On the Executability of Interactive Computation

0.13 MB·

by  Bas Luttik

On the Executability of Interactive Computation
BasLuttikandFeiYang
EindhovenUniversityofTechnology,TheNetherlands
6
1
Abstract. ThemodelofinteractiveTuringmachines(ITMs)hasbeenproposed
0
tocharacterisewhichstreamtranslationsareinteractivelycomputable;themodel
2
ofreactiveTuringmachines(RTMs)hasbeenproposedtocharacterisewhichbe-

n havioursarereactivelyexecutable.Inthisarticleweprovideacomparisonofthe
a two models. We show, on the one hand, that the behaviour exhibited by ITMs
J
isreactivelyexecutable,and,ontheotherhand,thatthestreamtranslationsnatu-

1 rallyassociatedwithRTMsareinteractivelycomputable.Weconcludefromthese
1
resultsthatthetheoryofreactiveexecutabilitysubsumesthetheoryofinteractive

computability. Inspiredbytheexistingmodel of ITMswithadvice,whichpro-
]
O videsamodelofevolvingcomputation,wealsoconsiderRTMswithadviceand
weestablishthatafacilityofadviceconsiderablyupgrades thebehaviouralex-
L
pressiveness of RTMs: every countable transition system can be simulated by
.
s someRTMwithadviceuptoafinenotionofbehaviouralequivalence.
c
[

1 Introduction
2
v
6 AccordingtotheChurch-Turingthesis,theclassicalTuringmachinemodeladequately
4 formalises which functions from natural numbers to natural numbers are effectively
5 computable. There is, however, a considerable semantic gap between computing the
1
resultofafunctionappliedtoanaturalnumberandthewaycomputingsystemsoperate
0
nowadays.Moderncomputingsystemsarereactive,theyareincontinuousinteraction
.
1
withtheirenvironment,andtheiroperationisnotsupposedtoterminate.Quiteanumber
0
ofextendedmodelsofcomputationhavebeenproposedinrecentdecadestostudythe
6
combination of computation and interaction (see, e.g., the collection in [7]). In this
1
: paperwecompareinteractiveTuringmachinesandreactiveTuringmachines.
v
VanLeeuwenandWiedermannhavedevelopedatheoryofinteractivecomputation
i
X from the stance that an interactive computation can be viewed as a never-endingex-
r changeofsymbolsbetweenacomponentanditsunpredictableinteractiveenvironment
a
[8].Semantically,thisamountstostudyingtherecognition,generationandtranslation
of infinitestreamsof symbols.In[9], the notionof interactiveTuringmachine(ITM)
is put forward as a tool to formally characterise which stream translations are inter-
actively computable. The notion is subsequently extended with an (non-computable)
advicemechanisminordertoobtainanon-uniformmachinemodel.VanLeeuwenand
WiedermannarguethattheresultingmodelofinteractiveTuringmachineswithadvice
isaspowerfulastheirmodelofevolvingfiniteautomata,andtheyconcludefromthis,
onintuitivegrounds,thatITMswithadviceareadequatetomodelevolvingsystemsuch
astheInternet[15].
ThemodelofinteractiveTuringmachinesfocussesoncapturingthecomputational
contentof sequentialinteractivebehaviour. The included mechanismof interaction is
therefore limited to achieving this goal, and does not easily generalise to more than
one distributed component, nor does it allow for more fine-grained considerations of
the behaviourof reactive systems. The behaviouraltheoryof reactivesystems, onthe
other hand, has focussed on aspects of modelling, specification and verification (see,
e.g.,[1]).
To integrate computability theory and the behaviouraltheory of reactive systems,
the notion of reactive Turing machine (RTM) has been proposed in [2,3]. It extends
Turingmachineswithconcurrency-styleinteraction.Semantically,theoperationalbe-
haviourofanRTMisgivenbyatransitionsystem.Fromthistransitionsystemonemay
extractasetofcomputations,orstreamtranslations,butamorerefinedanalysisisalso
possible.Infact,tostudytheeffectofinteractionofmultiplecomponentsmanyrefined
notionsofbehaviouralequivalencehavebeendevelopedintheconcurrencytheoryliter-
ature[6].ThenotionofRTMgivesrisetoageneraltheoryofexecutability:atransition
systemisexecutable(usuallyuptosomepreferrednotionofbehaviouralequivalence)
ifthereexistsanRTM thathasthetransitionsystemasitssemantics.(We referto[3]
formoreaelaboratemotivationofthenotionofRTM.)
