Reachability in pushdown register automata

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Original citation:

Murawski, Andrzej S., Ramsay , S. J. and Tzevelekos, N.. (2017) Reachability in pushdown

register automata. Journal of Computer and System Sciences, 87. pp. 58-83.

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Contents lists available atScienceDirect

Journal

of

Computer

and

System

Sciences

www.elsevier.com/locate/jcss

Reachability

in

pushdown

register

automata

✩

A.S. Murawski

a

,

S.J. Ramsay

a

,

∗

,

N. Tzevelekos

b

a_{DepartmentofComputerScience,UniversityofWarwick,CoventryCV47AL,UK}

b_{SchoolofElectronicEngineeringandComputerScience,QueenMaryUniversityofLondon,LondonE14NS,UK}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received 19 November 2014

Received in revised form 9 January 2017 Accepted 13 February 2017

Available online 8 March 2017

Keywords: Register automata Pushdown automata Reachability Infinite alphabets Complexity

Weinvestigatereachabilityinpushdownautomataoverinfinitealphabets.Weshowthat, in terms of reachability/emptiness, these machines can be faithfully represented using only 3r elements of the alphabet, where r is the number of registers. We settle the complexityofassociatedreachability/emptinessproblems.Incontrasttoregisterautomata, theemptinessproblemforpushdownregisterautomataisEXPTIME-complete,independent of the register storage policy used. We also solve the global reachability problem by representing pushdown configurations with a special register automaton. Finally, we examineextensionsofpushdown storagetohigherordersand show thatreachability is undecidableatorder2.

©2017ElsevierInc.Allrightsreserved.

1. Introduction

Recent yearshaveseenlively interestinautomata overinfinitealphabets.Thishaslargely beendrivenby applications which, althoughdiverse, hadacommonthread indealing withalphabets ofpotentiallyunbounded size forwhich finite-domain abstractions were deemed unsatisfactory.A casein point are markuplanguages [20,4], mostnotably XML. Here, documentscontaindatavalueswhoserangeispotentiallyunboundedandqueriesareallowedtoperformcomparisontests on such data.Asimilar scenariooccursinreference-based programminglanguages,suchasobject-oriented [6,2,12,17]or ML-like languages[18,19]. Insuch languages,memoryis managed withthehelp ofreferencenames thatcan be created afreshandcomparedforequalitybutareotherwise abstract.Other examplesincludearray-accessing programs[1],which areallowed tousethearraytostoreandcompareelementsofanunboundeddomain,aswellasprogramswithrestricted integerparameters[7].

Suchapplicationscallforarobusttheoryofautomataoverinfinitealphabetswhichcanleadtoanunderstanding compa-rabletothatofthefinite-alphabetsetting.Suchatheorywillexposethelimitsofthismodelofcomputationandestablisha complexity-theoreticguideforapplications.Alotofthegroundwork,surveyedin[22,3],wasalreadydedicatedto uncover-inganotionof“regularity”intheinfinite-alphabetcase.Veryearlyinthatwork,awaywasproposedtoextendtheconcept offinitememorytoinfinitealphabetswhichconsistedin introducingafixednumberofregistersforstoringelementsofthe alphabet[13].Theconstructedautomata,calledFinite-MemoryAutomata[13]orRegisterAutomata[20],wouldotherwiselook justasfinite-stateautomatawherethetransitionlabelswouldalsoinvolveindicesreferringtoregisteraddresses.Building on the successofthis work,another strandaimed to identifythe infinite-alphabet“context-free”languages. Tothisend,

✩ _{Research supported by the Engineering and Physical Sciences Research Council (EP/J019577/1) and the Royal Academy of Engineering (Tzevelekos,}

10216/111).

*

Corresponding author.

E-mailaddresses:[email protected](A.S. Murawski), [email protected](S.J. Ramsay), [email protected](N. Tzevelekos).

Table 1

Complexity of the emptiness problem.

Register discipline S F S#0 M F RA NL-c NP-c PSPACE-c PDRA EXPTIME-c EXPTIME-c EXPTIME-c

ChengandKaminski[8]introducedanotionofcontext-freegrammarsoverinfinitealphabetsanddefinedacorresponding notion ofpushdown automata.Later, Segoufinpresented asimilar definition in[22],albeit couched ina waysuitable to processdatawords.

Ourpaperisdevotedtostudyingexactlysuchcontext-freecomputationalscenariosthroughaninvestigationofpushdown registersystems(PDRS),devicesinwhichregistersare integratedwitha pushdownstore.Althoughoffoundationalnature, the work is largely motivated by the pertinence of such machines to software model checking [6,2]. A particular such applicationisingame-semantics-basedverification[19,16],wherebythesemanticsofprogramsisalgorithmicallygivenby means ofvariantsofpushdown registersystems,which inturnare used asmodels to befed inprocedures forchecking programequivalence.Wepresentseveralnewresultsonthecomplexityofreachabilitytesting, whichaltogetherfilla gap inthetheoryof“context-free”languagesoverinfinitealphabets.Morespecifically,wemakethefollowingcontributions.

Alphabetdistinguishability Afinite-memoryautomaton[13]withrregisterscanstorerelementsoftheinfinitealphabetat anyinstant.Infact,suchautomataareonlycapableofrememberingrelementsoftheinfinitealphabetoverthecourseofa run—foranyacceptingrunonecanconstructanotheroneinvolvingonlyr elementsofthealphabet.1 Hereweshowthat, eventhoughpushdownregistersystemshavenoboundonthenumberofelementsofthealphabetthattheycanstoreat anyinstant,overthecourseofaruntheycanreally“remember”atmost3r ofthem.Moreprecisely,weshowthatforany runofaPDRSwithrregistersthereexistsanequivalentrun,i.e.withthesameinitialandfinalconfigurations,butinwhich every configurationcontainsregister assignmentsdrawn fromonly 3r elements. Moreover,no smallernumberisenough: weexhibitafamilyofPDRSwhoserunsrequirerememberingatleast3relements.

Reachabilitytesting Theabove-mentionedresultyieldsanobviousmethodologyforreductionstothefinite-alphabetsetting, whichimmediatelyimpliesdecidabilityofassociatedreachabilityandlanguageemptinessproblems.Whilethedecidability of emptiness has already been proved in [8]using context-free grammars,we provide exact complexity bounds forthe problem,namely,EXPTIME-completeness.

