, (O 1 , O 2 ), ∆, q 0 , {q F })

^where

O 1 = {a ⁿ b ⁿ } _n≥1

^,

O 2 = {a ^m b ⁿ c ⁿ⁻¹ } _{n≥1, m≥0}

^and

∆

^onsistsin

(q 0 , a, (0, 0), q 1 )

^,

(q 1 , a, (0, 0), q 1 )

^,

(q 1 , b, (0, 0), q 2 )

^,

(q 2 , b, (0, 0), q 2 )

^,

(q 2 , b, (0, 1), q 2 ) (q 2 , c, (1, 0), q 3 )

^,

(q 2 , c, (1, 1), q F )

^,

(q 3 , c, (0, 0), q 3 )

^and

(q 3 , c, (0, 1), q F )

^.

Weassoiatewith eah automatonwith

m

^p-orales

A ∈

^FA

^{O ~} (A)

^{, anite au-}

tomaton

A ∈ e

^FA

(A × {0, 1} ^m )

^{alled soureof}

A

^{andonstruted bymoving}

thetestofeahtransitionintotheinputletterofthetransition: eahtransition

(p, a, ~o, q)

^istransformedin

(p, (a, ~o), q)

^{. Thelanguage}

L(A)

^{anbeobtainfrom}

thelanguage

L( A) e

^andthe"harateristilanguageof

O ~

ⁱⁿ

A

^.

Denition4. For every

O ~ = (O 1 , . . . , O m )

^,

O i ⊆ A

^{∗, the} harateristi lan- guageof

O ~

^isdenedby:

L ^O _χ ^~ = {(a 1 , ~o 1 ) . . . (a n , ~o n ) ∈ (A × {0, 1} ^m )

^∗

| ∀i ∈ [1, n], ~o i = χ ^O _A ^~

∗

(a 1 . . . a i−1 )}

^.

Observation 5. Forevery

O ~ = (O 1 , . . . , O m )

^,

O i ⊆ A

^∗:

REG

O ~ (A) = {π 1 (L ∩ L ^O _χ ^~ ) | L ∈

^REG

(A × {0, 1} ^m )}.

UsingtheKleene'stheorem, weobtaineasily:

Theorem6. Let

A

^{analphabet, and}

O ~

^{avetor ofsubsets of}

A

^∗,

1. REG

O ~ (A)

^{isthelass oflanguages reognizedby}deterministiautomatain

FA

O ~ (A)

^,

2. REG

O ~ (A)

^{islosedunderboolean}operations.

1.3. Iteratedpushdown stores

OriginallydenedbyGreibahin[22℄,iteratedpushdownstoresarestorage

struturesbuiltiteratively. Letusxaninnitesequene

A = A 1 , A 2 , . . . , A k , . . .

of disjoint andnite alphabets. Forall

k ≥ 1

^{, wedenote by}

A _k

the nite se- quene

A 1 , . . . , A k

^{and adoptthe onventionthat}

A ₀ = {⊥}

^andthat

A ₀ ∩ A i

isemptyofall

i ≥ 1

^.

Denition7. Wedene indutivelythe set

k

^-pds

( A _k )

^(or

k

^{-pdswhen alpha-}

bets ofstoreareunderstood)of

k

^-iteratedpushdown-storesover

A _k

:

0

^-pds

( A ₀ ) = {⊥}

^,

(k + 1)

^-pds

( A _k+1 ) = (A k+1 [k

^-pds

( A _k )])

^∗

⊥ [k

^-pds

( A _k )]

^.

Theset forall

k

^{-pushdowns for}

k ≥ 0

^{is denoted}

it

^-pds

( A )

^{. In therest of the}

paper, any

1

^-pds

a 1 [⊥]a 2 [⊥] · · · a n [⊥] ⊥ [⊥]

^{will bewritten simply}

a 1 . . . a n ⊥

and

∀k ≥ 0

^{. We denoteby}

⊥ k

^{the empty}

k

^{-pds ontainingonly symbols}

⊥

^:

⊥ 0 =⊥

^and

⊥ k+1 =⊥ [⊥ k ]

^.

From the denition, every

ω

ⁱⁿ

(k + 1)

^-pds

( A _k+1 )

^,

k ≥ 0

^{, has a unique}

deompositionas

ω = a[ω 1 ]ω

^′with

ω 1 ∈ k

^-pds

(A k )

^,

ω

^′

∈ (k + 1)(A k+1 )

^-pds

∪{ε}

and

a ∈ A k+1 ∪ {⊥}

^. Furthermore,

a =⊥

ⁱ

ω

^′

= ε

^.

Example8. Let

A 1 = {a 1 , b 1 }

^,

A 2 = {a 2 , b 2 }

^,

A 3 = {a 3 }

^{bestoragealphabets,}

ω ex = a 3 [b 2 [b 1 a 1 ⊥]a 2 [a 1 ⊥] ⊥ 2 ] ⊥ [a 2 [a 1 ⊥]a 2 [⊥] ⊥ 2 ] ∈ 3

^-pds

( A ₃ )

^.

Its deomposition orresponds to

a = a 3

^,

ω 1 = b 2 [b 1 a 1 ⊥]a 2 [a 1 ⊥] ⊥ 2

^and

ω

^′

=⊥ [a 2 [a 1 ⊥]a 2 [⊥] ⊥ 2 ]

^.

Thetwofollowingmapswillbeuseful.

