Modelling ordered collections

1. Modelling ordered collections Peter Gorm Larsen Modelling ordered collections

2. Agenda • Sequence Characteristics and Primitives • Revisiting the Minimum Safety Altitude Warning System • The Congestion Warning System Modelling ordered collections

3. Sequence Characteristics • Sequences are ordered collections of elements • There can be many copies of each element • The elements themselves can be arbitrary complex, e.g. they can be sequences as well • Sequences in VDM++ are finite • Sequence types in VDM++ are written as: • seq ofType • seq1 ofType (for non-empty sequences) Modelling ordered collections

4. Sequence Enumeration • A sequence enumeration consists of a comma-separated list enclosed between square brackets, ”[…]” • For example • [1,5,8,1,3] • [true, false] • [{}, {4,3},{2,4}] • [‘g’,’o’,’d’] • [3.567, 0.33455,7,7,7,7] Are all sequences • The empty sequence can be written as “[ ]” Modelling ordered collections

5. Sequence Length • The length of a sequence is the number of elements in the sequence i.e. its size • Multiple occurrences of the same value counts • The length of a sequence L is written as “ len L” • Quick examples: • len [1,2,3] • len [ ] • len [3,2,3,2] Modelling ordered collections

6. Sequence Equality • Two sequences are equal if both have the same length and for all indices in the sequences the respective index values are equal • Quick examples: • [2,4,1,2] = [4,1,2] • [true, true, false] = [false, true] • [1,1,1,1,1,1,1,1,1,1,1,1] = [1] • [{3,4,5},{4}] = [{3,5,4},{4,4,4}] Modelling ordered collections

7. Sequence Head and Tail • A non-empty sequence can be divided into its head (hd) and its tail (tl). • The head of a sequence is the first element • The tail of a sequence is the rest of the sequence • Quick examples: • hd [1,2,3,4,5] • tl[1,2,3,4,5] • hd [[5],[6,1],[4,4,4]] • tl [[5],[6,1],[4,4,4]] Modelling ordered collections

8. Sequence Elements • It is possible to extract the elements of a sequence using an elems operator • elems takes a sequence an yield a set of its elements (i.e. destroying the ordering information) • Quick examples: • elems [1,2,2] • elems [ ] • elems [[3],[2,3],[1]] Modelling ordered collections

9. Sequence Indices • It is possible to get hold of the indices for a sequence using the inds operator • In VDM++ indexing starts with 1 • Quick examples: • inds [1,2,21,6,5] • inds [{ }, {true}] • inds [ ] • inds [[3,2],[3],[1]] Modelling ordered collections

10. Sequence Application • Given a non-empty sequence it is possible to hold of its contents at the ith index • Sequence application is written as function application, i.e. sequence(index expression) • Quick examples: • [1,2,21,6,5](3) • [{ },{false}](2) • [[3,2],[3,1],[4]](1) Modelling ordered collections

11. Sequence Modification • Given a non-empty sequence it is possible to obtain a new sequence where the contents of certain indices are changed • A sequence modification expression looks as: sequence ++ modified mapping • The modified mapping goes from index to new value at that index • Quick examples • [{2,4},{3,1,2},{2,3,4,3}] ++ {1 |-> {}} • [[2,4],[3,1,1],[ ]] ++ {2 |-> [7,5],1 |-> [8]} • [{true},{false},{}] ++ {3 |-> {true,false}} Modelling ordered collections

12. Sequence Concatenation • Two sequences A and B can be concatenated together to form a new sequence where A’s elements are followed by B’s elements • Sequence concatenation is written as ”A ^ B” • Quick examples: • [1,2,2] ^ [1,6,5] • [ ] ^ [true] • [{3,2},{3},{1}] ^ [{4}] Modelling ordered collections

13. Distributed Sequence Concatenation • If we have a sequence of sequences then the elements can be concatenated together in a distributed fashion • Distributed sequence concatenation is written as ”conc SS” where SS is a sequence of sequences • Quick examples: • conc [[1,2,2], [1,6,5], [ ], [8,3]] • conc [[ ],[true],[false]] • conc [[{3,2},{3},{1}],[ ],[{9,5}],[{4}]] Modelling ordered collections