Inthepushdown-freesetting,languagenonemptinesswasknowntobeNL-,NP- andPSPACE-complete,dependingonthe registerdiscipline.Threeregisterassignmentdisciplines havebeenstudiedintheliterature,whichwe shallcall singleand full(S F),singleandinitiallyempty(S#0)andmultipleandfull(M F).Asingleassignmentdisciplinerequiresthatthecontents

ofallregistersbedistinct,whereasamultipleassignmentdisciplineallowsforduplicateregistercontents.Afullassignment discipline,ontheotherhand,requiresthatatall timesevery registermustcontainsome letterfromtheinfinitealphabet, whereas an initially empty discipline allows registers to be empty at the start of a run. The complexity of emptiness for register automata according to the assumed register discipline is given in the first row of Table 1: in the S F case the problemis NL-complete,asit coincides with emptiness ofthe underlying finite-state automaton; inthe S#0 caseit

becomes NP-complete[21],asone isabletousethe registerstoencode booleanassignments;whileinthe M F caseone isableto encodea linear-sizetape withthe registers,andthereforetheproblemisPSPACE-complete [10].In contrast,in thepushdowncase,weshowthatsuchdistinctionsdonotaffectthecomplexity:evenifidenticalelements canbekeptin differentregisters,theproblemcanstillbesolvedinEXPTIMEanditisEXPTIME-hardalreadyinthecasewhereonlydistinct elementsareallowed.Inthelattercase,thehardnessproofistechnicallyinvolvedsincesequencesofdistinctnamesdonot provide asupportive framework forrepresentingmemory content(as neededinreduction argumentsusing computation histories).

Globalreachability We give a simple,exponential-time algorithmfor globalreachability analysis. This analysisasks fora representationofallconfigurationsfromwhichaspecifiedsetoftargetconfigurationscanbereached.Inthefinite-alphabet caseit iswell known that, ifthetarget set isregular, the setof configurations thatreach it can becaptured by afinite automaton [5]. We prove an analogousresult in theinfinite-alphabet setting: givena PDRS

S

and a registerautomaton definingasetoftargetconfigurationsT of

S

,ouralgorithmconstructsanewregisterautomatonthatrepresentsexactlythe setofconfigurationsthatcanreachsomeconfigurationinT.Toensurethealgorithmisasmoothanalogyofthesaturation algorithm of[5],it manipulates a particularlysuccinct variant ofregister automatawhich we call a registermanipulating registerautomaton(RMRA),butweshowthatsuchmachinescanalwaysbetransformedtoamachineoftheusualkindfor atmostanexponentialincreaseinsize.

Higher-order Higher-order pushdown automata [15] take the idea of pushdown storage further by allowing for nesting. Standardpushdownstoreisconsideredtobeorder1,whiletheelementsstoredinan order-k (k

>

1)pushdownstoreare

(

k

−

1

)

-pushdownstores.Inthefinitealphabetsettingthisleadstoaninfinitehierarchyofdecidablemodelsofcomputation witha

(

k

−

1

)

-EXPTIME-completeproblematorderk.Weexaminehowthemodelbehavesintheinfinitealphabetsetting, aftertheadditionofafixednumberofregistersforstoringelementsoftheinfinitealphabet.

We firstobservethat onecanno longerestablishauniformbound onthenumberofsymbolsoftheinfinitealphabet that sufficetorepresentarbitraryruns. Theexistence ofsucha boundwouldimplydecidabilityoftheassociated reacha-bilityproblems,butthelackofa boundisnotsufficientforestablishingundecidability:indeed,thedecidableclassofdata automatafrom[4]containsanautomatonthatcanrecognise allwordsconsistingofdistinctletters.Still,weshowthatthe reachabilityproblemforhigher-orderregisterpushdownautomataisundecidable,alreadyatorder2 andwithoneregister.

2. Basicdefinitions

Letusassume acountably infinitealphabet

D

ofdatavaluesornames.Weintroduce asimpleformalism for compu-tations basedona finitenumberof

_D

-valuedregistersanda pushdownstore. Writing

[

r

]

for

{

1

,

· · ·

,

r

}

, byan r-register assignmentwemeananinjectivemapfrom

[

r

]

to

D

.WewriteRegr forthesetofallsuchassignments.2

Definition1.Apushdownr-registersystem(r-PDRS)isatuple

S

=

Q

,

qI

,

τ

I

,

δ

,where:

•

Q isafinitesetofstates,withqI

∈

Q beinginitial,

•

τ

I

∈

Regr istheinitialr-registerassignment,

•

and

δ

⊆

Q

×

Opr

×

Q isthetransitionrelation,

withOpr

= {

i•

,

push

(

i

),

pop

(

i

)

|

1

≤

i

≤

r

}

∪ {

pop•

}

.3

Theoperationsexecutedineachtransitionhavethefollowingmeaning:thei•operationrefreshesthecontentofthei-th register;push

(

i

)

pushesthesymbolcurrentlyinthei-thregisteronthestack;pop

(

i

)

popsthestackifthetopsymbolisthe same asthat storedinthe i-thregister;pop• popsthestack ifthetop ofthestackiscurrentlynotpresentinanyofthe registers.Thissemanticsisgivenformallybelow.

Definition2.Aconfigurationofanr-PDRS

S

isatriple

(

q

,

τ

,

s

)

∈

Q

×

Reg_r

×

_D

∗.Wesaythat

(

q2

,

τ

2

,

s2

)

isasuccessorof

(

q1

,

τ

1

,

s1

)

,written

(

q1

,

τ

1

,

s1

)

(

q2

,

τ

2

,

s2

)

,if

(

q1

,

op

,

q2

)

∈

δ

forsomeop

∈

Oprandoneofthefollowingconditionsholds.

•

op

=

i•,

∀

j

.

τ

2

(

i

)

=

τ

1

(

j

)

,

∀

j

=

i

.

τ

2

(

j

)

=

τ

1

(

j

)

ands2

=

s1.

•

op

=

push

(

i

)

,

τ

2

=

τ

1ands2

=

τ

1

(

i

)

s1.

•

op

=

pop

(

i

)

,

τ

2

=

τ

1 and

τ

1

(

i

)

s2

=

s1.

•

op

=

pop•,

τ

2

=

τ

1 and,forsomed

∈

D

,

∀

j

.

τ

1

(

j

)

=

dandds2

=

s1.

Atransitionsequenceof

S

isasequence

ρ

=

κ

0

,

· · ·

,

κ

k ofconfigurationswith

κ

j

κ

j+1,forall0

≤

j

<

k.Wesaythat

ρ

endsinastateqifqk

=

q,whereqkisthestatein

κ

k.Wecall

ρ

arunif

κ

0

=

(

qI

,

τ

I

,

)

.

