A Timed Calculus for Mobile Ad Hoc Networks

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by  Mengying Wang

A Timed Calculus for Mobile Ad Hoc Networks
MengyingWang YangLu
SoftwareEngineeringInstitute DepartmentofComputerScience
EastChinaNormalUniversity ShanghaiJiaotongUniversity
Shanghai,China Shanghai,China
[email protected] [email protected]
We develop a timed calculus for Mobile Ad Hoc Networks embodying the peculiarities of local
broadcast,nodemobilityandcommunicationinterference. WepresentaReductionSemanticsanda
LabelledTransitionSemanticsandprovetheequivalencebetweenthem. Wethenapplyourcalculus
tomodelandstudysomeMAC-layerprotocolswithspecialemphasisonnodemobilityandcommu-
nicationinterference.
Amainpurposeofthesemanticsistodescribethevariousformsofinterferencewhilenodeschange
theirlocationsinthenetwork. Suchinterferenceonlyoccurswhenanodeissimultaneouslyreached
bymorethanoneongoingtransmissionoverthesamechannel.
1 Introduction
Mobileadhocnetworks(MANETs)arecomplexdistributedsystemsthatconsistofacollectionof
wirelessmobilenodesthatcandynamicallyself-organizeintoarbitrarynetworktopologies,soastoallow
peopleanddevicestoseamlesslyinterworkinareaswithoutpre-existingcommunicationinfrastructures
[1]. Owing to the flexibility and convenience, their applications have been extended from traditional
military domain to a variety of commercial areas, e.g., ambient intelligence [2], personal area networks
[4]andlocation-basedservices[3].
Wireless nodes use radio frequency channels to broadcast messages. Compared to the conventional
wired-based broadcasts like Ethernet networks, this form of broadcast has some special features. First,
broadcasting is local, i.e., a transmission covers only a limited area, called a cell, and hence reaches a
(possiblyempty)subsetofthenodesinthenetwork. Second,channelsarehalf-duplex: onagivenchan-
nel,anodecaneithertransmitorreceive,butcannotdobothsimultaneously. Asaresult,communication
interference can only be detected at the destination. Further, nodes in MANETs can move arbitrarily,
which makes the network easily suffer from interference. Since interference plays an important role in
evaluating the performance of a network, it becomes a delicate aspect of MANETs that is handled by a
greatquantityofprotocols(e.g.,MACA/R-T[5]).
Over the last two decades, a number of process calculi have been proposed to model MANETs
[6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. These calculi can be divided into two categories according to their
attentionstothenetwork. ThefirstgroupcontainsCBS#[6],CMAN[7,10],RBPT[8],CNT[9],CMN
[11], ω-calculus [13] and CSDT [15]. They attempt to depict local broadcast and node mobility. Take
CMNasanexample, eachnodeisequippedwithalocationandaradiusthatdefinethecelloverwhich
the node can transmit. When a sender broadcasts messages, only nodes that are within its transmission
cell could receive. Furthermore, nodes are marked mobile or stationary, and mobile nodes can change
their locations randomly. Then CWS [12] and TCWS [14] constitute the second group. They focus on
local broadcast and communication interference. The former abstracts the transmission into two state
change events: begin transmission and end transmission, while the latter regards the transmission as a
(cid:13)c M.Wang&Y.Lu
C.ArthoandP.C.O¨lveczky(Eds.):FTSCS2012
Thisworkislicensedunderthe
EPTCS105,2012,pp.118–134,doi:10.4204/EPTCS.105.9
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M.Wang&Y.Lu 119
time consuming procedure. To our knowledge, no calculus has integrated all of the three peculiarities,
especiallyincludingnodemobilityandcommunicationinterference.
Inthispaper,wepresentatimedcalculusformobileadhocnetworks(TCMN),whichextendsCWS
[12] and deals with all of the three issues. A central concern of our calculus is to describe the forms of
interferencewhilenodesmovetheirlocationsinthenetwork. Towardslocalbroadcast, wewriten[Q]c
l,r
to stand for a node identified by n, located at l, executing process Q, and which can transmit data over
channel c in a cell centered at l with radius r. As for node mobility, measures vary according to the
specificsituation. Forinstance,nodesthatpresentlyparticipateinnotransmissioncouldmovearbitrarily
without any impact on the environment. However, the movement of an active transmitter may affect
the receptions of active receivers: some may get an error or interference, since they passively leave or
enterthetransmitter’stransmissioncell. Finally,withregardtocommunicationinterference,weassume
all wireless nodes have been synchronized by some clock synchronization protocol [16, 17]. Then we
consider a transmission proceeds in discrete steps which are represented by occurrences of a simple
action σ to denote passing of one time unit. And if a receiver is exposed to more than one ongoing
transmissionoverthesamechannel,itdetectsaninterference.
Inconcurrencytheory,LabelledTransitionSemantics(LTS)isthemostpopularwayofgivingopera-
tionalsemanticssincethetransitionsofaLTSexposethefullbehaviorofthesystem(itsinternalactivities
aswellastheinteractionswiththeenvironment)whichisrequiredfordefiningbehavioralequivalences
and providing powerful proof techniques. However, sometimes the rules of a LTS may be difficult to
understandparticularlywhenthecalculirelatestonodemobilitylike[7,10,11]. Hence,adifferentform
ofoperationalsemantics,namedReductionSemantics(RS),isintroduced. RSonlyconcernstheinternal
activitiesofasystem,soitiseasiertograsp. Besides,RScanbeusedtocheckthecorrectnessofaLTS,
by proving consistency with the LTS. For these reasons, we define both RS and LTS semantics for our
TCMNandprovethattheycoincide.
