1. TIVDM1Modelling unordered collections1 Peter Gorm Larsen

2. TIVDM1Modelling unordered collections2 Agenda  Set Characteristics and Primitives The Minimum Safety Altitude Warning System The Robot Controller

3. TIVDM1Modelling unordered collections3 Set Characteristics Sets are unordered collections of elements There is only one copy of each element The elements themselves can be arbitrary complex, e.g. they can be sets as well Sets in VDM++ are finite Set types in VDM++ are written as: set of Type

4. TIVDM1Modelling unordered collections4 Set Membership If an object x is a member (an element) of a set A, then we write “ x  A ”; if it is not a member then we write “ x  A ”. “ x  A ” can be written as “ x in set A ” “ x  A ” can be written as “ x not in set A ”

5. TIVDM1Modelling unordered collections5 Set Enumeration A set enumeration consists of a comma- separated list enclosed between curly braces, ”{…}” 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 sets The empty set can be written as “{ }” or “  ”

6. TIVDM1Modelling unordered collections6 The Subset Relation The set A is said to be a subset of the set B if every element of A is also an element of B. The subset relation is written as ” A  B ” or as ” A subset B ” Quick examples: {1,2,3}  {1,2,3,4,5} { }  {1,2,3} {3,2,3,2}  {2,3}

7. TIVDM1Modelling unordered collections7 Set Equality Two sets are equal if both are subsets of each other i.e. A  B and B  A implies that A = B 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} = {3,5,5}

8. TIVDM1Modelling unordered collections8 Proper Subsets The set A is said to be a proper subset of the set B if every element of A is also an element of B and B has at least member that is not a member of A. The subset relation is written as ” A  B ” or as ” A psubset B ” Quick examples: {1,2,3}  {1,2,3,4,5} { }  {1,2,3} {3,2,3,2}  {2,3}

9. TIVDM1Modelling unordered collections9 Set Cardinality The cardinality of a set is the number of distinct elements i.e. its size The cardinality of a set S is written as “card S” Quick examples: card {1,2,3} card { } card {3,2,3,2}

10. TIVDM1Modelling unordered collections10 Powersets If S is a set then the power set of S is the set of all subsets of S. The powerset of a set S is written as “ P S” or “power S” Quick examples: power {1,2,2} power { } power {3,2,3,1} power power {6,7}

11. TIVDM1Modelling unordered collections11 Set Union The union of two sets combines all their elements into one set The union of two sets A and B is written as ”A  B” or ”A union B” Quick examples: {1,2,2} union {1,6,5} { } union {true} {3,2,3,1} union {4}

12. TIVDM1Modelling unordered collections12 Set Intersection The intersection of two sets is the set of all elements that are in both of the original sets The intersection of two sets A and B is written as ”A  B” or ”A inter B” Quick examples: {1,2,2} inter {1,6,5} { } inter {true} {3,2,3,1} inter {4}

13. TIVDM1Modelling unordered collections13 Distributed Set Operators Union and intersection can be distributed over a set of sets Distributed set union To be written as  (or dunion in ASCII) dunion {{ 2,4},{3,1,2},{2,3,4,3}} dunion {{ 2,4},{3,1,1},{}} dunion {{true},{false},{}} Distributed set intersection To be written as  (or dinter in ASCII) dinter {{ 2,4},{3,1,2},{2,3,4,3}} dinter {{ 2,4},{3,1,1},{}} dinter {{true},{false},{}}

14. TIVDM1Modelling unordered collections14 Set Difference The set difference of two sets A and B is the set of elements from A which is not in B The set difference of two sets A and B is written as ”A \ B” Quick examples: {1,2,2} \ {1,6,5} { } \ {true} {3,2,3,1} \ {4}