Example3.Inthisexamplewewillgoalittlewaybeyondourdefinitionandconsideraregisterpushdownautomatonrather thanaregisterpushdownsystem,sothatwecanmotivatethedefinitionofPDRSbylookingataparticularlanguageofwords thatisaccepted.Anr-PDRAisanr-PDRSequippedwithanadditionalfamilyoftransitions

{

i

}

i∈[r]andadistinguishedsubset

offinal states.Atransitioni can betakenonlyifthenext letteroftheinput wordmatches thecontentsofregisteri and the action of taking the transitionis to consume the letter. The following example is a 2-PDRAaccepting the language

{

w wR

|

w

∈

D

∗

}

ofwords wfollowedbytheirownreverse.

q0

q1

q2

q3

q4

q5

q6

q7 q8 q9

q10

2• 2• push

(

2

₎

1•

1•

1

push

(

1

)

ε

1• 1•

1

pop

(

₁

)

pop

(

2

)

2 _{Thus, register assignments here are singleand full(S F).}

Wedraw themachine’sonly finalstate circled.Theinitial registerassignmentis unimportant,butletussaythat itis

{

1

→

a

,

2

→

b

}

.Thebehaviour ofthemachineis asfollows.Starting fromstate q0,theautomatonfirst guessesaletter c

from the infinitealphabet

D

to use as a ‘bottom of stack’marker and storesit inregister 2. To ensure that the guess isotherwiseunconstrained,thisisimplemented bytwo consecutive2• transitions(asingle 2• transitionwouldallow for guessinganyletterof

D

undertheconstraintthatit isdifferentfroma andb).Themachine thenpushesthis‘bottomof stack’markerontothestackandmovesintothelooponstateq3.Duringthisloop,thewordwisconsumedfromtheinput

by firstguessing thenext input letterusingtwo consecutive1• transitionsandthen readingitinusing the1 transition. The last part ofthe loop has the machine store each consumed letteron the stack. At the point atwhich the machine nondeterministically chooses to enter q7 ,the configuration has the form

(

q7

,

{

1

→

d

,

2

→

c

}

,

wc

)

where d is the final

letter of w.In the loop on q7 themachine readsin wR and, asin the finite-alphabetcase, the reversalof the wordis

ensuredbypoppingw offthestack.Finally,whenthestackhasbeenexhaustedexceptforthe‘bottomofstack’symbolc, thenthemachinemaytakethetransitiontoq2andthusaccept.

Remark4. r-PDRS is meantto be a minimalistic modelallowing usto studyreachability inthe infinite-alphabetsetting withregisters andpushdownstorage.Existingrelatedmodels[8,22]featuretransitionsofamorecompoundshape,which canbereadilytranslatedintosequencesofPDRStransitions.

Forinstance,atransitionofaninfinite-alphabetpushdownautomaton[8]typicallyinvolvesarefreshment(i•)followed by pop(pop

(

j

)

) anda sequenceofpushes(push

(

j

)

).Thisdecompositionleads toalinearblow-up insizefortranslations ofreachabilityquestionsintother-PDRSsetting.Forregisterpushdownautomata[22],anadditionalcomplicationistheir use of non-injective register assignments. Observe, though, that transitions in the non-injective framework can be easily mimicked usinginjective registerassignments provided we keep trackof thepartitions determined by duplicated values inthe originalautomaton.The book-keepingcan be implementedinside thecontrol state,which leadsto an exponential blow-up inthesize ofthesystem, becausethe numberofallpossible partitionsisexponential.Notethat thenumber of registersdoesnotchangeduringsuchasimulation.Anotherdifferenceisthatregisterpushdownautomata[22]aretailored towards datalanguages, i.e.a stacksymbol is an element of

_D

paired up witha tag drawnfrom a finiteset. From this perspective,r-PDRSsuseasingletonsetoftags.Still,richertagsetscouldbeencodedviasequences ofelementsof

D

(for example,tosimulatethei-thout ofk tags,we couldpushsequences oftheformd₁id2 ford1

,

d2

∈

D

withd1

=

d2). This

reductionisachievableinpolynomialtime.

Following [13,8,20], we mostly useinjective registerassignments. Thisis done toallow usto explore whetherthe re-striction still leads to asymptotically moreefficient reachabilitytesting, as inthe pushdown-free case. Ona foundational note,injectivity givesa moreessentialtreatmentoffreshnesswithrespecttoasetofregisters: non-injectiveassignments canbe easily usedtoencode PSPACEcomputationsthat havelittletodowiththeinteraction betweenfinite control(and pushdown)andfreshness.

Namepermutations There is a natural action of the group of permutations of

D

on stacks, assignments, runs, etc. For instance,givena permutation

π

:

D

→

D

andan assignment

τ

,the resultofapplying

π

to

τ

is theregisterassignment

π

·

τ

givenby

{

(

i

,

π

(

d

))

|

(

i

,

d

)

∈

τ

}

.Similarly,

π

·

s

=

π

(

dn

)

· · ·

π

(

d1

)

foranystacks

=

dn

· · ·

d1 while,ontheother hand,

π

·

q

=

q forall statesq.Hence,

π

·

(

q

,

τ

,

s

)

=

(

q

,

π

·

τ

,

π

·

s

)

and, for

ρ

=

κ

0

· · ·

κ

n a transitionsequence,

π

·

ρ

isthe

sequence

π

·

κ

0

,

· · ·

,

π

·

κ

n.

Notethat,aslongasourconstructionsinvolvefinitelymanynames,theywillalwayshaveafinitesupport:wesaythata setS

⊆

D

supportssome(nominal)elementxif,forallpermutations

π

,if

π

(

n

)

=

nforalln

∈

Sthen

π

·

x

=

x.Accordingly,

thesupport

ν

(

x

)

ofxisthesmallestsetS supportingx.Forexample,

ν

(

τ

)

= {

τ

(

i

)

|

i

∈ [

r

]}

,forallassignments

τ

.The sup-portofarun

ρ

=

κ

0

· · ·

κ

n is

ν

(

ρ

)

=

n

j=0

ν

(

κ

j

)

,i.e.itconsistsofallelementsof

D

thatoccurinit.Thefinite-support

settingcanbeformallydescribedbymeansofnominalsets[11]andclosureresultssuchasthefollowinghold.

Fact5(Closureunderpermutations).Fixanr-PDRSandlet

ρ

beatransitionsequenceand

π

:

D

→

D

apermutation.Then

π

·

ρ

isalsoatransitionsequence.