We end this section with an outline of the paper. In Section 2, we define the syntax of our core
language. TheninSection3,weprovideaRSforourcalculuswhichspecifieshowanunboundednumber
ofsystemcomponentscanbeinvolvedinanatomicinteraction. NextaLTSthatcapturesallthepossible
interactions of a term with its environment is proposed in Section 4. The equivalence between the RS
andtheLTSsemanticsisprovedinSection5. InSection6and7,weextendourcorelanguagebyadding
some newoperators to modelsome MAC-layer collision avoidance protocols: CSMAand MACA/R-T.
WeprovethattheCSMAprotocoldoesn’tsolvetheissueofnodemobilitywhiletheMACA/R-Tprotocol
is robust against node mobility. Finally, in section 8, we summarize our contributions and present the
futurework.
2 The Core Language
In Table 1, we present the core of TCMN. The syntax is defined in a two-level structure: a lower
oneforprocesseswhichdescribesthepossiblestatusofanode,andanupperonefornetworks. Foreasy
understanding, in this section we only focus on those operators that are necessary for communication
whiletheextendedlanguagewillbepresentedinSection6.
Generally,weuselettersa...cforchannels,m...oforidentifiers,x...zforvariables,uforvaluesthat
canbetransmittedoverchannels: theseincludevariablesandclosedvalues,andvforclosedvalues,i.e.
values that contain no variables. u is a unary function designed to estimate the number of time units
(cid:72) (cid:73)
requiredfor thetransmissionof thevalueu. Since onlyclosedvalues willbeused intransmissions, we
assume the existence of an evaluation function [[[.]]] to return the closed form of a value. Finally, we do
not set how locations should be specified, the only assumption is that they should be comparable, so to
120 ATimedCalculusforMobileAdHocNetworks
Table 1. The Syntax
Networks:
Nd=ef 0 empty network | n[Q]c node
l,r
| N|N parallel composition
Processes:
def
Q = P non-active process | A active process
def
P = 0 termination | out(cid:104)u(cid:105).P output
| in(x).P input
Ad=ef (cid:104)v(cid:105)δ.P active output | (x)δ.P active input
v
Values:
def
u = x variable | v closed value
Functions:
def
f = u time function | [[[u]]] evaluation function
(cid:72) (cid:73)
| d(l ,l ) distance function
1 2
where δ is a positive integer greater than 0
determinewhetheranodeisinoroutofthetransmissioncellofanothernode. Wedosobyintroducing
afunctiondwhichtakestwolocationsasparametersandreturnsthedistancebetweenthem.
Networks are collections of nodes (which actually represent devices) that run in parallel and use
same channels to communicate with each other. We use the symbol 0 to stand for the empty network,
and n[Q]c to denote a node identified by n, located at l, executing process Q, and which can transmit
l,r
dataoverchannelcinacellcenteredatlwithradiusr. WewriteN|Ntoindicateaparallelcomposition
oftwosub-networksN.
Processes, living within the nodes, are sequential. For convenience, we divide processes into two
categories: non-activeandactive. Anactiveprocessisaprocessthatiscurrentlytransmittingorreceiving
data,e.g.,anactiveoutputprocess(cid:104)v(cid:105)δ.Pdenotesatransmittingprocess,anditstransmissionofvaluev
willcompleteafterδ timeunits. Similarly,anactiveinputprocess(x)δ.Prepresentsareceivingprocess,
v
anditsreceptionofvaluevwilllastforthenextδ instantsoftime. Inthenon-activeprocessconstructs,
thesymbol0standsforaterminatedprocess. out(cid:104)u(cid:105).Pisanoutputprocesswillingtobroadcastthevalue
v=[[[u]]], and once the transmission starts, the process evolves into the active output process (cid:104)v(cid:105)δ.P,
where δ = u is the time necessary to transmit the value v. in(x).P indicates an input process willing
(cid:72) (cid:73)
to receive data, and when the beginning of a transmission v in the following δ time units is captured
clearly(i.e. withoutinterference), theprocessbecomestheactiveinputprocess(x)δ.P. Anodewithan
v
activeoutputprocessinsideisnamedactivetransmitter. Similarly,activeinputprocessesandnon-active
processesareincludedseparatelyinactivereceiversandnon-activenodes.
Weassumethateachnodehasauniqueidentifier,anddifferentnodescannotbelocatedatthesame
position at the same time. We consider such networks well-formed. Since nodes cannot be created or
destroyed,thewell-formednessofanetworkisalwayspreservedasthenetworkevolves. Intheremainder
of the paper, all networks are well formed, and we use a number of notational conventions. Process Q
standsforeitheranon-activeoranactiveprocesswhilePandArepresentnon-activeandactiveprocesses
separately. We identify (cid:104)v(cid:105)δ.P=P and (x)δ.P=P{v/x} if δ =0. We write out(cid:104)u(cid:105) for out(cid:104)u(cid:105).0, and
v
(cid:104)v(cid:105)δ for(cid:104)v(cid:105)δ.0.