15. TIVDM1Modelling unordered collections15 Overview of Set Operators e in set s1 Membership (  ) A * set of A -> bool e not in set s1 Not membership (  ) A * set of A -> bool s1 union s2 Union (  ) set of A * set of A -> set of A s1 inter s2 Intersection (  ) set of A * set of A -> set of A s1 \ s2 Difference (\) set of A * set of A -> set of A s1 subset s2 Subset (  ) set of A * set of A -> bool s1 psubset s2 Proper subset (  ) set of A * set of A -> bool s1 = s2 Equality (=) set of A * set of A -> bool s1 <> s2 Inequality (≠) set of A * set of A -> bool card s1 Cardinality set of A -> nat dunion s1 Distr. Union (  ) set of set of A -> set of A dinter s1 Distr. Intersection (  ) set of set of A -> set of A power s1 Finite power set ( P ) set of A -> set of set of A

16. TIVDM1Modelling unordered collections16 Set Comprehensions Using predicates to define sets implicitly In VDM++ formulated like: {element | list of bindings & predicate} The predicate part is optional Quick examples: {3 * x | x : nat & x < 3} or {3 * x | x in set {0,…,2}} {x | x : nat & x < 5} or {x | x in set {0,…,4}}

17. TIVDM1Modelling unordered collections17 Questions What are the set enumerations for: {x|x : nat & x < 3} {x|x : nat & x > 3 and x < 6} {{y}| y in set {3,1,7,3}} {x+y| x in set {1,2}, y in set {7,8}} {mk_(x,y)| x in set {1,2,7}, y in set {2,7,8} & x > y} {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}

18. TIVDM1Modelling unordered collections18 Set Range Expressions The set range expression is a special case of a set comprehension. It has the form {e1,..., e2} where e1 and e2 are numeric expressions. The set range expression denotes the set of integers from e1 to e2 inclusive. If e2 is smaller than e1 the set range expression denotes the empty set. Examples: {2.718,...,3.141} {3.141,...,2.718} {1,...,5} {8,...,6}

19. TIVDM1Modelling unordered collections19 Agenda Set Characteristics and Primitives  The Minimum Safety Altitude Warning System The Robot Controller

20. TIVDM1Modelling unordered collections20 MSAW General Monitoring 500´ Threshold Terrain Clearance Altitude Minimum Safe Altitude (MSA)

21. TIVDM1Modelling unordered collections21 MSAW Approach Path Monitoring Runway Glideslope Path 1 nm Alarm Trigger Area (100´ below glideslope path)

22. TIVDM1Modelling unordered collections22 UK Civil Aviation Authority Minimum Safe Altitude Warning (MSAW) utilises secondary surveillance radar (SSR) responses from aircraft transponders and trajectory tracking to determine whether it is likely that the aircraft may be exposed to an unacceptable risk of Controlled Flight Into Terrain (CFIT). MSAW is normally implemented locally within the radar display system software and compares predicted aircraft trajectories with a database of levels at which an alert will be triggered within specific geographic areas. The system is technically complex (due to the need to compensate for radar processing delays) and requires careful installation, commissioning and operation to ensure that false alert occurrences do not present a hazard to operations.

23. TIVDM1Modelling unordered collections23 MSAW Requirements Radar(s) must track flying objects using their transponders Height of obstacles must be known statically Flying objects must be warned against obstacles close to their flight path New areas with obstacles can be defined The MSAW system must ensure the safety of flying objects against static obstacles Other flying objects (dynamic) is NOT a part of MSAW (dealt with using TCAS)

24. TIVDM1Modelling unordered collections24 UML Class Diagram

25. TIVDM1Modelling unordered collections25 A Collection of Flying Objects What instance variables should the FO class have? How should the airspace association between the Airspace and FO be made? class FO instance variables id : Id; coord : Coordinates; alt : Altitude; end FO class Airspace instance variables airspace : set of FO; inv forall x,y in set airspace & x <> y => x.getId() <> y.getId() end Airspace

26. TIVDM1Modelling unordered collections26 Adding New Flying Objects It must be possible to add new flying objects to an airspace: public addFO : FO ==> () addFO(fo) == airspace := airspace union {fo} pre fo.getId() not in set {f.getId() | f in set airspace}