The aim of this paper is to make a connection between the theory of interactive
computabililtyandthetheoryofreactivesystems,providingacomparisonofthemod-
els of ITMs and RTMs in both their semantic domains. We shall first, in Section 2,
recapitulatebothmodels.Then,inSection3wepresentatransition-systemsemantics
forITMs; the transitionsystem associated with an ITMis executableupto a fine no-
tionofbehaviouralequivalence.InSection4weshallidentifyasubclassofRTMsthat
canbeconsideredsuitableforstreamtranslation,andprovethatthestreamtranslation
associated with an RTM in this subclass is interactively computable.In Section 5 we
consider an extension of RTMs with an advice mechanism adapted from the advice
mechanismconsideredforITMs.RTMswithadvicecanexecuteeverycountabletran-
sitionsystem,atthecostofintroducingdivergenceinthecomputation.Thepaperends
withaconclusioninSection6.
2 Preliminaries
2.1 TheTheoryofInteractiveComputation
In[11],vanLeeuwenandWiedermannpresentananalysisofinteractivecomputation
onthebasisofacomponentC(thoughttobehaveaccordingtoadeterministicprogram)
interactingwith an unpredictableenvironment E. Theydiscuss the consequencesof a
fewgeneralpostulatespertainingtothebehaviourandinteractionofC andE forinter-
active recognition,interactive generationand interactive translation. In their analysis,
the componentC acts asa stream transducer,transformingan infinite inputstream of
data symbols from Σ = {0,1} presented by E at its input port into an infinite output
streamofsymbolsfromΣ producedatitsoutputport.Henceforth,byanω-translation
we mean a mappingφ : Σω → Σω (with Σω denotingthe set of streams, i.e., infinite
sequences,overΣ).
Interactivecomputationis a step-wise process. It is not requiredthat the environ-
mentoffersasymbolineverystep,northatthecomponentproducesasymbolinevery
step.Forthepurposeofmodellingcomponents,however,itisconvenienttorecordthat
nothingis offered or produced.The symbolλ is used to indicate the situation that no
symbolisofferedattheinputportorproducedattheoutputport,andweletΣ =Σ∪{λ}.
λ
Itisassumedthatwhen E offersanon-λsymbolinsomestep, thenthecomponentC
producesanon-λsymbolatits outputportwithinfinitelymanysteps, andviceversa;
this assumption is referred to as the interactiveness (or finite delay) condition in the
workofvanLeeuwenandWiedermann.
Inordertoformallydefinewhichω-translationsareinteractivelycomputablebya
computationaldevice,vanLeeuwenandWiedermannproposedthenotionofinteractive
Turingmachine[9,10].ItextendstheclassicalnotionofTuringmachinewithaninput
portandanoutputport,throughwhichitexchangesaninfinite,neverendingstreamof
datasymbolswithitsenvironment.InteractiveTuringmachinesuseatwo-wayinfinite
tape as memory on which they can write symbols from some presupposedset D(cid:3) of
tape symbols, not necessarily disjoint from Σ and including the special (cid:3) symbol to
denote an empty tape cell. Our formal definition below is adapted from [14] (but we
leaveoutthedistinctionbetweeninternalandexternalstates).
Definition1. A (deterministic) interactive Turing machine (ITM) with a single work
tapeisatripleI=(Q,−→ ,q ),where
I in
1. Qisitssetof states;
2. −→I: Q×D(cid:3)×Σλ →Q×D(cid:3)×{L,R}×Σλ isatransitionfunction;and
3. q ∈ Qisitsinitialstate.
in
ThecontentsofthetapeofanITMmayberepresentedbyanelementof(D(cid:3))∗.We
denotebyDˇ(cid:3) = {dˇ| d ∈ D(cid:3)}thesetofmarkedsymbols;atapeinstanceisasequence
δ ∈ (D(cid:3)∪Dˇ(cid:3))∗ suchthatδcontainsexactlyoneelementofDˇ(cid:3).Themarkerindicates
thepositionofthetapehead.