Projetion: themap assoiatingeah

k

^{-pdsto itstop}

i

^-pds,

1 ≤ i ≤ k

^is

p k,i

^:

k

^-pds

(A 1 , . . . , A k ) → i

^-pds

(A 1 , . . . , A i )

^,where

∀ω = a[ω 1 ]ω

^′

∈ k

^-pds,

p k,k (ω) = ω

^and

p k,i (ω) = p _k−1,i (ω 1 )

^if

1 ≤ i ≤ k − 1

^.

The double subsript notation will be used to handle inverse funtions,

therestofthetime,wewillnote

p i

^for

p k,i

^.

Top symbols: themap assoiatingany

k

^-pds,

k ≥ 1

^{to its}

k

top-symbolsis

top : k

^-pds

(A 1 , . . . , A k ) → (A k ∪ {⊥}) · · · (A 2 ∪ {⊥})(A 1 ∪ {⊥})

^dened

∀ω = a[ω 1 ]ω

^′

∈ k

^-pdsby

top(ω) = a

^,if

k = 1

^,else

top(ω) = a · top(ω 1 )

^.

For

i ∈ [1, k]

^{, and}

ω ∈ k

^{-pds, we denote by}

top _i (ω)

^the

i

^{-th letter of}

top(ω)

^.

Example9. Let

ω ex

^bethe

3

^{-pdsgivenin Example8:}

p 2 (ω ex ) = b 2 [b 1 a 1 ⊥]a 2 [a 1 ⊥] ⊥ 2

^,

p 1 (ω ex ) = b 1 a 1 ⊥

^,and

top(ω ex ) = a 3 b 2 b 1

^,

top(p 2 (ω ex )) = b 2 b 1

^,

top(p 1 (ω ex )) = b 1

^.

An instrution on

it

^{-pds is a funtion from}

it

^{-pds to}

it

^{-pds whih does not}

modifythelevelofthe

k

^{-pushdowns(i.e.,ifinstrisaninstrutionthenforany}

k ≥ 0

^{and any}

ω ∈ k

^-pds,instr

(ω) ∈ k

^{-pds). An instrution of level}

i

^isan

instrutionwhihdoesnotmodifythelevelsgreaterthan

i

^ofany

it

^{-pds. Hene,}

giveninstraninstrutionoflevel

i

if

ω = a[ω 1 ]ω

^′

∈ k

^-pds,

k > i

^,theninstr

(ω) = a[

^instr

(ω 1 )]ω

^′

if

ω ∈ k

^-pds,

k < i

^,theninstr

(w) = w

^.

Therefore,to deneaninstrutionof level

i

^{, thereis onlyneedto deneitfor}

any

ω ∈ i

^-pds.

Threeinstrutionsoflevel

k

^{aregenerallyappliableto}

it

-pushdowns.

Denition10. Classial instrutions of level

i ≥ 1

^over

A

are dened for every

ω = b[ω 1 ]ω

^′

∈ i

^-pds

( A _i )

^by:

pop _i (ω) = ω

^{′ if}

b 6=⊥

^,else

pop _i (ω)

^isundened,

push _i,a (ω) = a[ω 1 ]ω

^,

change _i,a (ω) = a[ω 1 ]ω

^′,if

b 6=⊥

^else

change _i,a (ω)

^isundened.

For

k ≥ 1

^,

I k ( A _k ) = {pop _i | i ∈ [1, k]} ∪ {push _i,a , change _i,a | a ∈ A i , i ∈ [1, k]}

^.

istheset ofinstrutionsover

A _k

.

Thus, given

ω ∈ k

^{-pds and}

i ≤ k

^,

pop _i (ω)

^erases

p i (ω)

^{on the top of the}

store,

push _i,a

_i

(ω)

^{onsists in add}

a i [p i−1 (ω)]

^{on the top of the top}

i

^{-pds and}

change _i,a

_i

(ω)

^{onsistsin replae}

top _i (ω)

^by

a i

^.

Example11. Let

ω = b 3 [b 2 [b 1 ⊥] ⊥ 2 ] ⊥ 3

^bea

3

^-pds,

pop ₃ (ω) =⊥ 3

^,

pop ₂ (ω) = b 3 [⊥ 2 ] ⊥ 3

^,

pop ₁ (ω) = b 3 [b 2 [⊥] ⊥ 2 ] ⊥ 3

^,

push _2,a

₂

(ω) = b 3 [a 2 [b 1 ⊥]b 2 [b 1 ⊥] ⊥ 2 ] ⊥ 3

^,

push _1,a

₁

(ω) = b 3 [b 2 [a 1 b 1 ⊥] ⊥ 2 ] ⊥ 3

^,

change _3,a

₃

(ω) = a 3 [b 2 [b 1 ⊥] ⊥ 2 ] ⊥ 3

^,

change _1,a

₁

(ω) = b 3 [b 2 [a 1 ⊥] ⊥ 2 ] ⊥ 3

^.

Wealso dene theinverse instrutionof

push _i,a

^{whih will beused to enode}

the

k

^{-pushdownsaswords.}

Denition12. Forany

i ≥ 1

^and

a ∈ A i

^{, theinstrutionof level}

i push _i,a

^is

denedforany

ω ∈ i

^-pds

( A _i )

^by

push _i,a (ω) = ω

^{′ ifthereexists}

ω

^′

∈ i

^-pdssuhthat

ω = push _i,a (ω

^′

) push _i,a (ω)

^{isundenedelse.}

Inotherwords,

∀ω ∈ k

^-pds,

push _k,a (ω) = ω

^{′ i}

ω = a[ω 1 ]b[ω 1 ]ω

^′′and

ω

^′

= b[ω 1 ]ω

^′′

.