M.Wang&Y.Lu 121
3 Reduction Semantics
In this section, we study the reduction semantics (RS) for TCMN. In the literature [12], the only
internalactivityisabroadcastwhichismodelledbytwoevents: begintransmissioneventandendtrans-
missionevent. Yetinoursystem,anewtypeofinternalactivity: amigrationisappendedtodepictnode
movement. Inourmodel,thebroadcastwillbedescribedbyabegintransmissioneventandseveraltime
passingevents(asshowninTable2),whilethemigrationwillberepresentedbyanodemovementevent
from a specific node (as shown in Table 3). Among these three types of events, the begin transmission
event (i.e. a node initiates a transmission) has the same meaning as that in [12], while the time passing
event (i.e.,aunitoftimedelays)isimportedtoreplacetheendtransmissioneventin[12],andthenode
movementevent(i.e.,anodemovesfromonelocationtoanother)isanewlyaddedevent.
In our RS for core TCMN, a reduction denotes either a begin transmission event, or a time passing
event, or a node movement event. In order to handle the interaction among an unbounded number of
processes, we use rule schemas instead of simple rules to demonstrate the reductions. Also, since a
reduction, e.g. a begin transmission event, cannot be performed inside arbitrary contexts: one should
guaranteethatthecurrentcontextmeetsthespecificconditions,theminimalinformationaboutthecom-
municationisattachedtothereduction. Further,inordertomodelcommunicationinterference,westore
alltheneededactivetransmitters’informationinaglobalsetTwhichdisplaysinanyreductiontodeter-
mine whether a node is simultaneously reached by more than one transmission over the same channel.
Forthisreason,thereductionsemanticsisnamedRST:RSwithparameterT. ThecomponentTisaset
oftriples(l,r,c)witheachl,r,cinatriplerepresentslocation,radiusandchannelofanactivetransmitter
separately. For simplicity, the semantics does not automatically update the set T. Therefore, when a
reduction is performed, the new T which will be used in the next one has to be manually computed.
However,itisnotdifficulttomodifytherulessothattheyalsoproducethenewT.
As usual in process calculi, the reduction semantics relies on an auxiliary relation, called structural
congruence,denotedby≡,toallowthemanipulationofthetermstructuresoastobringtheparticipants
of a potential interaction into contiguous positions. Here we define a smallest congruence including
associativity,commutativityandidentityovertheemptynetwork:
N|(N(cid:48)|N(cid:48)(cid:48))≡(N|N(cid:48))|N(cid:48)(cid:48) N|N(cid:48)≡N(cid:48)|N N|0≡N
NextaresomeusefulnotationsthatwillbeusedinRST:
• T| isthesubsetoftheactivetransmittersTwhosetransmissionsaresynchronizedonchannelc
l,c
andcanreachanodelocatedatl. Formally,
T| ={(l(cid:48),r(cid:48),c(cid:48))|(l(cid:48),r(cid:48),c(cid:48))∈T∧d(l(cid:48),l)≤r(cid:48)∧c(cid:48)=c}
l,c
• (l,r,c)⇓/N holds if Network N contains no input nodes n[in(x).P]c or n[(x)δ.P]c for which
i l(cid:48),r(cid:48) v l(cid:48),r(cid:48)
d(l,l(cid:48))≤r is true (i.e., a transmission from a node located at l, with radius r, synchronized on c
reachesnoinputnodesinN).
• (l,r,c)⇓/ N holds if Network N contains no active input nodes n[(x)δ.P]c for which d(l,l(cid:48))≤r
ai v l(cid:48),r(cid:48)
is true (i.e., a transmission from a node located at l, with radius r, synchronized on c reaches no
activeinputnodesinN).
Let’s explain the rules in Table 2 and 3. Rule RST-BEGIN is used to derive begin transmission
reduction. Asin[12],itrewritesatomicallyanoutputnoden[out(cid:104)u(cid:105).P]c whichisintendingtoinitiatea
l,r
transmissionandallthereceivernodesthatarenotonlyinitstransmissioncellbutalsosynchronizedon
122 ATimedCalculusforMobileAdHocNetworks
Table 2. Reduction Semantics - Begin transmission and time passing event
[RST-BEGIN] [RST-PASS-NULL]
∀h∈I∪J∪K.d(l,lh)≤r ∀i∈I.T|li,c=0/ ∀j∈J.T|lj,c(cid:54)=0/ T(cid:66)0(cid:44)→σ0
T(cid:66)n[out(cid:104)u(cid:105).P]cl,r|h∈∏I∪Jnh[in(xh).Ph]clh,rh|k∏∈Knk[(xk)δvkk.Pk]clk,rk(cid:44)→cl,r
n[(cid:104)[[[u]]](cid:105)(cid:72)u(cid:73).P]cl,r|i∏∈Ini[(xi)[(cid:72)[[uu(cid:73)]]].Pi]cli,ri|j∏∈Jnj[in(xj).Pj]clj,rj|k∏∈Knk[Pk{⊥/xk}]clk,rk