AcomputationofanITMI=(Q,−→ ,q )isaninfinitesequenceoftransitions
I in
(q ,(cid:3)ˇ)=(q ,δ )−i0/→o0 (q ,δ )−i1→/o1 ···(q ,δ )−ik→/ok ··· . (1)
in 0 0 I 1 1 I k k I
Theinputstream associatedwiththecomputationin(1)isobtainedfromi ,i ,... by
0 1
omittingalloccurrencesofλ,andtheoutputstreamassociatedwiththecomputationin
(1)isobtainedfromo ,o ,... byomittingalloccurrencesofλ.Apair(x,y)∈Σω×Σω
0 1
isaninteractionpairassociatedwithIifthereexistsacomputationofIwithxasinput
streamandyasoutputstream.ThesetofallinteractionpairsassociatedwithanITMI
iscalleditsinteractivebehaviour.(InSection3weshallpresentamorerefinedviewon
itsbehaviourwhenweassociatewitheveryITMatransitionsystem.)Thecomputation
in(1)isinteractiveif,forallk ∈ N,ifi , λ,thenthereexistsℓ ≥ k suchthato , λ.
k ℓ
Thecomputationin(1)isinput-activeifi ,λforallk∈N.
k
An ITM satisfies the interactiveness condition if all its computations are interac-
tive. Clearly, if a deterministic ITM I satisfies the interactiveness condition, then its
interactive behaviour is total, in the sense that for every x ∈ Σω there is at least one
y ∈ Σω such that (x,y) is an interaction pair of I. By confining our attention to the
input-activecomputations—which,intheterminologyof[11],correspondstoadopting
thefullenvironmentalactivitypostulate—,wemaythenassociatewitheverysuchITM
an ω-translation:we say that ITM I produces y on input x if (x,y) is the interaction
pairassociatedwithaninput-activecomputationofI.
Definition2. Anω-translationφ:Σω →Σω isinteractivelycomputableifthereexists
adeterministicITMthatsatisfyingtheinteractivenessconditionthatproducesφ(x)on
inputxforallx∈Σω.
Van Leeuwen and Wiedermann present in [11] a characterisation of the interac-
tively computable ω-translations by showing that they can be approximated by clas-
sically computablepartialfunctionson finite sequencesoverΣ. For finite and infinite
sequencesxandy,wewritex≺ yifxisafiniteandstrictprefixofy,andx(cid:22) yifx≺ y
or x = y.Weusethefollowingdefinitionofmonotonicfunctionsandlimit-continuous
functions.
Definition3. 1. Apartialfunction f :Σ∗ ⇀Σ∗ismonotonicifforallx,y∈Σ∗ such
thatx≺ yand f(y)isdefined,itholdsthat f(x)isdefinedaswelland f(x)(cid:22) f(y).
2. Apartialfunctionφ : Σω → Σω iscalled limit-continuousifthereexistsaclassi-
callycomputablemonotonicpartialfunction f :Σ∗ →Σ∗suchthatφ(lim x )=
k→∞ k
lim f(x ) for all strictly increasing chains x ≺ x ≺ ··· ≺ x ≺ ··· with
k→∞ k 1 2 k
x ∈Σ∗.
k
In [11] a criterion of the interactively computable ω-translations is presented by
usinglimit-continuousfunctions.
Theorem1. Atotalω-translationisinteractivelycomputableiffitislimit-continuous.
2.2 TheTheoryofExecutability
Thetheoryofexecutabilitycombinescomputationandconcurrency-styleinteractionin
suchawaythatbotharetreatedonequalfooting;thus,anintegrationofcomputability
andconcurrencytheoryisrealised.
Thetransition system is the centralnotionin the mathematicaltheoryof discrete-
event behaviour. It is parameterised by a set A of action symbols, denoting the ob-
servableeventsofa system. We extendAwith a specialsymbolτ, whichintuitively
denotesunobservableinternalactivity.WeshallabbreviateA∪{τ}byA .
τ
Definition4. AnA -labelledtransitionsystemT isatriple(S,−→,↑),where,
τ
1. Sisasetof states,
2. −→⊆S×A ×SisanA -labelledtransitionrelation,
τ τ
3. ↑∈Sistheinitialstate.