3. Distinguishability

Deviceswithrregistersbutwithoutpushdownstorage,suchasfinite-memoryautomata[13],cantakeadvantageofthe registerstodistinguishr elementsof

D

fromtherest.Consequently,anyruncan bereplacedwitharunthatendsinthe samestate,yetissupportedbymerelyrelementsoftheinfinitealphabet[13,Proposition 4].

With extra pushdown storage, an r-PDRS is capable of storing unboundedly many elements of

D

. Nevertheless, the restrictednature ofthestack makes itpossible toplacea finitebound on thesize ofthesupport neededforarun to a givenstate,whichisagainafunctionofthenumberofregisters.

Lemma6(Limiteddistinguishability).Fixanr-PDRS.Foreverytransitionsequence

ρ

=

(

q0

,

τ

0

,

)

n

(

qn

,

τ

n

,

)

,thereisatransition

Proof. Theproofisbyinductiononn.Whenn

≤

1,theresultistrivial.Otherwise,wedistinguishtwocases. Inthefirstcase,thetransitionsequenceisoftheform:

(

q0

,

τ

0

,

)

(

q1

,

τ

1

,

d

)

n−2

(

qn−1

,

τ

n−1

,

d

)

(

qn

,

τ

n

,

)

in whichthe first transitionisby push

(

i

)

(sod

=

τ

1

(

i

)

), thelast transitionis by pop

(

j

)

or pop• andthe stackdoesnot

empty until thefinal transition. Sinced isnever poppedfromthestack during themiddlesegment, also

(

q1

,

τ

1

,

)

n−2

(

qn−1

,

τ

n−1

,

)

isa validtransitionsequenceandhence,fromthe inductionhypothesis,thereis atransitionsequence

be-tweenthesametwoconfigurationsusingnomorethan3rnames.Byaddingdtothebottomofeverystackinthissequence one obtains another validtransitionsequence:

(

q1

,

τ

₁

,

d

)

n−2

(

qn−1

,

τ

_n−1

,

d

)

with

τ

₁

=

τ

1 and

τ

_n−1

=

τ

n−1, andthenew

sequencefeatures

≤

3rnames.Itfollowsthatthelattercanbeextendedtotherequired:

(

q0

,

τ

0

,

)

(

q1

,

τ

1

,

d

)

n−2

(

qn−1

,

τ

n−1

,

d

)

(

qn

,

τ

n

,

)

sinceneither push

(

i

)

,norpop

(

j

)/

pop•changetheregisters. Otherwise,thetransitionsequenceisoftheform:

(

q0

,

τ

0

,

)

k

(

qk

,

τ

k

,

)

n−k

(

qn

,

τ

n

,

)

with0

<

k

<

n.Itfollowsfromtheinductionhypothesisthattherearesequences:

ρ

1

=

(

q0

,

τ

0

,

)

k

(

qk

,

τ

k

,

)

ρ

2

=

(

qk

,

τ

k

,

)

n−k

(

qn

,

τ

n

,

)

with

τ

₀

=

τ

0,

τ

n

=

τ

n,

τ

_k

=

τ

kandwhicheach,individually,usenomorethan3r names.LetN

⊇

ν

(

τ

0

)

∪

ν

(

τ

k

)

∪

ν

(

τ

n

)

bea

setofnamesofsize3r.Weaimtomap

ν

(

ρ

1

)

and

ν

(

ρ

2

)

intoN byinjectionsiand j respectively.Foriweseti

(

a

)

=

afor

anya

∈

(

ν

(

τ

0

)

∪

ν

(

τ

k

))

andotherwisechoosesome distinctb

∈

N

\

(

ν

(

τ

0

)

∪

ν

(

τ

k

))

.Similarly,for j we set j

(

a

)

=

a forany

a

∈

(

ν

(

τ

k

)

∪

ν

(

τ

n

))

andotherwisechoosesomedistinctb

∈

N

\

(

ν

(

τ

k

)

∪

ν

(

τ

n

))

.Notethat thesechoicesare alwayspossible

because

|

ν

(

ρ

1

)

|

≤ |

N

|

≥ |

ν

(

ρ

2

)

|

.Finally,weextendiand j topermutations

π

i and

π

jon

D

.Sincetransitionsequencesare

closedunderpermutations(Fact 5):

(

q0

,

π

i

·

τ

0

,

)

k

(

qk

,

π

i

·

τ

k

=

π

j

·

τ

k

,

)

n−k

(

qn

,

π

j

·

τ

n

,

)

isavalidtransitionsequencewith

π

i

·

τ

0

=

τ

0,

π

j

·

τ

n

=

τ

n andwhichissupportedbyasubsetofN.

2

Corollary7.Fixanr-PDRS

S

andastateq of

S

.Ifthereisarunof

S

endinginq thenthereisarunof

S

endinginq thatissupported byatmost3r distinctnames.

The 3rboundgivenabove isoptimalinthesense thatthereexists anr-PDRSsuchthatall runsto acertainstate will havetorelyon3relementsof

D

.

Lemma8(Mostdiscriminatingr-PDRS).Thereexistsanr-PDRS

Q

,

qI

,

τ

I

,

andq

∈

Q suchthat

|

ν

(

ρ

)

|

=

3r foranyrun

ρ

ending

inq.

Proof. Considerthefollowinghigh-leveldescriptionofanr-PDRS.Themachineproceedsasfollows:

1. Pushregistersinnumericalorder,twice,toobtainstack

τ

I

(

r

)

· · ·

τ

I

(

1

)

τ

I

(

r

)

· · ·

τ

I

(

1

)

.

2. Refreshregistersbyperformingi• forall1

≤

i

≤

r.Letthenewassignmentbe

τ

1.

3. Performpop•r-times,thusensuringthat,foreach1

≤

i

,

j

≤

r,

τ

I

(

i

)

=

τ

1

(

j

)

.

4. Pushallregistersinnumericalorder,toobtainstack

τ

1

(

r

)

· · ·

τ

1

(

1

)

τ

I

(

r

)

· · ·

τ

I

(

1

)

.

5. Refreshallregisters.Letthenewassignmentbe

τ

2.

6. Performpop• 2r-times,thusensuringthat,foreachi

,

j,

τ

2

(

i

)

=

τ

1

(

j

)

and

τ

2

(

i

)

=

τ

I

(

j

)

.

7. Silentlytransitiontostateq.

Now observe that the conditions in steps 3 and 6 and the fact that register assignments are injective ensure that

|

ν

(

τ

I

)

∪

ν

(

τ

1

)

∪

ν

(

τ

2

)

| =

3r.Hence,anyrunreachingqissupportedbyexactly3rdistinctnames.