[RST-SENDING] [RST-PASS-NA]
δ>0 ∀i∈I.d(l,li)≤r T(cid:66)n[P]c (cid:44)→σn[P]c
T(cid:66)n[(cid:104)v(cid:105)δ.P]cl,r|i∏∈Ini[(xi)δv.Pi]cli,ri(cid:44)→σn[(cid:104)v(cid:105)δ−1.P]cl,r|i∏∈Ini[(xi)vδ−1.Pi]cli,ri l,r l,r
[RST-CONT] [RST-CONT-PASS] [RST-CONGR]
T(cid:66)N(cid:44)→cl,rN(cid:48) (l,r,c)⇓/iN(cid:48)(cid:48) T(cid:66)N(cid:44)→σN(cid:48) T(cid:66)N(cid:48)(cid:48)(cid:44)→σN(cid:48)(cid:48)(cid:48) N≡N(cid:48) T(cid:66)N(cid:48)(cid:44)→&N(cid:48)(cid:48) N(cid:48)(cid:48)≡N(cid:48)(cid:48)(cid:48)
T(cid:66)N|N(cid:48)(cid:48)(cid:44)→c N(cid:48)|N(cid:48)(cid:48) T(cid:66)N|N(cid:48)(cid:48)(cid:44)→σN(cid:48)|N(cid:48)(cid:48)(cid:48) T(cid:66)N(cid:44)→&N(cid:48)(cid:48)(cid:48)
l,r
thesamechannelc. Afterthisreduction,theoutputprocessevolvesinto(cid:104)[[[u]]](cid:105) u .Pindicatinganactive
(cid:72) (cid:73)
outputprocessthatwilltransmittheevaluationresult[[[u]]]ofvalueuinthefollowing u timeunits. The
(cid:72) (cid:73)
effectofthebegintransmissioneventoneachreceivervariesaccordingtothestructureofeachreceiver
and the set T. There are three different situations, corresponding to the sets I, J and K. Processes in I
represent normal inputs. Since their environments are silent (T| =0/), they become active inputs of
li,c
u
the form (x) .P and start receiving data [[[u]]] for the next u time units. By contrast, for processes
i (cid:72)[[[u(cid:73)]]] i
(cid:72) (cid:73)
in J, as they are currently reached by at least one other transmission (T| (cid:54)=0/), they could not receive
lj,c
the begin transmission event clearly and stay idle. Finally, processes in K are active inputs, i.e., they
arereceivinganothertransmission,sothenewbegintransmissioneventcausesinterference,denotedby
receivingsymbol⊥.
Rule RST-SENDING deals with the time passing event for active processes. Initially, the active
output process (cid:104)v(cid:105)δ.P requires δ time units to complete the date transmission. After a time interval,
the remaining time would be δ−1 units for both sender and receivers. Meanwhile rule RST-PASS-
NA and RST-PASS-NULL handle the time passing event for non-active processes and empty networks
respectively. Nomatterhowtimeflies,theyremainunchanged.
Rule RST-MOVE-AO, RST-MOVE-AI1, RST-MOVE-AI2, RST-MOVE-AI3, and RST-MOVE-NA
are all used to describe node movements. In RST-MOVE-AO, an active transmitter moves from l to l(cid:48).
Then for active receivers in set I, as they are always reachable no matter from l or l(cid:48), they continue to
receive data normally. As for active receivers in set J, since they are reachable from l but not from l(cid:48),
theygetanerror, representedbyaspecialsignε. Finally, activereceiversinsetK, whicharereachable
froml(cid:48)butnotfroml,arereceivinganothertransmission,sothenewlyjoinedtransmitterwillmakethem
get interference. Rule RST-MOVE-AI1, RST-MOVE-AI2 and RST-MOVE-AI3 depict all the different
movementsofanactivereceiver. InRST-MOVE-AI1,theactivereceivermovesfromltol(cid:48) whichmakes
theoriginaltransmissionnolongerreceivable,henceitgetsanerror. WhileinRST-MOVE-AI2,although
the active receiver moves from l to l(cid:48), it has always been in the transmitter’s transmission cell and there
is no more active transmitter in l(cid:48), so the active receiver remains unchanged. On the contrary, in RST-
MOVE-AI3,whentheactivereceiverarrivesatl(cid:48),someothertransmissionsinl(cid:48)interferewithitsoriginal
one. Asaresult,theactivereceiverobtainsaninterference. RuleRST-MOVE-NAisstraightforward,for
M.Wang&Y.Lu 123
Table 3. Reduction Semantics - Node movement event
[RST-MOVE-AO]
∀i∈I.d(l,li)≤r∧d(l(cid:48),li)≤r ∀j∈J.d(l,lj)≤r∧d(l(cid:48),lj)>r ∀k∈K.d(l,lk)>r∧d(l(cid:48),lk)≤r
T(cid:66)n[(cid:104)v(cid:105)δ.P]cl,r|h∈∏I∪Jnh[(xh)δv.Ph]clh,rh|k∏∈Knk[(xk)δvkk.Pk]clk,rk(cid:44)→cl:l(cid:48),r
n[(cid:104)v(cid:105)δ.P]cl(cid:48),r|i∏∈Ini[(xi)δv.Pi]cli,ri|j∏∈Jnj[Pj{ε/xj}]clj,rj|k∏∈Knk[Pk{⊥/xk}]clk,rk
[RST-MOVE-AI1]
d(l,li)≤ri∧d(l(cid:48),li)>ri
T(cid:66)n[(x)δv.P]cl,r|ni[(cid:104)v(cid:105)δ.P]cli,ri(cid:44)→n[P{ε/x}]cl(cid:48),r|ni[(cid:104)v(cid:105)δ.P]cli,ri