Transition systems can be used to give semantics to programminglanguages and
processcalculi.Thestandardmethodistofirstassociatewitheveryprogramorprocess
expressionatransitionsystem(itsoperationalsemantics),andthenconsiderprograms
andprocessexpressionsmodulooneofthemanybehaviouralequivalencesontransition
systems thathave been studiedin the literature. In this paper,we shall use the notion
of(divergence-preserving)branchingbisimilarity[4,5],whichisthefinestbehavioural
equivalenceinvanGlabbeek’slineartime-branchingtimespectrum[6]thatabstracts
frominternalcomputationsteps(representedinthetransitionsystembytransitionsla-
belledwithτ)..
Inthedefinitionof(divergence-preserving)branchingbisimilarityweneedthefol-
lowingnotation:let−→beanA -labelledtransitionrelationonasetS,andleta∈A ;
τ τ
(a) a
we write s −→ t for “s −→ t” or “a = τ and s = t”. Furthermore, we denote the
transitiveclosureof−→τ by−→+ andthereflexive-transitiveclosureof−→τ by−→∗.
Definition5 (Branching Bisimilarity). Let T = (S ,−→ ,↑ ) and T = (S ,−→
1 1 1 1 2 2 2
,↑ )betransitionsystems.AbranchingbisimulationfromT toT isabinaryrelation
2 1 2
R⊆S ×S suchthatforallstatess ands , s Rs implies
1 2 1 2 1 2
1. ifs −→a s′,thenthereexists′,s′′ ∈S ,s.t.s −→∗ s′′ −(→a) s′,s Rs′′ands′Rs′;
1 1 1 2 2 2 2 2 2 2 1 2 1 2
2. ifs −→a s′,thenthereexists′,s′′ ∈S ,s.t.s −→∗ s′′ −(→a) s′,s′′Rs ands′Rs′.
2 2 2 1 1 1 1 1 1 1 1 2 1 2
ThetransitionsystemsT andT arebranchingbisimilar(notation:T ↔ T )ifthere
1 2 1 b 2
existsabranchingbisimulationRfromT toT s.t.↑ R↑ .
1 2 1 2
AbranchingbisimulationRfromT toT isdivergence-preservingif,forallstates
1 2
s ands , s Rs implies
1 2 1 2
τ
3. ifthereexistsaninfinitesequence(s1,i)i∈N s.t. s1 = s1,0, s1,i −→ s1,i+1 and s1,iRs2
foralli∈N,thenthereexistsastate s′ s.t. s −→+ s′ ands Rs′ forsomei∈N;
2 2 2 1,i 2
and
τ
4. ifthereexistsaninfinitesequence(s2,i)i∈N s.t. s2 = s2,0, s2,i −→ s2,i+1 and s1Rs2,i
foralli∈N,thenthereexistsastate s′ s.t. s −→+ s′ ands′Rs forsomei∈N.
1 1 1 1 2,i
ThetransitionsystemsT andT aredivergence-preservingbranchingbisimilar(nota-
1 2
tion:T ↔∆ T )ifthereexistsadivergence-preservingbranchingbisimulationRfrom
1 b 2
T toT s.t.↑ R↑ .
1 2 1 2
ThenotionofreactiveTuringmachine(RTM)wasputforwardin[3]tomathemat-
icallycharacterisewhichbehaviourisexecutablebyaconventionalcomputingsystem.
WerecallthedefinitionofRTMsandtheensuednotionofexecutabletransitionsystem.
Definition6. AreactiveTuringmachine(RTM)Misatriple(S,−→,↑),where
1. Sisafinitesetof states,
2. −→⊆S×D(cid:3)×Aτ×D(cid:3)×{L,R}×Sisa(D(cid:3)×Aτ×D(cid:3)×{L,R})-labelledtransition
a[d/e]M
relation(wewrite s −→ tfor(s,d,a,e,M,t)∈−→),
3. ↑∈Sisadistinguishedinitialstate.
a[d/e]M
Intuitively,the meaningofatransition s −→ t isthatwheneverM isin state s,
andd isthesymbolcurrentlyreadbythe tapehead,thenitmayexecutethe actiona,
writesymboleonthetape(replacingd),movetheread/writeheadonepositiontothe
leftortherightonthetape,andthenendupinstatet.