27. TIVDM1Modelling unordered collections27 Get Hold of a Particular FO Given a particular identifier we need to be able to find the flying object with that transponder public getFO : Id ==> FO getFO(id) == find that value fo in the set airspace where fo.getId() equals id VDM++ Construct (let-be-such-that expression): let x in set s be st predicate on x in expression using x

28. TIVDM1Modelling unordered collections28 Get Hold of a Particular FO Using the let-be-such-that expression we get public getFO : Id ==> FO getFO(id) == let fo in set airspace be st fo.getId() = id in return fo pre FOExists(id,airspace); and functions FOExists: Id * set of FO -> bool FOExists(id,space) == exists fo in set space & fo.getId() = id

29. TIVDM1Modelling unordered collections29 Removing Existing Flying Objects It must also be possible to remove existing flying objects from an airspace: public removeFO : Id ==> () removeFO(id) == airspace := airspace \ {getFO(id)} pre FOExists(id,airspace) where we reuse the getFO operation

30. TIVDM1Modelling unordered collections30 Complete AirSpace Class This completes the AirSpace class Visibility shown with icons Stereotypes used to seperate operations and functions Signatures can be listed

31. TIVDM1Modelling unordered collections31 Constructor for Flying Objects Constructors in VDM++ use operation syntax Return type is implicit, so no return is needed public FO : Id * Coordinates * Altitude ==> FO FO(i,co,al) == (id := i; coord := co; alt := al; );

32. TIVDM1Modelling unordered collections32 What Instance Variables in Radar? What information is needed for each radar? instance variables location : Coordinates; range : nat1; detected : set of FO

33. TIVDM1Modelling unordered collections33 What can a radar see? Scanning from a radar public Scan : AirSpace ==> () Scan(as) == detected := { x | x in set as.airspace & InRange(x) }; private InRange : FO ==> bool InRange(obj) == let foLocation = obj.getCoordinates() in return isPointInRange(location,range,foLocation);

34. TIVDM1Modelling unordered collections34 A circle from a given point In the GLOBAL class general functionality is present functions protected isPointInRange : Coordinates * nat1 * Coordinates -> bool isPointInRange(center,range,point) == (center.X - point.X)**2 + (center.Y - point.Y)**2 <= range**2;

35. TIVDM1Modelling unordered collections35 The Obstacles Class What information do we need about an obstacle? instance variables MSA : MinimumSafetyAltitude ; location : Coordinates; radius : nat1; securityRadius : nat; type : ObstacleType; Where we inherit the following types public ObstacleType = | | | ; public FOWarning = ObstacleType; public RadarWarning = ; public MinimumSafetyAltitude = nat | ;

36. TIVDM1Modelling unordered collections36 The AirTrafficController Class class AirTrafficController is subclass of GLOBAL instance variables radars : set of Radar := {}; obstacles : set of Obstacle := {}; operations public addRadar : Radar ==> () addRadar(r) == radars := {r} union radars; public addObstacle : Obstacle ==> () addObstacle(ob) == obstacles := {ob} union obstacles;

37. TIVDM1Modelling unordered collections37 Finding Treats for FOs 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 isFOinVicinities(ob,fo) and not isFOatSafeAltitude(ob,fo) then writeObjectWarning(ob,fo); for all r in set radars do if r.saturatedRadar() then writeRadarWarning(r) );

38. TIVDM1Modelling unordered collections38 Conditions for Warnings isFOinVicinities : Obstacle * FO -> bool isFOinVicinities(obs,fo) == let obsloc = obs.getCoordinates(), secureRange = obs.getSecureRange(), foloc = fo.getCoordinates() in isPointInRange(obsloc,secureRange,foloc); isFOatSafeAltitude : Obstacle * FO -> bool isFOatSafeAltitude(obs,fo) == let msa = obs.getMSA() in if msa = then false else msa < fo.getAltitude();