[RST-MOVE-AI2]
d(l,li)≤ri∧d(l(cid:48),li)≤ri T|l,c=T|l(cid:48),c
T(cid:66)n[(x)δv.P]cl,r|n1[(cid:104)v(cid:105)δ.P]cli,ri(cid:44)→n[(x)δv.P]cl(cid:48),r|ni[(cid:104)v(cid:105)δ.P]cli,ri
[RST-MOVE-AI3]
d(l,li)≤ri∧d(l(cid:48),li)≤ri ∀j∈J.d(l,lj)>rj∧d(l(cid:48),lj)≤rj T|l(cid:48),c=T|l,c∪J
T(cid:66)n[(x)δv.P]cl,r|ni[(cid:104)v(cid:105)δ.P]cli,ri|j∏∈Jnj[(cid:104)vj(cid:105)δj.Pj]clj,rj(cid:44)→n[P{⊥/x}]cl(cid:48),r|ni[(cid:104)v(cid:105)δ.P]cli,ri|j∏∈Jnj[(cid:104)vj(cid:105)δj.Pj]clj,rj
[RST-MOVE-NA] [RST-CONT-MOVE] [RST-CONT-INT]
T(cid:66)n[P]cl,r(cid:44)→n[P]cl(cid:48),r T(cid:66)N(cid:44)→cl:l(cid:48),TrN(cid:66)(cid:48)N|(Nl,(cid:48)r(cid:48),(cid:44)→c)⇓cl/:la(cid:48),irNN(cid:48)(cid:48)(cid:48)∧|N((cid:48)l(cid:48)(cid:48),r,c)⇓/aiN(cid:48)(cid:48) T(cid:66)TN(cid:66)|NN(cid:48)(cid:48)(cid:44)(cid:44)→→NN(cid:48)(cid:48)|N(cid:48)(cid:48)
non-active nodes, their movement will not affect the environment, therefore they can move arbitrarily
withoutanylimitationsandchanges.
Rule RST-CONT, RST-CONT-PASS, RST-CONT-MOVE and RST-CONT-INT are closure rules
withregardtodifferentreductionforms((cid:44)→c ,(cid:44)→σ,(cid:44)→c ,and(cid:44)→). InRST-CONT,itprovidesaclosure
l,r l:l(cid:48),r
under contexts that do not contain receivers in the transmission cell of the transmitter. Similarly, rule
RST-CONT-MOVE presents a closure under contexts that have never contained active receivers in the
transmission cell of the transmitter when the transmitter moves from l to l(cid:48). Rule RST-CONT-INT is
analogoustotheprevioustwoexceptthatitconcernsinternalevents(i.e.,anodemovementeventfrom
anactivereceiveroranon-activenode). RuleRST-CONT-PASSisthetimesynchronization,itdefinesa
closureundercontextsthatarealsoaffectedbythetimepassingevent.
The last rule, RST-CONGR is a closure rule under structural congruence, where (cid:44)→& ranges over
(cid:44)→c ,(cid:44)→σ,(cid:44)→c and(cid:44)→forsomec,l,l(cid:48) andr.
l,r l:l(cid:48),r
4 Labelled Transition Semantics
We divide our Labelled Transition Semantics (LTS) into two set of rules corresponding to the two-
levelstructureofourlanguage. Table4containstherulesfortheprocesses,whileTable5and6presents
thoseforthenetworks.
Intheprocesssemantics,atransitionhastheformQ−→α Q(cid:48),wherethegrammarforα is:
α := !v:δ |?v:δ |?⊥|?ε |σ
124 ATimedCalculusforMobileAdHocNetworks
Table 4. Labelled Transitions for Processes
[[[u]]]=v u =δ [PS-OUT ] δ>0 [PS-OUT ]
out(cid:104)u(cid:105).P−!−v→:(cid:72)δ (cid:73)(cid:104)v(cid:105)δ.P begin (cid:104)v(cid:105)δ.P−→σ (cid:104)v(cid:105)δ−1.P send
− [PS-IN ] δ>0 [PS-IN ]
in(x).P−?−v→:δ (x)δv.P begin (x)δv.P−→σ (x)vδ−1.P receive
− [PS-IN ] − [PS-IN ]
in(x).P−?→⊥ in(x).P wait (x)δ.P−?→⊥ P{⊥/x} interfere
v
− [PS-IN ] − [PS-PASS] α∈{?v:δ,?ε,?⊥} Q∈/IQ[PS-NOIN]
(x)δ.P−→?ε P{ε/x} err P−→σ P Q−→α Q
v
where IQ is the set of processes of the form in(x).P or (x)δ.P
v
Label!v:δ representsabegintransmissionevent(i.e.,atransmissionofvaluevinthefollowingδ time
units) is initiated by Q which then evolves into Q(cid:48); ?v:δ indicates a begin transmission event reaches
Q and makes the process transform into Q(cid:48); Analogously, ?⊥ and ?ε stand for an interference or error
arrives;finally,σ meansatimepassingevent.
Explanations for the rules in Table 4 are as follows: in PS-OUT , the output process calculates
begin
thevalueuandinitiatesthetransmissionoftheresultv inthenextδ timeunits;inPS-IN ,theinput
begin
process successfully becomes involved with the transmission of value v for the next δ instants of time;
inPS-OUT andPS-IN ,withthetimepassingby,theremainingtransmissiontimeisdecreasing;
send receive
inPS-IN , theinputprocessstaysidlesinceitcouldnotreceivethebegintransmissioneventclearly;
wait
inPS-IN andPS-IN ,anactiveinputprocessencountersaninterferenceorerrorinitsreception,
interfere err
andhencestopsreceiving;RulePS-PASSshowsthatthenon-activeprocesswouldneverchangeastime
goes by, and PS-NOIN demonstrates that the non-input processes would never respond to the reception
ofevents.