ToformalisetheintuitiveunderstandingoftheoperationalbehaviourofRTMs,we
associate with every RTM M an A -labelled transition system T(M). The states of
τ
T(M)aretheconfigurationsofM,pairsconsistingofastateandatapeinstance.
Definition7. LetM= (S,−→,↑)beanRTM.ThetransitionsystemT(M)associated
withMisdefinedasfollows:
1. itssetofstatesSconsistsofthesetofallconfigurationsofM;
2. itstransitionrelation−→istheleastrelationsatisfying,foralla ∈ Aτ, d,e ∈ D(cid:3)
andδ ,δ ∈D∗:
L R (cid:3)
– (s,δ dˇδ )−→a (t,δ<eδ )iffsa[−d→/e]Lt,and
L R L R
– (s,δ dˇδ )−→a (t,δ e>δ )iffsa[−d→/e]Rt
L R L R
(δ< isobtainedfromδ byplacingthetapeheadmarkerontheright-mostsymbol
L L
inδ ,and>δ isobtainedanalogouslyfromδ );
L R R
3. itsinitialstateistheconfiguration(↑,(cid:3)ˇ).
Turingintroducedhismachinestodefinethenotionofeffectivelycomputablefunc-
tionin[13].Byanalogy,wehaveanotionofeffectivelyexecutablebehaviour[3].
Definition8. Atransitionsystem isexecutableifitisthetransitionsystemassociated
withsomeRTM.
3 Executability ofInteractive Turing Machines
InthissectionweassociateatransitionsystemwitheveryITM,andthenprovethat
it is executablemodulo divergence-preservingbranchingbisimilarity. It is convenient
toconsiderinputandoutputasseparateactionsinthetransitionsystemassociatedwith
anITM.Wedenoteby?itheactionofinputtingthesymboli∈ Σ,andby!otheaction
ofoutputtingthesymbolo∈Σ.
Definition9. LetI=(Q,−→ ,q )beanITM.ThetransitionsystemT(I)associated
I in
withIisdefinedasfollows:
1. itssetofstatesistheset{(s,δ)| s∈ Q∪{s |o∈Σ ,s∈ Q}, δisatapeinstance};
o λ
2. itstransitionrelation−→istheleastrelationsatisfying,foralli,o∈Σλ,d,e∈D(cid:3),
andδ ,δ ∈D∗:
L R (cid:3)
– (s,δ dˇδ )−?→i (t ,δ<eδ )iff(s,d,i)−→ (t,e,L,o)andi∈Σ,
L R o L R I
– (s,δ dˇδ )−?→i (t ,δ e>δ )iff(s,d,i)−→ (t,e,R,o)andi∈Σ,
L R o L R I
– (s,δ dˇδ )−→τ (t ,δ<eδ )iff(s,d,i)−→ (t,e,L,o)andi=λ,
L R o L R I
– (s,δ dˇδ )−→τ (t ,δ e>δ )iff(s,d,i)−→ (t,e,R,o)andi=λ,
L R o L R I
!o τ
– (s ,δ)−→(s,δ)iffo∈Σ,and(s ,δ)−→(s,δ)iffo=λ.
o o
3. itsinitialstateistheconfiguration(q ,(cid:3)ˇ).
in
ThefollowingtheoremshowsthateverytransitionsystemsassociatedwithanITM
can be simulated by an RTM. In the proof it is convenient to allow RTMs to have
a[d/e]S
transitionsoftheforms −→ t,whereS isastaytransitionwithnomovementofthe
tape head. We refer to such machinesas RTMs with stay transitions. The operational
semantics of RTMs can be extended to an operationalsemantics for RTMs with stay
transitionsby addingthe clause:(s,δ dˇδ ) −→a (t,δ eˇδ ) iff s a[−d→/e]S t. Thetransition
L R L R
systemofanRTMwithstaytransitionscanbesimulatedbyanRTMuptodivergence-
preservingbranchingbisimilarity.
Lemma1. ThetransitionsystemassociatedwithanRTMwithstaytransitionsisexe-
cutableuptodivergence-preservingbranchingbisimilarity.