2

Remark9.The3rboundgivenabovecanbeadaptedtotheautomatapresentationsof[8,22]yieldingbounds3r

+

(

1

)

.An adaptationofLemma 8improvesuponExample6of[8],wherealanguagerequiring2r

−

1 differentsymbolswaspresented.

4. ReachabilityisEXPTIME-complete

Weconsiderthefollowingdecisionproblem.

r-PDRSReach:Givenanr-PDRS

S

andq

∈

Q,istherearunof

S

endinginq?

Weshallshowthattheproblem(anditscounterpartsforalltheothercloselyrelatedmachinemodels)isEXPTIME-complete. Notethatreachabilityisequivalenttolanguagenon-emptinessintheautomatacase.

4.1. ReachabilityisEXPTIME-solvable

Theorem10. r-PDRS Reachand languageemptinessforinfinite-alphabet pushdownautomata[8]andregister pushdown au-tomata[22]aresolvableinexponentialtime.

Proof. Lemma 6 yields an exponential-time reductionof r-PDRSReach to theclassic reachability problemforpushdown systemsoverfinitealphabets[5]:onecanreplacether

D

-valuedregisterswithr

[

3r

]

-valuedregisters,andthenincorporate them intothe finitecontrol (for asingly-exponential blow-up ofthestate space). Since thelatter problemissolvablein polynomialtime,itfollowsthatr-PDRSReachisinEXPTIME.

By Remark 4, the emptiness problem for infinite-alphabet pushdown automata [8] can be reduced to r-PDRSReach in polynomial time, immediatelyyielding the EXPTIMEupper bound.4 Forregisterpushdown automata [22] we have an exponential-timereductiontor-PDRSReach,whichdoesnotyieldtherequiredbound.However,recallthatthetranslation intor-PDRSpreservesthenumberofregisters,soLemma 6 stillimpliesalinearupperboundforthenumberof

D

-values neededforfindinganacceptingrun.Consequently,wecanreducelanguageemptiness ofregisterpushdownautomatatoa reachabilityproblemforpushdownsystemsatanexponentialcost.SincethelatterisinP,theformerisinEXPTIME.

2

4.2. ReachabilityisEXPTIME-hard

Thebound givenaboveis tight:we simulateapolynomial-spaceTuringmachine withastack (i.e.apolynomial-space auxiliarypushdownautomaton[9]),whichhasanEXPTIME-completehaltingproblem. Areductionfromthemorefamiliar alternating polynomial-spaceTuringmachineswouldalsobe possible,butCook’smodeliscloserto r-PDRS,whichallows ustoconcentrateonthemainissueofencodingbinarymemorycontentwithouttheneedtomodelalternation.

Theorem11.r-PDRSReachisEXPTIME-hard.

Wefirstgiveanoutlineoftheargumentbeforedescribingtheproofindetail.

4.2.1. Argumentoutline

Letusassumeanauxiliarypushdownautomaton

M

workingoverabinarytapeofsizenandastackalphabetofsizek. We canassume WLOGthat every transitionof

M

performs asingle pushorpop (butnot both)along witharead/write bytheheadfollowedbyaheadmovement.Moreover,letusassumethatthestackalphabetof

M

includesadistinguished symbol$ markingthebottomofthestackandthat,additionally,

M

startswith$ onthestackandhaltswhenitispopped ($ isnotusedotherwise).Finally,weletthetapeof

M

initiallycontainonly0’s.

Weshallconstructa

(

6n

+

k

+

1

)

-PDRS

S

suchthat

M

terminatesiff

S

reachesadesignated“final”state.Theregisters of

S

willbedividedinto4groups:

τ

=

τ

0

τ

1

τ

2

τ

3

ofsizesk

+

1,2n,2nand2nrespectively.

•

τ

0 will contain

D

-valuescoding the k stack symbolsand an auxiliary symbol # (we write # for

ˆ

the letter which encodes#,etc.).Itwillbeleftuntouchedthroughoutthesimulation.

•

τ

1and

τ

2willbeusedtoencodethen-bittapeof

M

duringpush- andpop-transitionsrespectively.

•

τ

3 will have an auxiliary role andin particular will ensure the integrity of the tape when changing frompop- to push-transitions.

The simulation must maintain a representation of the current tape and stack content of

M

(as well as the current state, headpositionetc.). Since

S

is itselfequippedwitha pushdownstack,we shalluseits stackto representthestack of

M

by puttingsome distinguishedk-elementsubset of

D

inbijection withthe stackalphabet.A naturalcandidatefor

representing the tape content of

M

is theregisters of

S

.However, the requirement that the registerassignment be an injective map whose rangeis

_D

makes constructingan encoding difficult:in anygivenregister assignment,there are no non-trivialrelationshipsbetweentheelements!Hence,toencodeinformationweareforcedtousetheregisterassignment inconjunctionwithotherdata.

Let usconsider two registerassignments

τ

and

τ

of length 2n.We say that such a pairis compatible just if, forall 1

≤

j

≤

n:

{

τ

(

2j

−

1

),

τ

(

2j

)

} = {

τ

(

2j

−

1

),

τ

(

2j

)

}

.

(1)

Although anygiven registerassignment cannot encode anymeaningful information,a pairof compatibleregister assign-mentscanbeusedtogethertorepresentatapetbyageneralisedxor,written

τ

[

τ

]

:

τ

[

τ

]

(

j

)

=

0 iff

τ

(

2j

)

=

τ

(

2j

).

Inotherwords,thetapecontentatposition1

≤

j

≤

nis0 if

τ

and

τ

agreeontheorderofthedatavalues

{

τ

(

2j

−

1

),

τ

(

2j

)

}

and1 ifthey arethetranspositionofeach other.We canthinkofthenotation

τ

[

τ

]

asabinary functionof

τ

and

τ

,in whichcaseithasthepropertythatanyequation:

τ

[

τ

] =

t

canbe solveduniquelyfor

τ

,

τ

ort (giventheotherparameters).Thisfactbecomesessentialtoensuringtheintegrityof the tape encoding whenchanging fromsimulatinga pop tosimulating a pushtransition. Notethat although thisbinary function iscommutative,theasymmetricnotationreflectsthefact that,byconvention,wewillview thefirstargumentas beingprimary,oftenwritingthat

τ

isanencodingoft withrespecttothemask

τ

.