FollowingaresomeusefulmathematicalsymbolsinLTS:
• d(l,l(cid:48))≤r(cid:48)(cid:12)d(l,l(cid:48)(cid:48))≤r(cid:48)=(d(l,l(cid:48))≤r(cid:48)∧d(l,l(cid:48)(cid:48))≤r(cid:48))∨(d(l,l(cid:48))>r(cid:48)∧d(l,l(cid:48)(cid:48))>r(cid:48)).
• T|l,c−T|l(cid:48),c isthesetofelementsthatarecontainedinT|l,c butnotinT|l(cid:48),c.
• T| ⊂T| holdsonlyifT| isapropersubsetofT| .
l,c l(cid:48),c l,c l(cid:48),c
Inthenetworksemantics,transitionsareoftheformT(cid:66)N−→µ N(cid:48) whereTisthesameasinSection
3. Let’s comment on the rules in Table 5 and 6. Rule NS-OUT, NS-IN , NS-IN and NS-IN concern
1 2 3
the communication between a transmitter and its receivers. Rule NS-OUT shows that a node initiates a
transmission. ThenruleNS-IN describesthebehaviorofanodethatiswithinthetransmissioncelland
1
couldhearthebegintransmissioneventclearly,whereasNS-IN handlesthosethatdetectconflicts. Rule
2
NS-IN demonstratesthatanodewouldnotreacttotransmissionsthatarebeyonditsreceptionrangeor
3
notinitslisteningchannel.
NextrulesNS-MOVE , NS-MOVE , NS-MOVE , NS-MOVE , NS-MOVE , NS-MOVE ,
ao in1 in2 in3 ai1 ai2
NS-MOVE andNS-MOVE areallusedtodealwiththenodemovementeventsfromdifferentkinds
ai3 na
ofnodes. Forexample,ruleNS-MOVE depictsthatanactivetransmitterlocatedatlmovestol(cid:48) during
ao
its transmission over channel c with radius r. Then the behaviors of surrounding nodes can be divided
M.Wang&Y.Lu 125
Table 5. Labelled Transitions for Networks - Begin transmission and time passing event
P−!−v→:δ A [NS-OUT] Q−?−v→:δ Q(cid:48) d(l,l(cid:48))≤r(cid:48) T|l,c=0/[NS-IN ]
T(cid:66)n[P]c −−c!−v:−δ[l−,r→] n[A]c T(cid:66)n[Q]c −−c?−v:δ−[l(cid:48)−,r(cid:48)→] n[Q(cid:48)]c 1
l,r l,r l,r l,r
Q−?→⊥ Q(cid:48) d(l,l(cid:48))≤r(cid:48) T|l,c(cid:54)=0/[NS-IN ] d(l,l(cid:48))>r(cid:48)∨c(cid:54)=c(cid:48) [NS-IN ]
T(cid:66)n[Q]c −−c?−v:δ−[l(cid:48)−,r(cid:48)→] n[Q(cid:48)]c 2 T(cid:66)n[Q]c −−c(cid:48)−?v−:δ−[l(cid:48),−r(cid:48)→] n[Q]c 3
l,r l,r l,r l,r
Q→σQ(cid:48) [NS-PASS] − [NS-NULL ]
T(cid:66)n[Q]cl,r−→σ n[Q(cid:48)]cl,r T(cid:66)0−−c?−v:−δ[−l,r→] 0 in1
− [NS-NULL ] T(cid:66)N1−−c?−v:−δ[−l,r→] N(cid:48)1 T(cid:66)N2−−c!−v:−δ[l−,r→] N(cid:48)2[NS-COM]
T(cid:66)0−→σ 0 pass T(cid:66)N1|N2−−c!−v:−δ[l−,r→] N(cid:48)1|N(cid:48)2
T(cid:66)N2|N1−c−!v−:δ−[−l,→r] N(cid:48)2|N(cid:48)1
T(cid:66)N1−−c?−v:−δ[−l,r→] N(cid:48)1 T(cid:66)N2−−c?−v:−δ[−l,r→] N(cid:48)2[NS-COM ] T(cid:66)N1−→σ N(cid:48)1 T(cid:66)N2−→σ N(cid:48)2[NS-SYN]
T(cid:66)N1|N2−−c?−v:−δ[−l,r→] N(cid:48)1|N(cid:48)2 in T(cid:66)N1|N2−→σ N(cid:48)1|N(cid:48)2
into three different cases corresponding to NS-MOVE , NS-MOVE and NS-MOVE respectively.
in1 in2 in3
(1)Fornon-activenodesandactivereceiversthatarereceivingoverotherchannelsorthatarealwaysin
or out of the transmission cell, they will remain unchanged. (2) For active receivers that are originally
within the transmission cell, but later beyond it, they will receive an error. (3) For active receivers that
are in the reverse situation, they will get interference. Analogously, rule NS-MOVE , NS-MOVE
ai1 ai3
andNS-MOVE havedescribedthepossiblescenariosofanactivereceiverthatmovesfromltol(cid:48): (1)
ai2
if the active receiver moves out of the transmission cell, it will obtain an error; (2) if the active receiver
has always been within the transmission cell and there is no more transmission in l(cid:48), it will continue
to receive data normally; (3) if there are more transmissions in l(cid:48) apart from its original one, the active
receiverwillgetinterference. Finally,wecanseefromNS-MOVE thatfornon-activenodes,theycan
na
movearbitrarilywithoutconditionsandlimitations.