Proof. WesupposethatM = (S,−→,↑)isanRTMwithstaytransitions,anditstran-
sition system is T(M). We define a normal RTM M′ = (S ,−→ ,↑ ) that simulates
1 1 1
T(M)asfollows:
1. S =S∪{s | s,t∈S};
1 t
a[d/e]L a[d/e]L
2. s −→ tiff s −→ t;
1
a[d/e]R a[d/e]R
3. s −→ tiffs −→ t;
1
a[d/e]L τ[d/d]R a[d/e]S
4. s −→ s ands −→ tiff s −→ t;and
1 t t 1
5. ↑ =↑.
1
ThenitisstraightforwardtoT(M′) ↔∆ T(M).
b
Theorem2. ForeveryITMIthereexistsanRTMM,suchthatT(I) ↔∆ T(M).
b
We let I = (Q,−→ ,q ) be an ITM. By Lemma 1, it is enough to show that there
I in
existsanRTMwithstaytransitionsMsatisfyingT(M) ↔∆ T(I).WeconstructM=
b
(S,−→,↑)asfollows:
1. S=I∪O,whereI =QandO={s |o∈Σ ,s∈Q};
o λ
in(i)[d/e]M
2. the transition relation −→ is defined by: s −→ t if (s,d,i) −→ (t,e,M,o),
o I
out(o)[e/e]S
ands −→ sforalls∈S,o∈Σ ;and
o λ
3. ↑=q .
in
ThenaccordingtoDefinitions7and9,wegetatransitionsystemT(M)=T(I),where
‘=’isthepointwiseequality,whichalsoimpliesT(M) ↔∆ T(I). Asaconsequence
b
wehavethefollowingcorollary.
Corollary1. ThetransitionsystemassociatedwithanITMisexecutablemodulodivergence-
preservingbranchingbisimilarity.
4 Executable ω-Translations
Recallthatanω-translationisdefinedtobeinteractivelycomputableif,andonlyif,it
canberealisedbyanITM.RTMsaredesignedforexhibitingtheexpressivepowerof
executabletransitionsystems,ratherthanω-translations,andnoteveryRTMnaturally
hasanω-translationassociatedwithit.Imposingsomerestrictionsontheformalismof
RTMs,however,weshalldefineasubclassofRTMswithwhichanω-translationisnat-
urallyassociated.Theω-translationrealisedbysuchanRTMisthencalledexecutable,
andweshallestablishthatanω-translationisinteractivelycomputableif,andonlyif,
itisexecutable.
By analogy to the systems described in the theory of interactive computation,we
let the RTMsfor ω-translationsexecutein steps, in such a way thatwith everystep a
pairof inputandoutputactionscanbe associated.With everyinfinite computationof
theRTMwecanthenassociateainteractionpair,andtheRTMwillthusgiverisetoan
ω-translation.
Definition10. Let A = {?i,!o | i,o ∈ {0,1}}∪{τ}, and let M = (S,−→,↑) be an
τ
RTMwithA asitssetoflabels.ThenMisanRTMforω-translationifitsatisfiesthe
τ
followingproperties:
1. thesetofstatesSispartitionedintodisjointsets∈ofinputstatesandEofexecution
states,i.e.,S=I∪EandI∩E=∅;
2. theinitialstate↑isaninputstate,i.e.,↑∈I;
a[d/e]M
3. for a transition s −→ t, if s ∈ I, then a ∈ {?0,?1} and t ∈ E; if s ∈ E, then
a∈{!0,!1,τ}andt∈I;and
a[d/e]M
4. forall(s,d)∈E×D(cid:3),thereisatmostonetransitionoftheform s −→ t;and
a[d/e]M
5. forall(s,d) ∈ I×D(cid:3),thereareexactlytwotransitionsoftheform s −→ t,one
witha=?0andonewitha=?1.
Inthefollowinglemmaweestablishsomepropertiesofthetransitionsystemasso-
ciatedwithanRTMforω-translation.
Lemma2. LetMbeanRTMforω-translation.ThenT(M)=(S ,−→ ,↑ )satis-
M M M
fiesthefollowingproperties:
1. (Alternation)ThesetofstatesS ispartitionedintoasetofinputstatesI anda
M M
setofoutputstatesE ,i.e.,S =I ∪E andI ∩E =∅.Foreverytransition
M M M M M M
s−→a s′,ifs∈I ,thena∈{?0,?1}ands′ ∈E ;ifs∈E ,thena∈{!0,!1,τ}and
M M M
s′ ∈I .