Weconstruct

S

suchthat,wheninaconfigurationwithregisterassignment

τ

whichisfaithfullysimulatingapush tran-sitionof

M

withtapecontentt,wehave

τ

1

[

τ

I1

]

=

t.Whensimulatingapoptransitioninsuchaconfiguration,wehave

τ

2

[

τ

I2

]

=

t.In other words,themasksusedare just theinitial registerassignments togroups

1

and

2

respectively.

Consequently,wehavetheinvariantthat,inanyfaithfulsimulationof

M

reachingaconfigurationwithassignment

τ

,

τ

1

and

τ

I1arecompatibleand

τ

2and

τ

I2arecompatible.

At eachstepofthesimulation,

S

needsto queryitsrepresentationofthetapeinordertodeterminewhichtransition of

M

toapply,whichentailscomputing,e.g.

τ

1

[

τ

I1

]

.However, duetotheinjectivityconstraintonregisterassignments,

for

S

tobeabletoaccessapairofcompatibleregisterassignments,atleastonemustbestoredonthestack.Hence,

S

will use thestacknot onlytoencode the stackcontentof

M

,butalsoaspartoftheencoding ofits tapecontent bystoring copiesofthemasks

τ

I1 and

τ

I2 sothatthey arealwaysaccessiblewhenneeded.Thisdualuseforthestackrequiresa

considerableamountofbookkeeping.

Thekeyproblemtoovercomeistoensurethatthemasks

τ

I1and

τ

I2arealwaysavailableatthetopofthestackwhen

they are neededfordecoding.Letusfirstconsider

τ

I2 whichis somewhateasierbecauseit isusedforpop transitions.

Since

M

haltsonlywithanemptystack,weknowthatforeverypoptransitionmade,theremusthavealreadybeenmade a correspondingpush transition.Hence, wearrangefor

S

,whensimulatingapushtransitionwithastackofheighth,to additionallypushan extracopyof

τ

I2 (whichithasstoredinits registers)so thatitisavailable atthetopofthestack whenthemachinenextreturnstoastackofheighth,shoulditberequiredtosimulatethecorrespondingpoptransition.

Ensuringtheavailabilityof

τ

I1 forthepurposeofsimulatingpushtransitionsismuchmoredifficult.Itisnotpossible

to“pre-load”thestackwithcopiesof

τ

I1 inthesamewayasfor

τ

I2because(i)thelengthofsequencesofconsecutive

pushtransitionsisnotknownand(ii)thedualuseofthestackrequiresthat copiesof

τ

I1mustbeinterleavedwithdata

valuessimulatingthelettersofthestackalphabetpushedby

M

.Mattersarefurthercomplicatedbythefactthat looking up thevalueof

τ

I1inordertocompute

τ

1

[

τ

I1

]

requirespoppingitoffthetopofthestack.Considerthesimulationof

twoconsecutivepushtransitions.Aspartofthesimulationofthefirsttransition,

τ

I1 mustbepoppedandhencelost.The

injectivity constraintontheregistersensuresthat theonlywaythatitcouldbe preserved(bytemporarilypoppingitinto theregisters)wouldbeatthecostoflosing

τ

1 instead.However,bythetimethesecondtransitionissimulated,another

copyisrequiredtobeaccessible(i.e.beatthetopofthestackagain).Wecannot“pre-load”thestackwithcopiesof

τ

I1,

we cannot temporarilystore itin theregistersandaccessing acopyconsumes it.Hence we areforcedto construct

S

in suchawaythatitcomputes

τ

1

[

τ

I1

]

withoutaccessing

τ

I1atall.Toachievethis,wearrangefor

S

toguessamask

τ

1,

compute

τ

1

[

τ

₁

]

withrespecttothisguessandplacetheguessedmaskonthestack.Duringthelatersimulationofpop transitions,when

S

hassomesparecapacityinits registersbecause

τ

1 isnolongerusedtoencode tapecontent,itwill

beabletodischargetheobligationofverifyingthatitguessed

τ

₁

=

τ

I1.

So far, we havediscussed theneed tostore three kindsofentities on thestack, namely:representations ofthe stack letters of

M

,copies ofthe mask

τ

I2 andunverified guesses ofthe mask

τ

I1.In fact, to be ableto switch from tape

encoding using register group

2

to tape encoding usingregister group

1

(which happens whenever

S

simulates

M

performingapopfollowedbyapush)withoutlossofinformation,weneedtomaintainadditionalentities.

s

=

si₀ si₁

τ

I2 si3

τ

I2 si3

si−₀1 si−₁1

τ

I2 si−31

τ

I2 si−31

..

.

..

.

..

.

..

.

..

.

..

.

(2)

with

|

si₁

|

= |

τ

I2

|

= |

si3

|

=

2nand

|

s i

0

|

=

1 (andarbitraryi).Thetopstacksymbolisinthetop-leftcorner,theelement

belowitonthestackisonitsright,si−₀1isbelowsi

3etc.Eachsj0abovecorrespondstoanon-# stacksymbol(asspecified

by

τ

0).

Theguessedmasks, tobeverifiedlater,willbestoredassj₁.Torepresentthetapeatpops,wewillusethemask

τ

I2.

Oursimulationwillmakesurethat thisiscorrectlystoredon thestackduringpushes.Finally,theelements sj₃

,

τ

I2

,

sj₃ willbeusedtosupportthesimulationwhenitswitchesfrompopstopushes.Theelementsj₃ willalsobeamask.

Therewillalsobe“exceptional”rows,whichhavetheform:

si_#

= ˆ

# si₃

τ

I2 si1

τ

I2 si3 (3)

where# is

ˆ

theencodingof# in

τ

I0.Theserowswillbeusedtoverifythecorrectnessofsj3,aswediscusslater.

Thus, ineach rowwe store astack symboland 5 masks.The purposeof themasksis tohelp usdetermine thetape content of

M

.Onthe other hand,thecurrentstack of

M

can be recoveredby projecting out thefirst columnofs and erasinganyoccurrenceof#.

ˆ

Wenextdefinethedifferentstepsofthesimulation.Initially,thereisaninitialisationsteppushingarows0_onthestack;

S

reachesitsfinalstatepreciselywhenitsuccessfullypops s0_{.Recallthateverytransitionof}

_M

_{includesapushorapop}

action.Accordingly,thestatesof

S

havetwomodes:apush-modeandpop-mode.Eachtransitionof

S

non-deterministically guessesthemode ofthenext state(of

S

).