Moreover,ruleNS-PASSrepresentstheresponsesofnodesastimegoesby. RuleNS-NULL ,NS-
in1
NULL and NS-NULL allow the empty network to receive data and evolve with time. At last, the
in2 pass
propagationofeventsthroughnetworksisportrayedbyruleNS-COM,NS-MOVE,NS-INT,NS-COM ,
in
NS-MOVE ,andNS-SYN.Thefirstthreedenotethataneventgeneratedinanetworkispropagatedto
in
theparallelnetwork;whilethelateronesindicatethattwoparallelnetworksreceivethesameevent.
5 Harmony Theorem
TheHarmonyTheoremaimsatprovingthattheLTS-basedsemanticscoincideswiththeRST-based
semantics. Withthisobjective,thetheoremhasthreeparts. First,itshowsthatthestructuralcongruence
respectstheLTS,i.e.,applicationofstructuralcongruencewillnotchangethepossibletransitions. Then
it demonstrates that the RST behaves the same as the LTS, i.e., each reduction in the RST has a corre-
spondingtransitionintheLTSwhichmakestheresultingnetworksstructurallycongruent. Intheend,it
testifiestheconversealsoholds.
Beforeprovingthetheorem,therearesomeauxiliarylemmasthatportraytheshapeofprocessesable
toperformaparticularlabelledtransition,andtheshapeofthederivativeprocesses(seetheAppendix).
126 ATimedCalculusforMobileAdHocNetworks
Table 6. Labelled Transitions for Networks - Node movement event
− [NS-MOVE ]
T(cid:66)n[(cid:104)v(cid:105)δ.P]c −−c!−[(l−:l(cid:48))−,r→] n[(cid:104)v(cid:105)δ.P]c ao
l,r l(cid:48),r
(Q(cid:54)∈AIQ)∨(c(cid:54)=c(cid:48))∨(d(l,l(cid:48))≤r(cid:48)(cid:12)d(l,l(cid:48)(cid:48))≤r(cid:48))
[NS-MOVE ]
T(cid:66)n[Q]c −−c(cid:48)?−[(−l(cid:48):l−(cid:48)(cid:48))−,r(cid:48)→] n[Q]c in1
l,r l,r
Q∈AIQ Q−→?ε Q(cid:48) d(l,l(cid:48))≤r(cid:48)∧d(l,l(cid:48)(cid:48))>r(cid:48)
[NS-MOVE ]
T(cid:66)n[Q]c −−c?−[(l−(cid:48):l−(cid:48)(cid:48))−,r(cid:48)→] n[Q(cid:48)]c in2
l,r l,r
Q∈AIQ Q−?→⊥ Q(cid:48) d(l,l(cid:48))>r(cid:48)∧d(l,l(cid:48)(cid:48))≤r(cid:48)
[NS-MOVE ]
T(cid:66)n[Q]c −−c?−[(l−(cid:48):l−(cid:48)(cid:48))−,r(cid:48)→] n[Q(cid:48)]c in3
l,r l,r
Q∈AIQ Q−→?ε Q(cid:48) T|l,c−T|l(cid:48),c=T|l,c[NS-MOVE ]
T(cid:66)n[Q]c →−n[Q(cid:48)]c ai1
l,r l(cid:48),r
Q∈AIQ Q−?→⊥ Q(cid:48) T|l,c⊂T|l(cid:48),c[NS-MOVE ] Q∈AIQ T|l,c=T|l(cid:48),c[NS-MOVE ]
T(cid:66)n[Q]c →−n[Q(cid:48)]c ai2 T(cid:66)n[Q]c →−n[Q]c ai3
l,r l(cid:48),r l,r l(cid:48),r
− [NS-MOVE ] − [NS-NULL ]
T(cid:66)n[P]cl,r→−n[P]cl(cid:48),r na T(cid:66)0−−c?−[(l−:l(cid:48)−),r→] 0 in2
T(cid:66)N1−−c?−[(l−:l(cid:48)−),r→] N(cid:48)1 T(cid:66)N2−−c!−[(l−:l(cid:48))−,r→] N(cid:48)2[NS-MOVE] T(cid:66)N1→−N(cid:48)1 [NS-INT]
T(cid:66)N1|N2−−c!−[(l−:l(cid:48))−,r→] N(cid:48)1|N(cid:48)2 T(cid:66)N1|N2→−N(cid:48)1|N2
T(cid:66)N2|N1−c−![−(l:−l(cid:48)−),→r] N(cid:48)2|N(cid:48)1 T(cid:66)N2|N1→− N2|N(cid:48)1
T(cid:66)N1−−c?−[(l−:l(cid:48)−),r→] N(cid:48)1 T(cid:66)N2−−c?−[(l−:l(cid:48)−),r→] N(cid:48)2[NS-MOVE ]
T(cid:66)N1|N2−−c?−[(l−:l(cid:48)−),r→] N(cid:48)1|N(cid:48)2 in
where AIQ is the set of processes of the form (x)δ.P
v
Theorem1(HarmonyTheorem). LetNbeanetwork,andTasetofactivetransmitters.