M
2. (Unambiguity)Forevery s ∈ E ,thereisexactlyoneoutgoingtransition s−→a s′
M
witha∈{!0,!1,τ}.
3. (Totality) For every s ∈ I , there are exactly two outgoing transitions, labelled
M
with?0and?1,respectively.
Proof. AstateinS isaconfiguration(s,δ)ofM,andwecanmakeapartitionofthe
M
set of all configurationsaccording to the control states. If s ∈ I, then (s,δ) ∈ I ; if
M
s∈E,then(s,δ)∈E ,whereI andEaredefinedinDefinition10.
M
1. (Alternation)By condition1in Definition10,we haveS = I ∪E andI ∩E = ∅,
which infersS = I ∪E and I ∩E = ∅;moreover,by condition2, for a
M M M M M
a[d/e]M
transitions −→ t,ifs∈I,thena∈{?0,?1}andt∈E;ifs∈E,thena∈{!0,!1,τ}
andt∈I,whichinfersthatforeverytransitions−→a s′,ifs∈I ,thena∈{?0,?1}
M
ands′ ∈E ;ifs∈E ,thena∈{!0,!1,τ}ands′ ∈I .
M M M
2. (Unambiguity) By condition 3 in Definition 10, for all (s,d) where s ∈ E and
o[d/e]M
d ∈ D(cid:3), there is at most one transition s −→ t, which infers that for every
s∈E ,thereisexactlyoneoutgoingtransitions−→a s′ witha∈{!0,!1,τ}.
M
3. (Totality) By condition4 in Definition 10, forall (s,d) where s ∈ I and d ∈ D(cid:3),
i[d/e]M
thereareexactlytwotransitionsoftheforms −→ t,with?0and?1astherelabels,
respectively,whichinfersthatforevery s ∈ I ,therearetwooutgoingtransitions
M
labelledby?0and?1,respectively.
We call a transition that satisfies the conditionsof Lemma 2 an i/o transition sys-
tem. Moreover, by analogy to the interactiveness condition for ITMs, we impose an
interactivenessconditiononRTMsforω-translation.
?i
Definition11. An i/o transitionsystem is interactive, iffor every s ∈ S and s −→ s
0
withi∈{0,1},andforeverysequences −→ s −→···,thereexistsanaturalnumber
0 1
!o
i,suchthats −→ s witho∈{0,1}.
i i+1
AnRTMforω-translationisinteractiveiftheassociatedi/otransitionsystemis.
We define the ω-translation realized by an RTM by defining the ω-translation re-
alized by the i/o transition system associated with it. Let T = (S,−→,↑) be an i/o
σ
transitionsystem,let s∈S,andletσ∈Aω,sayσ=a ,a ,...;wewrites−→ifthere
0 1
exists ,s′,s ,s′,...∈Ssuchthats= s ,ands −→∗ s′ −a→i s foralli≥0.(By−→∗
0 0 1 1 0 i i i+1
wedenotethereflexive-transitiveclosureoftherelation−→τ .)Ifσ∈Aωands−σ→,then
σ is a weak infinite trace from s. We denoteby Tr∞(s) the set of weak infinite traces
w
froms.
Definition12. Let T be an i/o transition system, and s be the initial state. For σ ∈
0
Tr∞(s ), the input stream realised by σ is the stream x ∈ Σω such that x = x x ...,
w 0 1 2
where x = i if ?i is the j-th input action in σ, and similarly for the output stream
j
realizedbyσ.WesaythatT realizesω-translationφ:Σω →Σω iff,forevery x∈Σω,
thereexistsatraceσ∈Tr∞(s )with xasitsinputstream,andforeverysuchtrace,its
w 0
outputstreamisy=φ(x).
Wecannowdefinewhenanω-translationisexecutable.
Definition13. Anω-translationisexecutableifitcanberealizedbyanexecutablei/o
transitionsystem.
Thefollowinglemmaestablishesthatanω-translationcanbeassociatedwithevery
interactivei/otransitionsystem.
Lemma3. Ifani/otransitionsystemisinteractive,thenitrealisesanω-translation.