14. Sequence Operators hd l Head seq1 of A -> A tl l Tail seq1 of A -> seq of A len l Length seq of A -> nat elems l Elements seq of A -> set of A inds l Indexes seq of A -> set of nat1 l1 ^ l2 Concatenation seq of A * seq of A -> seq of A conc ll Distr. conc. seq of seq of A -> seq of A l(i)Seq. applicationseq1 of A * nat1 -> A l ++ mSeq. modificationseq1 of A * map nat1 to A -> seq1 of A l1 = l2 Equality seq of A * seq of A -> bool l1 <> l2 Inequality seq of A * seq of A -> bool Modelling ordered collections

15. Sequence Comprehensions • Using predicates to define sequences implicitly • In VDM++ formulated like: • [element | numeric setbinding & predicate] • The predicate part is optional • The numeric order of the binding is used to determine the order in the sequence • The smallest number is taken to be the first index • Quick examples • [3 * x | x in set {0,…,2}] • [x | x in set {0,…,4} & x > 2] Modelling ordered collections

16. Questions • What are the sequence enumerations for: • [x|x in set {8,…,1} & x < 3] • [x|x in set {1,…,10} & x > 3 and x < 6] • [{y}| y in set {3,1,7,3}] • [x+6| x in set {1,2}] • [mk_(x,8)| x in set {1,2,7} & x > 4] • [y|y in set {0,1,2} & exists x in set {0,…,3} & x = 2 * y] • [x = 7| x in set {1,…,10} & x < 6] Modelling ordered collections

17. Sub-sequence Expressions • A subsequence of a sequence L is a sequence formed from consecutive elements of L; from index n1 up to and including index n2. It has the form: • L(n1, ..., n2) • where n1 and n2 are integer expressions. • Quick Examples • [5,4,3,7,8,2](2,…,4) • [5,4,3,7,8,2](-6,…,4) • [5,4,3,7,8,2](2,…,8) • [5,4,3,7,8,2](6,…,4) Modelling ordered collections

18. Agenda • Sequence Characteristics and Primitives • Revisiting the Minimum Safety Altitude Warning System • The Congestion Warning System Modelling ordered collections

19. Adding Predictions and Priorities • In order to warn flying objects before they crash into an obstacle we need to be able to predict flight path • To deal with saturated radars we could introduce priorities • The flying objects that arrive in the airspace after the capacity is exceeded with be warned Modelling ordered collections

20. An Updated Class Diagram Modelling ordered collections

21. Adding a History Type • How can we define a history type? Class GLOBAL public History = seqof Position end GLOBAL Modelling ordered collections

22. Flying Objects Needs a History class FO is subclass of GLOBAL instance variables id : Id; coord : Coordinates; alt : Altitude; hist : History := []; inv len hist <= 3; operations public registerPosition : () ==> () registerPosition() == iflen hist < 3 then hist := hist ^ [mk_Position(coord,alt)] else hist := tl hist ^ [mk_Position(coord,alt)]; Modelling ordered collections

23. Introducing Vectors class GLOBAL … types public Vector :: X : real Y : real; operations protected vectorSum : Vector * Vector -> Vector vectorSum(v1,v2) == mk_Vector(v1.X + v2.X, v1.Y + v2.Y); … end GLOBAL Modelling ordered collections

24. Using Vectors class FO … operations public getDirectionHistory : () ==> seq of Vector getDirectionHistory() == let p1 = hist(1), p2 = hist(2), p3 = hist(3) in return [mk_Vector(p1.coord.X - p2.coord.X, p1.coord.Y - p2.coord.Y), mk_Vector(p2.coord.X - p3.coord.X, p2.coord.Y - p3.coord.Y)] pre len hist = 3; end FO Modelling ordered collections

25. Updating ATC Threads public findThreats : () ==> () findThreats() == let allFOs = dunion { r.getDetected() | r in set radars } in (for all fo in set allFOs do for all ob in set obstacles do if not isFOSafe(ob,fo.getPosition()) then writeObjectWarning(ob,fo) else if len fo.getHistory() = 3 then willFObeSafe(ob,fo); for all r in set radars do if r.saturatedRadar() then writeRadarWarning(r) ); Modelling ordered collections

26. Will a Flying Object be Safe? willFObeSafe : Obstacle * FO ==> () willFObeSafe(obs,fo) == let pred = isPredictPossible(fo) in for all p in set pred do if not isFOSafe(obs,p) then let id = fo.getId(), cs = fo.getCoordinates(), alt = fo.getAltitude(), type = <EstimationWarning>, msa = obs.getMSA(), t = World`timerRef.GetTime() in World`env.handleFOWarningEvent(id, cs, alt, type, msa, t) pre len fo.getHistory() = 3; Modelling ordered collections