39. TIVDM1Modelling unordered collections39 Saturating a radar There is a limit to how many FO´s a radar can deal with at one time. We call this saturation of a radar. class Radar values maxFOs : nat1 = 4; instance variables range : nat1; detected : set of FO … operations public saturatedRadar : () ==> bool saturatedRadar() == return card detected > range / maxFOs; end Radar

40. TIVDM1Modelling unordered collections40 Detecting FOs with multiple radars Some radars will have overlap so it may be interesting to collect the FOs that are detected by at least 2 radars: public detectedByTwoRadars : set of Radar -> set of FO detectedByTwoRadars(radars) == dunion {a.getDetected() inter b.getDetected() | a,b in set radars & a <> b}; FOs that are detected by all radars may also be interesting: public detectedByAllRadars : set of Radar -> set of FO detectedByAllRadars(radars) == dinter {x.getDetected() | x in set radars};

41. TIVDM1Modelling unordered collections41 The World Class class World instance variables public static env : [Environment] := nil; public static timerRef : Timer := new Timer(); operations public World : () ==> World World() == (env := new Environment("scenario.txt"); env.setAirSpace(MSAW`airspace); MSAW`atc.addRadar(MSAW`radar1); MSAW`atc.addRadar(MSAW`radar2); MSAW`atc.addObstacle(MSAW`militaryZone)); public Run : () ==> () Run() == env.Run(); end World

42. TIVDM1Modelling unordered collections42 The Environment Class (1) class Environment is subclass of GLOBAL operations public Environment : String ==> Environment Environment(fname) == def mk_(-,input) = io.freadval[seq of inline](fname) in inlines := input; public Run : () ==> () Run() == (while not isFinished() do (updateFOs(); MSAW`atc.Step(); World`timerRef.StepTime(); ); showResult() ); … end Environment

43. TIVDM1Modelling unordered collections43 The Environment Class (2) class Environment is subclass of GLOBAL operations 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 (airspace.updateFO(id,mk_Coordinates(x,y),altitude); inlines := tl inlines; done := len inlines = 0 ) else done := true) else busy := false ); … end Environment

44. TIVDM1Modelling unordered collections44 Updating a Flying Objects Since flying objects move we need to be able to update them: class AirSpace public updateFO : Id * Coordinates * Altitude ==> () updateFO(id,coord,alt) == if FOExists(id,airspace) then let fo = getFO(id) in (fo.setCoordinates(coord); fo.setAltitude(alt)) else let newfo = new FO(id,coord,alt) in airspace := airspace union {newfo} … end AirSpace where we reuse the getFO operation again

45. TIVDM1Modelling unordered collections45 Stepping in ATC Now all radars needs to have a chance to scan: class AirTrafficController is subclass of GLOBAL … public Step : () ==> () Step() == (for all r in set radars do r.Scan(MSAW`airspace); findThreats(); ); end AirTrafficController

46. TIVDM1Modelling unordered collections46 Agenda Set Characteristics and Primitives The Minimum Safety Altitude Warning System  The Robot Controller

47. TIVDM1Modelling unordered collections47 The Robot Controller A system for navigating a robot from a start point, via a collection of waypoints to a final destination, where it performs some task, e.g., delivering a payload.

48. TIVDM1Modelling unordered collections48 Existing Subsystems Position Sensor: This is used to find the robot's current location and the direction in which it is moving. Steering Controller: This controls the direction in which the robot travels. Steering Monitor: A system used to ensure that the steering controller is operating within known safe boundaries.

49. TIVDM1Modelling unordered collections49 Controller Requirements 1.The robot's current position is always available to the controller from a position sensor. 2.The robot has a predetermined journey plan based on a collection of waypoints. 3.The robot must navigate from waypoint to waypoint without missing any. 4.The robot moves only horizontally or vertically in the Cartesian plane. It is not physically capable of changing direction with an angle greater than 90 o. Attempts to do so should be logged. 5.If the robot is off-course, i.e., it cannot find a route to the next waypoint, it should stop in its current position. 6.The robot is able to detect obstacles in its path.