Proof. Let T be an i/o interactive transition system, and let s be the initial state of
0
T.ByDefinition12,weneedtoshowthatthereexistsanω-translationφsuchthatfor
every x ∈ Σω,thereexistsatraceσ ∈ Tr∞(s )withinputstream x,andforeverytrace
w 0
withinputstreamx,itsoutputstreamis y=φ(x).
BythealternationconditioninLemma2,everyσ∈Tr∞(s )isoftheformi o i o ...
w 0 0 0 1 1
wherei ∈{?0,?1}ando ∈{!0,!1,τ}.Letxbeanarbitraryinputstream,bythetotality
j j
conditioninLemma2,wecanfindatraceσ∈Tr∞(s )withinputstream x.
w 0
Moreover, given an trace σ with an infinite input stream x, by interactiveness, it
wouldalwaysproduceaninfiniteoutputstream y.
Finally,byunambiguity,theredonotexisttwotracessharingthesameinputstream.
Itfollowsthatforeverytracewithinputstreamx,itsoutputstreamisy.Hence,werelate
witheveryinputstreamauniqueoutputstream,inaway,wegetaω-translationfrom
T.
Itisnothardtoshowthefollowinglemmas,
Lemma4. Let T and T be two i/o transition systems, and T ↔ T . Then they
1 2 1 b 2
realizethesameω-translation.
Proof. We let s and s betheinitialstatesofT andT ,respectively.AsT ↔ T ,
1 2 1 2 1 b 2
we havethatfor everyσ ∈ Tr∞(s ), there existsa trace σ′ ∈ Tr∞(s ), andthey share
w 1 w 2
thesameinputandoutputstream,andviceversa.ItfollowsthatT andT realizethe
1 2
sameω-translation.
Lemma5. LetT beaninteractivei/otransitionsystem,andlet s beitsinitialstate,
0
thenthefollowingfunctioniscomputable:g:Σ∗ →Σ∗,satisfyingthatifg(x)=y,then
foreveryσ∈Tr∞(s )withinputandoutputstreamxandy,ifx≺ x,theny≺ y.
w 0
Proof. We considerafinitetracefrom s ,wecanassociatewithsucha traceitsinput
0
andoutputsequencesinasimilarwayasdefinedinDefinition12.ByLemma2,there
is only one finite trace with x as its input sequence, and its output sequence is y. By
totality,itholdsforeveryx∈Σ∗.Asthetransitionrelationofi/otransitionsystemsare
computable,gisalsocomputable.
Moreover,wehavethefollowingtheorem.
Theorem3. Anω-translationisanexecutableiffitisalimit-continuoustotalfunction.
Proof. Weletφbeanω-translation.
1. Forthe“onlyif”part,weneedtoshowthatthereexistsacomputabletotalfunction
g : Σ∗ → Σ∗, suchthatg is monotonicand forallstrictly increasingchainsu ≺
1
u ≺...≺u ≺...withu ∈Σ∗ (t≥1),onehasφ(lim u)=lim g(u).
2 t t t→∞ t t→∞ t
We assume thatφisrealizedbyaninteractivei/otransitionsystemT, andwe let
s be the initial state of T. By Lemma 5 the following function is computable:
0
g :Σ∗ →Σ∗,satisfyingthatifg(x)= y,thenforeveryσ ∈ Tr∞(s )withinputand
w 0
output stream x and y, if x ≺ x, then y ≺ y. By unambiguityand totality, g is a
monotonicandtotalcomputablefunction.
Moreover,forastrictlyincreasingchainu ≺ u ≺ ... ≺ u ≺ ...withu ∈ Σ∗ for
1 2 t t
t ≥ 1, the computationoflim g(u) is the executionof a trace σ receivingthe
t→∞ t
inputstreamlim u.Hencewehaveφ(lim u)=lim g(u).
t→∞ t t→∞ t t→∞ t
Thus, g is the computabletotal functionwe need, and it followsthat φ is a com-
putablelimit-continuoustotalfunction.
2. Forthe“if”part,weassumethatφisatotallimit-continuousfunction,anddesign
anRTMMtorealizethistranslation.ByTheorem1,φisinteractivelycomputable
bysomeITMM′.AccordingtoDefinition9andLemma2, thetransitionsystem
associatedwithM′isani/otransitionsystem,moreover,accordingtoCorollary1,
it is an executable i/o transition system. Therefore, we have shown that φ is an
executableω-translationbyLemma4.