27. Adding priorities to Radar class Radar is subclass of GLOBAL instance variables … priority : seq of FO := []; operations private addNewlyDetected : set of FO ==> () addNewlyDetected(newlyDetect) == priority := priority ^ set2seqFO(newlyDetect); functions set2seqFO : set of FO -> seq of FO set2seqFO(fos) == if fos = {} then [] else let fo in set fos in [fo] ^ set2seqFO(fos\{fo}) Modelling ordered collections

28. Updating priorities in Radar class Radar is subclass of GLOBAL instance variables … priority : seq of FO := []; operations private removeNotDetected : set of FO ==> () removeNotDetected(fos) == priority := [priority(i) | i in set inds priority & priority(i) not in set fos]; private UpdatePriorityList : () ==> () UpdatePriorityList() == let notDetect = elems priority \ detected, newlyDet = detected \ elems priority in ( removeNotDetected(notDetect); addNewlyDetected(newlyDet) ); Modelling ordered collections

29. Using Sequences in Environment class Environment is subclass of GLOBAL types inline = Id * int * int * Altitude * Time; outline = FOOut | RadarOut; FOOut = Id * Coordinates * Altitude * FOWarning * MinimumSafetyAltitude * Time; RadarOut = Coordinates * nat1 * RadarWarning * nat * Time; instance variables inlines : seq of inline := []; outlines : seq of outline := []; operations public Environment : String ==> Environment Environment(fname) == defmk_(-,input) = io.freadval[seq of inline](fname) in inlines := input; Modelling ordered collections

30. Updating Flying Objects class Environment … operations private updateFOs : () ==> () updateFOs() == (if len inlines > 0 then (dcl curtime : Time := World`timerRef.GetTime(), done : bool := false; while not done do def mk_(id,x,y,altitude,pt) = hd inlines in if pt <= curtime then let p = mk_Coordinates(x,y) in (airspace.updateFO(id,p,altitude); inlines := tl inlines; done := len inlines = 0 ) else done := true ) else busy := false ); Modelling ordered collections

31. Agenda • Sequence Characteristics and Primitives • Revisiting the Minimum Safety Altitude Warning System • The Congestion Warning System Modelling ordered collections

32. History for Altitude class FO public getAltitudeHistory : () ==> seq of nat getAltitudeHistory() == let lastHist = hist(2,...,3) in return [lastHist(i).altitude | i in set inds lastHist] end FO Modelling ordered collections

33. The Congestion Warning System • A system for warning drivers of upcoming congestion on highways with lower speed limits to reduce the likelihood of collisions. Modelling ordered collections

34. The Main CWS Components • Sensors: These are used to derive status information about the traffic. Sensors include video cameras, radar and human observers. • Traffic Controls: This interpret the data coming from sensors and take appropriate action. • Actuators: These are used to signal to the drivers about potential congestions. Here traffic signs will be used but different technologies could be envisaged as well. Modelling ordered collections

35. Overview of the CWS System Modelling ordered collections

36. UML Class Diagram for CWS Modelling ordered collections

37. Example Journey Plan class CWS … instance variables roadNetwork: seq of CongestionMonitor := []; sensors : seq of PassageSensor := []; invlen roadNetwork = len sensors; am: ActuatorManager := new ActuatorManager(); op: OperatorControl := new OperatorControl(); types Location = nat1 end CWS Modelling ordered collections

38. Multiple Assignment Statements • We somehow need to update the roadNetwork and the sensors instance variables synchronously to ensure the invariant • VDM++ Construct: atomic(assignment statement 1; assignment statement 2; ... assignment statement n ) Modelling ordered collections

39. The AddCongestionMonitor Operation public AddCongestionMonitor: Location ==> () AddCongestionMonitor(loc) == (def sensor = new PassageSensor(loc); cm = new CongestionMonitor(loc, sensor, am, op) in let numberOfWarners = len roadNetwork in atomic(roadNetwork := roadNetwork(1,...,loc) ^ [cm] ^ roadNetwork(loc+1,..., numberOfWarners); sensors := sensors(1,...,loc) ^ [sensor] ^ sensors(loc+1,...,numberOfWarners) ); am.AddActuator(loc) ) Modelling ordered collections