S

simulatesthepush-transitionsof

M

using

τ

1

[

si₁

]

asitstape;itsimulates thepoponesusing

τ

2

[

τ

I2

]

asitstape.Thesimulationonlygoesthroughifthemasksoftheseencodingsarecorrect,that

is,ifsi₁

=

τ

I1foralli.Put otherwise,inanacceptingcomputationof

S

,allmaskssi₁ thatappearonthestackmustbe equalto

τ

I1.

Initialisation

S

startsoffbypushingthefollowingrow:

s0

= ˆ

$

τ

I1

τ

I2

τ

I3

τ

I2

τ

I3 (4)

Recallherethatthetapeof

M

isassumedtobeinitiallyempty, andthisiscapturedby ourinitialencoding:

τ

I1

[

s01

]

=

0

· · ·

0 (since

τ

I1

=

s01).

Push-mode Duringthisphase,

S

uses

τ

1 to representthetape of

M

,while

τ

0

,

τ

2

,

τ

3 stayunchanged. Assuming

S

hasstackcontentasin(2)or(3)andthecurrentregisterassignmentis:

τ

=

τ

I0

τ

1

τ

2

τ

3

thecurrenttapecontent(accordingtothesimulation)is:

t

=

τ

1

[

si1

]

.

Moreover,byconstruction,weshallhave

τ

2

=

τ

I2.

Nowsuppose

M

istoperformapush-transitionq

−−−−−−−−→

x,y,z,push(C) q,wherexisthecurrentlyscannedsymboltobe over-writtenwith y,theheadmovement isindicatedby z

∈ {

L

,

R

,

N

}

andC isthestacksymboltobepushed.Assumethatthe currentheadpositionis1

≤

j

≤

n(

S

willkeeptrackofitinitsstate).Inordertosimulatethetransition,

S

firstneedsto retrieve

τ

1

[

si₁

]

(

j

)

.Onewaytodothatwouldbetosimplypopsi₁offthestack.However,thatwouldbecatastrophicfor oursimulation sincethere isnoguarantee atthispointthat si

1 isacorrect mask(i.e. equalto

τ

I1);poppingit would

destroyit,thus annihilatinganypossibilityofverifyingits correctness.5 _{Thus,instead,}

_S

_{will guesswhat thevalueof s}i

1

mightbeandoperateaccordingtotheguess. Moreover,theguesswillbepushedonthestackforsubsequentverification inthepop-mode.

Moreprecisely,

S

firstpushestheword

τ

2

τ

3

τ

2

τ

3.Itthenproducesaguesssi+₁1ofthemasksi₁whichisconsistent with reading x at the current head position,in that

τ

1

[

si1+1

]

(

j

)

=

x. Thisis achievedby non-deterministically pushing

5 _{Note that pushing initially τ}

consecutivepairsofelements of

τ

1,subjecttothepreviousrequirementandtheconstraintexpressedby equation(1).It nextperformstheoperationsof

M

(asinstructedby yandz)accordingtothetape

τ

1

[

si+₁1

]

.Thus,aftertheseoperations, theregisterassignmentandtopofthestackread

τ

=

τ

I0

τ

1

τ

I2

τ

3 si+1

=

si+11

τ

I2

τ

3

τ

I2

τ

3

where

τ

₁

[

si+₁1

]

isthenewtapecontent(inparticular,wehavewritten yinbit j).

Finally,

S

choosesthekindofthenexttransition.Ifitisapush,thenC

ˆ

ispushedonthestack(thenamecorresponding to C). Otherwise,beforeswitchingto pop-mode,ithastotransfer itstape-encoding from

τ

1 to

τ

2.Weachieve thisby simultaneouslymoving si+₁1 fromthestackinto

τ

₁andchanging

τ

I2into

τ

₂ sothat

τ

₂

[

τ

I2

]

=

τ

1

[

si+11

]

.Thiscanbe

executed bypoppingsi+₁1

(

2j

)

andcomparingitwith

τ

₁

(

2j

)

foreachbit1

≤

j

≤

nand,then,depending ontheoutcome, swappingthecontentsofregisters2j

−

1 and2jineachof

τ

₁and

τ

2inordertoreflectthedesiredchange.Finally,we

pushsi+₁1 backonthestack,followedby C

ˆ

toarriveat

τ

=

τ

I0 si+11

τ

2

τ

3 s

i+1

_{= ˆ}

_{C s}i+1

1

τ

I2

τ

3

τ

I2

τ

3

with

τ

₂

[

τ

I2

]

=

τ

1

[

si+11

]

.

Pop-mode

S

nowuses

τ

2 asthetape.Letthecurrentconfigurationof

S

havestackasin(2)andlet

τ

=

τ

I0

τ

1

τ

2

τ

3

sothat

τ

2

[

τ

I2

]

istherepresentedtapecontent.

Suppose that

M

’s headscans position j andthe next transitionisto be q

−−−−−−−→

x,y,z,pop(C) q, where si₀

= ˆ

C.Recallthat

τ

1

,

si1

,

· · ·

,

s11areguessesfrompreviouspushtransitions(orfromexceptionalpushes,stilltobediscussed),whiles01

=

τ

I1by(4).Toverifytheguesses,itsufficestoverifythat

τ

1

=

si₁and,atthenextpop,verifythat

τ

1

=

si−₁1,etc.Thus,

S

will firstpop C

ˆ

fromthe stackandthenpop si

1,simultaneously checkingthat it equals

τ

1 (otherwiseit willblock).

Nextwe pop

τ

I2 todetermine

τ

2

[

τ

I2

]

andperformtheinstruction

(

x

,

y

,

z

)

bychanging

τ

2 to

τ

2(andupdating the

headpositioninthestate).Thus,weobtain

τ

=

τ

I0

τ

1

τ

2

τ

3 si

=

si₃

τ

I2 si3

havingverifiedthat

τ

1

=

si1.

Now

S

guesses thekindofthenext transition. Ifitis tobe apop,

S

pops the remainderof si.Otherwise,

S

should switch topush-mode andinparticular changeitstape-storing routinefrom

τ

2 to

τ

1.Thatis,

τ

1 needsto beupdated

to

τ

₁ such that

τ

₁

[ ˆ

τ

1

]

=

t, for an appropriate mask

τ

ˆ

1, where t

=

τ

₂

[

τ

I2

]

is the currenttape. Also, having now

preserveditsvalueaccordingtotheencoding,

τ

₂ shouldbechangedbackto

τ

I2.