(1) IfT(cid:66)N−→µ N(cid:48) andN≡N ,thenthereexistsN(cid:48) suchthatT(cid:66)N −→µ N(cid:48) ≡N(cid:48).
1 1 1 1
(2) (a) IfT(cid:66)N(cid:44)→c N(cid:48),thenT(cid:66)N−c−![−(l−:l(cid:48)−),→r] N(cid:48) ≡N(cid:48).
l:l(cid:48),r 1
(b) IfT(cid:66)N(cid:44)→c N(cid:48),thentherearevandδ suchthatT(cid:66)N−c−!v−:δ−[−l,→r] N(cid:48) ≡N(cid:48).
l,r 1
(c) IfT(cid:66)N(cid:44)→σ N(cid:48),thenT(cid:66)N−→σ N(cid:48) ≡N(cid:48).
1
(d) IfT(cid:66)N(cid:44)→N(cid:48),thenT(cid:66)N→− N(cid:48) ≡N(cid:48).
1
(3) Foreachitemin(2),thereversealsoholds.
Proof. Nowweprovethethreepointsinsequence.
(1) The equivalence is defined in terms of commutativity, associativity, and identity over the empty
network. First commutativity is guaranteed since rule NS-COM, NS-MOVE and NS-INT are
symmetric, and rule NS-COM , NS-MOVE and NS-SYS are self-symmetric. Identity over the
in in
emptynetworkconservessince,owingtoruleNS-NULL ,NS-NULL andNS-NULL ,the
in1 in2 pass
M.Wang&Y.Lu 127
Table 7. Extended Syntax for Processes
Pd=ef ... old processes | (cid:100)in(x).P(cid:101)tQ input with timeout
| σ.P delay | (cid:66)c.P channel switch
→−
| [e]Q ,Q choice | H(u) recursion
1 2
where t is a positive integer greater than 0
c?v:δ[l,r] c?[(l:l(cid:48)),r] σ
empty network can perform any labels of the form −−−−−→, −−−−−→, and −→, which serve as
neutral element of parallel composition. Finally as the operations for parallel composition are
associativeandthenetworkstructureisalwayspreserved,thereforeassociativityisalsoensured.
(2) Asproofsforthefourstatementsaresimilar,weonlytake(a)asanexample.
(a)TheproofisbyruleinductiononthederivationofT(cid:66)N(cid:44)→c N(cid:48).
l:l(cid:48),r
First we consider rule RST-MOVE-AO, the proof for this case is by induction on the size
of I∪J∪K. The base case is I∪J∪K=0/, using rule NS-MOVE . In the inductive case, we
ao
randomly choose an element h from I∪J∪K. Remember that by the inductive hypothesis, we
c![(l:l(cid:48)),r]
already have a transition with label −−−−−→. Below are different cases according to which set
h belongs to. Suppose h∈I, then we can apply rule NS-MOVE since d(l,l)≤r∧d(l(cid:48),l)≤r
in1 i i
from the premise of rule RST-MOVE-AO. Thus the desired transition can be proved using rule
NS-MOVE. Suppose now h∈J, we can use rule PS-IN to derive (x)δ.P −?→ε P{ε/x}, and
err j v j j j
c?[(l:l(cid:48)),r]
then rule NS-MOVE to derive a transition with label −−−−−→. Hence the desired format can
in2
be arrived using rule NS-MOVE. Finally suppose h∈K, we can use rule PS-IN to derive
interfere
(x )δk.P −?→⊥ P {⊥/x }. This transition can be lifted up to the network level using rule NS-
k vk k k k
MOVE because d(l,l )>r∧d(l(cid:48),l )≤r from the precondition of rule RST-MOVE-AO. Simi-
in3 k k
larly,usingruleNS-MOVE,wecangetthedesiredtransition.
TheproofisanalogousforruleRST-CONT-MOVE,since(l,r,c)⇓/ N(cid:48)(cid:48)∧
ai
(l(cid:48),r,c)⇓/ N(cid:48)(cid:48) ensuresthatalltheactiveinputnodessatisfytheconditionsofNS-MOVE . Asfor
ai in1
non-activeinputnodes,ruleNS-MOVE canalsobeapplied.
in1
RuleRST-CONGRcanbesimulatedbythefirstpartofthetheorem.
(3) The proof for (a) and (b) are based on Lemma 2 and Lemma 4 respectively. The proof for (d) is
straightforward. Nowweconsidertheprooffor(c).
The proof is based on Lemma 5. If I∪J∪K=0/, the desired reduction can be derived using rule
RST-PASS-NULLandRST-CONGR.Otherwise,foreachelementiinIandforeachelementkin
K,firstapplyruleRST-SENDINGandRST-PASS-NArespectively,thenemployruleRST-CONT-
PASSandRST-CONGRtogetthedesiredreduction. (cid:3)
6 The Extended Language
SofarwehaveconsideredthesubsetofTCMNwithonlytheoperatorsthatarenecessaryforcom-
munication. Now we present some extensions: a series of processes are added in Table 7, while the
syntaxforothersremainsthesame.
First of all, the input construct is replaced by the input with timeout construct in (cid:100)in(x).P(cid:101)tQ. This
processiswaitingforreceivingavalue,ifthevaluearrivesbeforetheendofthettimeunits,theprocess
evolves into an active receiver; otherwise, the process continues as Q. Process σ.P stands for sleeping