40. Different kinds of Sensors Modelling ordered collections

41. Sensors and PassageSensors class Sensor instance variables protected location: CWS`Location end Sensor class PassageSensor is subclass of Sensor instance variables passages: seq of CWS`Speed := [] … operations public PassageSensor: CWS`Location ==> PassageSensor PassageSensor(loc) == location := loc; end PassageSensor Modelling ordered collections

42. Finding the Average Speed class PassageSensor is subclass of Sensor … public AverageSpeed: nat1 ==> CWS`Speed AverageSpeed(numberOfPassages) == ( dcl accSpeed: CWS`Speed := 0; let passInAccount = passages(1,...,numberOfPassages) in ( for speed in passInAccount do accSpeed := accSpeed + speed; return (accSpeed/numberOfPassages) ) ) pre len passages >= numberOfPassages end PassageSensor Modelling ordered collections

43. The Congestion Sensor class CongestionSensor is subclass of Sensor types public CongestionStatus = <Congestion>|<NoCongestion>| <Doubt> operations public CongestionSensor: PassageSensor ==> CongestionSensor CongestionSensor(sensor) == passageSensor := sensor; public IssueCongestionStatus: () ==> CongestionStatus IssueCongestionStatus() == def averageSpeed = passageSensor.AverageSpeed(noPassages) in if averageSpeed < congestionThreshold then return <Congestion> elseif averageSpeed > noCongestionThreshold then return <NoCongestion> else return <Doubt> end CongestionSensor Modelling ordered collections

44. Actuator Structure public Signal = <NoWarning>| <PreAnnouncement>| <CongestionWarning>; as: seq of Actuator Modelling ordered collections

45. Show Signal in Actuation Manager class ActuationManager … public ShowSignal: CWS`Location * CongestionMonitor`Signal ==> () ShowSignal(location, signal) == (let downstream = as(location + 1), actuator = as(location), upstream = as(location - 1) in -- Set the right signal at the location itself (ShowSignalAtLoc(signal,downstream,actuator); -- Set the right signal upstream ShowSignalUpstream(signal,upstream) ) ) pre location in set {2,..., len as -1} and (signal = <NoWarning> or signal = <CongestionWarning>); end ActuationManager Modelling ordered collections

46. Show Signal at a given Location class ActuationManager … ShowSignalAtLoc: CongestionMonitor`Signal * Actuator * Actuator ==> () ShowSignalAtLoc(signal,downstream,actuator) == if signal = <NoWarning> thendef downstreamsignal = downstream.GetSignal() in if downstreamsignal = <CongestionWarning> then actuator.SetSignal(<PreAnnouncement>) else actuator.SetSignal(<NoWarning>) else def currentsignal = actuator.GetSignal() in let safest = MostRestrictive(currentsignal, signal) in actuator.SetSignal(safest); end ActuationManager Modelling ordered collections

47. Most Restrictive Signal class ActuationManager … functions MostRestrictive: CongestionMonitor`Signal * CongestionMonitor`Signal -> CongestionMonitor`Signal MostRestrictive(s1, s2) == if s1 = <CongestionWarning> or s2 = <CongestionWarning> then <CongestionWarning> elseif s1 = <PreAnnouncement> or s2 = <PreAnnouncement> then <PreAnnouncement> else <NoWarning> end ActuationManager Modelling ordered collections

48. Adding and Replacing Actuators class ActuationManager … public AddActuator: CWS`Location ==> () AddActuator(loc) == def act = new Actuator() in as := as(1,...,loc) ^ [act] ^ as(loc+1,..., len as) pre loc in setinds as; public ReplaceActuator: CWS`Location ==> () ReplaceActuator(loc) == def act = new Actuator() in as := as ++ {loc |-> act} pre loc in set inds as; end ActuationManager Modelling ordered collections

49. Operator Control class OperatorControl … instance variables messageLog: seq of seq1 of char := []; locations : seq of CWS`Location := []; inv len messageLog = len locations end OperatorControl Modelling ordered collections

50. Manipulating Log Messages class OperatorControl … operations public ResetLog: () ==> () ResetLog() == atomic (messageLog := []; locations :=[] ); public WriteLog: seq1 of char * CWS`Location ==> () WriteLog(message, location) == atomic (messageLog := messageLog ^ [message ^ ConvertNum2String(location)]; locations := locations ^ [location] ); end OperatorControl Notice that WriteLog has an error in the book. This is the right version. Modelling ordered collections