Fabio A. Schreiber

SCHREIBER, FABIO ALBERTO

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STATE AND TIME GRANULARITY INSYSTEM DESCRIPTION AN EXAMPLE Fabio A SchreiberDipartimento di Elettronica  Politecnico di MilanoP zza L da Vinci  I Milano Italyemail	 schreibeeletpolimiitAbstract It is shown how a complex system can be described at dierentlevels of granularity and how timedriven and eventdriven descriptions cancoexistKeywords ATC radar realtime systems sequential machines systemsdescription and specication  IntroductionReal world systems are complex objects which can be modeled at very dierent levels of detail and indeed we can dene a system recursively as a system isa composition of interacting component systems Therefore we need a conceptof system which allows us to treat it as atomic when we wish to stop decomposing but also allows us to continue decomposing when we wish to do so	 In orderto x the basic concepts in systems description we are giving in the followingsome denitions mostly taken from 	Three are the main types of levels we can consider  behavior level this is also called inputoutput description or blackboxdescription	 In fact the system is viewed as a black box and measuresdone on it are recorded in a chronological order	 This requires that atimebase be dened as a subset of R continuous time or of Z discretetime	 The behavior of the system is described as a set of trajectorieswhich are mappings from subintervals of the timebase to some sets ofvalues representing possible observation results	  state structure level the systems is described in terms of mechanismsfor its internal working	 Such a description is sucient to generate byiteration over time a set of trajectories	 The tools for such a descriptionare the state set which represents the possible congurations at any timethe state transition function which provides the rules for computing thefuture state from the current one and possibly an output function to mapthe internal state set to an observable output set	  composite structure level the system is described as the connectionof many black boxes therefore it can also be called a network description	 The black boxes are dened as the components with specied inputvariables and output variables and a coupling specication must be given IEEE Real Time Systems Newsletter Vol  n  which determines the interconnections of the components and the interfacing of the input with the output variables	The set of internal states represents the memory of the system and it isthe heart of the modeling of its internal structure	 The choice of the state setis not unique and even its dimensionality is not xed	 The single state can beconsidered as an instantaneous snapshot of the universe 	The dierent representations of a system can be related to each other both atthe behavioral and at the internal structure level	 At the behavioral level thebasic relation is that of some kind of equivalence at the structural level thebasic notions are those of homomorphism and of isomorphism	 Thereforeit is possible to reduce a structural description of a system to a simpler onewhich is homomorphic with the former description	 Ontology of states and events an exampleGiven the preceding picture of the system modeling tools and alternatives weare faced with the question of dening the right levels of description for a systemthe primitive objects their attributes the values set of each attribute the rulesconstituting the state transition function the input stimuli which trigger statetransitions	We can dene as a state the set of values taken by the attributes statevariables of the objects which are the primitives at the level of description weare considering	As an example we can consider the snaphsot of an ATC radar screen as thestate of the sky above us where the spatial coordinates of the light dots representing the airplanes objects are the state variables	 An equivalent descriptioncould be that of considering the distance matrix between any two dots as theunique state variable as we shall see in the following	Most authors link the concept of system evolution i	e	 the transition fromone state to another to the concept of event	 In many cases the consideredevent is just the owing of time	 Mc Dermott  states that every state has atime of occurrence and that states are arranged in totally ordered sets calledchronicles which are complete possible histories of the universe a concept analogous to what has been called trajectories before	On the other hand events can represent complex situations which in turnentail a state transition	 It seems therefore that a contradiction emerges fromthe two views the rst with an absolutistic avor events are marked with timethe second with a relativistic one events constitute time 	The contradiction can be resolved if we consider the two conceptions  i	 e	an event triggers a state change or a state change is an event  at dierent levelsof detail or if we see states and events as elements of a duality relation withrespect to dierent views of the system	At a lower level a relativistic view is taken where some simple events like e	g	 the coincidence of two marks constitute time and are taken as primitives webuild some simple systems which count the events and we call them clocks	 Ata higher level we consider clocks as components in a system and their outputsare slightly more complex events which cause state changes which in turncan be taken as complex events at yet another level of description	 In thisway it is possible to distinguish between the local states  as the internal statesof each object  and the global states of the whole system	 It should be noticedthat by adopting the recursive denition of system these descriptions can beAC CC j E j rulesj jFALSE FALSE j N j ACTRUE FALSE j A j AC CCTRUE TRUE j C j CCFALSE TRUE j  j