We are nowfaced withthefollowing obstacle.Updating

τ

1 to

τ

₁ wouldmake uslose thecurrent

τ

1.Thiswould breakoursimulationas, although

S

hasverified

τ

1

=

si₁,itremainsto checkthat

τ

1 isthesameasallthose guessed masks still on thestack, i.e.si−₁1

,

. . . ,

s1

1. Since entering push-moderequires that we overwritethe contentsof

τ

1, we

would like to preserve its currentassignment on the stack. However, doing thisdirectly is impossiblesince we need to obtain

τ

I2 fromlowerdown thestackinordertodecode thecurrenttapecontents

τ

2.Weovercomethisby storingin

thestackaguessfor

τ

1,alongwithauxiliarymaskswhichwillallowustoverifythisguesswhenpopping.Moreover,we

shallpick

τ

ˆ

1

=

τ

1.Hence,

S

operatesasfollows.

•

Itfirstpopssi₃andstoresitin

τ

3.

•

Then,itpops

τ

I2andstoresitin

τ

2and,atthesametime,itcopiest in

τ

1and

τ

3,thatis,itupdatesthemto

τ

₁

and

τ

₃respectively,suchthatt

=

τ

₁

[

τ

1

]

=

τ

3

[

s i

3

]

.

•

Itpushes

τ

I2backonthestack.

•

Itthenmakesaguess

τ

₁of

τ

1andpushesitonthestack.Whiledoingso,itupdates

τ

₃tothemasksi₃satisfying

τ

₁

[

τ

₁

]

=

τ

₃

[

si₃

]

.

•

Finally,itpushes#s

ˆ

i₃

τ

I2onthestack.

Thus,weendupwith

and the automaton switches to push-mode. Note that

τ

₁ uses

τ

1 as a mask, yet we have

τ

₁ on the stack instead. Nevertheless,byconstruction:

τ

₁

=

τ

1

iff

τ

₁

[

τ

₁

] =

τ

₁

[

τ

1

]

iff

τ

₃

[

si₃

] =

τ

₃

[

si₃

]

iff si₃

=

si₃

.

Thus, when popping the row si_#, the automaton will additionally check that si₃

=

si₃ andthus verify that the correct mask

τ

₁hasbeenstoredonthestack.

Finally,wediscussthecasewhen

S

isinpop-modeandthetopofthestackisasin(3).Insuchacase,thepoptransition of

S

istakenindependentlyof

M

.Inparticular,let

τ

=

τ

I0

τ

1

τ

2

τ

3 .

•

S

startsbypopping# and

ˆ

si₃,andstoresthelatterin

τ

3.

•

Next,itpops

τ

I2andsi1,andchecksthatthelatterisequalto

τ

1.

•

Finally,itpops

τ

I2andsi3,andchecksthatthelatterisequaltosi3(whichisnowstoredin

τ

3).

Thus,accordingtoourdiscussionabove,

S

correctlyverifiesthecontinuityofthemaskssi₁whileconsumingtherowsi_#. Altogether,theaboveconstructionyieldsa

(

6n

+

k

+

1

)

-PDRS

S

ofpolynomialsizewithrespectton.Moreover,

S

popss0 iffitmakesconsistentguessesformasksusedinitsfirstcomponentand,therefore,faithfullysimulatestheoperationsof

M

leadingtopoppingtheterminatingsymbolfromitsstack.

4.2.2. Argumentindetail

Wenowmaketheabovesketchmoreprecise.

Proof. Let

M

=

Q

,

qI

,

T

,

δ

beanauxiliarypushdownautomatonoperatingoverabinarytapeofsizenandstackalphabet

{

1

,

· · ·

,

k

}

,with$

∈ {

1

,

· · ·

,

k

}

beingadistinguished bottom-of-stacksymbol.

M

hasinitialstate qI,andlet usassume its

transitionrelationisofthefollowingtype,

δ

⊆

Q

×

Op_k

×

Q,with

Op_k

= {

0

,

1

}

2

× {

L

,

R

,

N

} × {

push

(

i

),

pop

(

i

)

|

1

≤

i

≤

k

}

.

Thatis,ineverytransition,

M

readsabitxfromitscurrentheadposition,writesback y,movesthehead(Left,Right,or No-move),andpushesor popsa symbol fromthestack. Moreover, weassume that

M

hasinitial tape0

· · ·

0 andinitial stack$,and

M

haltswhenitpops$ fromthestack($ isnotusedotherwise).

Weconstructa

(

k

+

1

+

6n

)

-PDRS

S

suchthat

M

pops$ fromthestackiff

S

reachesadesignatedstateq_F.Inparticular,

S

=

(

Q

× {

1

,

· · ·

,

n

} × {↑

,

↓}

)

∪ {

qI

} ∪

Q

,

qI

,

τ

I

, δ

where each state ofthe form

(

q

,

j

,

↓

)

[resp.

(

q

,

j

,

↑

)

] is said to be inpush-mode [pop-mode]. The index j indicates the positionoftheheadonthetapeof

M

.Thus,atitsinitialposition,

M

readsthefirstbitofthetapeandcanonlyperform apush. Qisasetofauxiliarystatesofpolynomialsizeinn,whichwegraduallyspecifybelow.Theinitialassignment

τ

I is

arbitrary;wedividetheregistersinto4groups(ofsizesk

+

1,2n,2nand2nrespectively)andwriteregisterassignments

τ

intheform:

τ

=

τ

0

::

τ

1

::

τ

2

::

τ

3

.

Wemoreover stipulatethat,throughouttheoperation of

S

,the valuesofitsgroup-0registers remainfixed and,foreach i

=

1

,

2

,

3 and1

≤

j

≤

n,

{

τ

i

(

2j

−

1

),

τ

i

(

2j

)

} = {

τ

Ii

(

2j

−

1

),

τ

Ii

(

2j

)

}

.

Thedisciplineimposedaboveisinstrumentedsoastoencodeann-sizetapeineachof

τ

1

,

τ

2

,

τ

3.Inparticular,foreach i

=

1

,

2

,

3 and2n-components

τ

i

,

τ

ˆ

i,wecandefineann-tape

τ

i

[ ˆ

τ

i

]

bysetting,

τ

i

[ ˆ

τ

i

]

(

j

)

=

0 iff

τ

i

(

2j

)

= ˆ

τ

i

(

2j

)

forall1

≤

j

≤

n.Wecall

τ

ˆ

i themaskoftheencoding. OurPDRS

S

willsimulatetheoperationsof

M

usingjustsuchan encoding.Ontheotherhand,thestackisgroupedintorowsoftwopossibleforms:

si

=

si₀

::

si₁

::

si₂

::

si₃

::

si₂

::

si₃ (5)