Figure  States tablestacked up and the global states of a lower level become local states of an objectat a higher level	Going back to the radar example we can think at the screen snapshot as amacro object composed of micro objects constituted by the light spots	 Thestate change of the micro world depends on the ow of time since at every sweepof the radar antenna the airplane positions on the screen change	 However fromthe ATC controller point of view we can distinguish at the macro level threedierent states of the screen the rst is the normal sky the second is analert situation the third is a collision scene	 Obviously under the statechanges we can nd the ow of time but a more fruitful representation at themacro level is to consider the state change to be induced by an event e	 g	 twoairplanes whose distance is below a minimal threshold	From this example we can notice that an event can be the result of anevaluation operation on some state variables	 Is this fact in contrast with theprevious denitions of event Transforming states into eventsLet us look closer at the example	 Let us call St fxtiytiztig the state of thesystem at time t where the state variables xi yi ziare the spatial coordinatesof each airplane aion the screen	 This is the lowest description level of oursystem and it can be modelled as a Moore synchronous sequential machine in which the output is the state Stitself and a state transition is triggered bya periodical signal every T the sweep period let us call it the MSM machine	Let us dene now the function dai ajdtij qxtj xti  ytj yti  ztj zti the spatial distance between any two airplanes ai aj and a particular valueof d d  DTH   called the danger threshold	Since we are mainly interested in aircrafts mutual positions we can assumeall the states having the same distance matrix to be equivalent therefore wecan transform the MSM machine into an internally equivalent machine MSM whose states Dt fdtijg are dened by the isomorphism but for rotationsor translations induced by d	Then the following predicates can be set  AC Alert Condition t  dt ij  DTH  t t  t  t dt i j 	 dti j  CC Collision Condition t  dti j  TTStSt+1St-1NAC¬AC¬AC¬ACAC∧¬CCAC∧¬CCAC∧¬CCCCCCFigureStatesdiagramNowwegroupthestatesoftheMSM machineintothreeequivalenceclassesfollowingtherulesshowningurewheretheequivalenceclassesarecalledNforNormalAforAlertandCforCollision	TracingbacktothephysicalworldthisclusteringamountstointerpretingthemeaningoftheroughdatabymeansofconditionsACandCC	AteachsweepoftheantennathedataareevaluatedandoneoftherulesbecomesTRUEthestateoftheMSM machinepossiblychangesmovingamongtheclusters	ThereforeatthebehaviorlevelthesystemcanbedescribedbyasequentialmachineBSMwhosestatesetisEfNACgandwhosestatetransitionsaretriggeredbyasetofeventsconstitutedbychangesinconditionsACandCC	ThesearenotexternalinputstothesystembuttheycomefromtheevaluationofthebooleanfunctiongACDTHDCCDACACCCCCwhichalsodeterminestheequivalenceclassesfNACg	Figuregivesaglobalpictureoftheradarsystemwhereallthedescriptionlevelsareshown	Sofarweshowedhowitispossibletotransformalowleveldescriptionwithapotentiallyinnitenumberofstatesintoahighleveldescriptionwithonlythreestates	LetusexaminenowthetimestructureoftheMSMandoftheBSMmachinesthisisimportantmainlyforrealtimesystemdescriptionwheresynchronousandasynchronousinputsriseratherdierentproblems	WesawabovethatMSMisasynchronousmachinedrivenbyaperiodicinputi	e	itsstatechangeateverysweepwhoseoutputdependsonlyontheinternalstateOtfSt	OnthecontraryBSMisdrivenbynontemporallydeterministiceventsconstitutedbypredicatesACorCCswitchingfromTRUEtoFALSEorviceversa	HoweverwecannoticethatswitchingisnottotallyasynchronoussinceitcanonlyhappenatsomemultiplenofthesweepperiodT	ThereforealsoBSMcanbedescribedasasynchronoussequentialmachinebutoftheMealytypethatiswhoseoutputisfunctionbothoftheinputIandofthestateEOtfItEt	InfactinaMealymachinetheoutputisproducedatthesametimeastheinputbutforinternaldelayswhileintheequivalentMooremachineaninputattimetproducesanoutputattimet	ThisfeatureseemsmoreimportantforrealtimesystemsthantheriskthatanoiseinputtotheMealymachinewhichcouldbelteredaheadofthedecisionunitcouldproduceaspuriousoutput	 ConclusionsIn this paper we showed by a simple but realistic example which can be extended to other application domains such as nuclear power plants or the gameof chess that the description tools for complex systems must be adapted to thedesired level of granularity and that the dierent levels can be formally relatedto each other	In an ATC radar system a change in the state value at the lower leveli	e	 airplanes position results in an event which in turn possibly triggers astate change at the upper level i	e	 the meaning of the radar snapshot	 Thisshows that not only the duality between events and state change can be fruitfulfor modeling complex systems but also that the concept of state evaluation isconsistent with such a model	 Figure  synthetically shows the main steps tothe description transformation	 References G	B	 Gerace  La logica dei sistemi di elaborazione  Editori Riuniti Roma D	 McDermott  A Temporal Logic for Reasoning about Processes andPlans  Cognitive Science Vol	   p	  F	A	 Schreiber  A Breviary to Time Concepts for Information Systems Rivista di Informatica Vol	 XX n	   pp	  J	A	 Spriet G	C	 Vansteenkiste  Computer Aided Modeling and Simulation Academic Press London 	BSMI   fACCCg  E   fNACg Ot fIt Etbehavior equivalencemap into equiv classfDDTHACCCgg fNAC gMSM I   fTig  D   fdijg Ot fDtstate isomorphismtransf state variabSd DMSMI   fTig  S   fxi yi zig Ot fStFigure  Transformation between the descriptions

State and Time Granularity in Systems Description: An Example

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State and Time Granularity in Systems Description: An Example

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