50. TIVDM1Modelling unordered collections50 Class Diagram for Robot Controller

51. TIVDM1Modelling unordered collections51 A Collection of Points What instance variables should the Point class have? How should the journeyPlan association between the Controller and Point be made? class Point instance variables x: nat; y: nat; index: nat end Point class Controller instance variables journeyPlan : set of Point; end Controller

52. TIVDM1Modelling unordered collections52 Example Journey Plan {new Point(1, 4, 1), new Point(4, 5, 2), new Point(6, 8, 3), new Point(10, 8, 4), new Point(9, 11, 5), new Point(8, 13, 6), new Point(11, 13, 7)}

53. TIVDM1Modelling unordered collections53 Getting a Point at a Particular Index public static GetPointAtIndex: set of Point * nat -> Point GetPointAtIndex(pts, index) == find that value p in the set pts where p.GetIndex() equals index VDM++ Construct: let x in set s be st predicate on x in expression using x

54. TIVDM1Modelling unordered collections54 The GetPointAtIndex Operation public static GetPointAtIndex: set of Point * nat -> Point GetPointAtIndex(pts, index) == let p in set pts be st p.GetIndex() = index in p pre exists p in set pts & p.GetIndex() = index;

55. TIVDM1Modelling unordered collections55 Checking Coordinates What is the value of: new Point(1,1,1) in set {new Point(1,1,1)} Assume we have an operation inside Point: GetCoord: () ==> nat * nat How can we then test whether a waypoint has been reached? wp.GetCoord() in set {o.GetCoord()|o in set obs}

56. TIVDM1Modelling unordered collections56 Arriving at a Waypoint journeyPlan desirable index properties 1.Next waypoint has index 1 2.Final waypoint has index equal to number of waypoints 3.Indices are numbered consecutively Modeled as invariant inside Controller: inv {p.GetIndex() | p in set journeyPlan} = {1,..., card journeyPlan};

57. TIVDM1Modelling unordered collections57 Taking a Step on a Journey Inside the Point class: public TakeStep: () ==> Point TakeStep() == ( index := index - 1; return self ) pre index > 1; Inside Route: static public TakeStep: set of Point -> set of Point TakeStep(pts) == let laterPoints = {pt | pt in set pts & pt.GetIndex() <> 1} in {p.TakeStep() | p in set laterPoints};

58. TIVDM1Modelling unordered collections58 Controlling the Robot 1.Find out the robot's current position. 2.Find out the next waypoint that the robot must visit. 3.If this waypoint has the same location as the current position then there are two possibilities: Either this is the last waypoint, i.e., the robot has reached its final destination and can therefore complete its journey or there are further waypoints to visit, in which case the journey plan must be updated. Otherwise do nothing. 4.Calculate the commands needed by the steering controller to get the robot to this next waypoint. 5.Give these commands to the steering controller.

59. TIVDM1Modelling unordered collections59 The Update Operation Update: () ==> () Update() == let currentPosition = ins.GetPosition() in ( if Route`GetPointAtIndex(journeyPlan,1).GetCoord() = currentPosition.GetCoord() then if card journeyPlan = 1 then CompleteJourney() else ( journeyPlan := Route`TakeStep(journeyPlan); let obstacles = obs.GetData(), route = PlotCourse(obstacles) in if route = nil then emergencyBrake.Enable() else def dfps = ComputeDesiredSteerPosition( ins.GetDirection(), route.GetPoint(2), str.GetPosition()) in AdjustSteering(dfps) ) );

60. TIVDM1Modelling unordered collections60 Neighbours of a Journey Point class Point … public Neighbour: () ==> set of Point Neighbour () == return {new Point(x, y1, index + 1) | y1 in set {y-1,y+1} & y1 >= 0} union {new Point(x1, y, index + 1) | x1 in set {x-1,x+1} & x1 >= 0}; end Point

61. TIVDM1Modelling unordered collections61 Plotting a Course class Controller … PlotCourse: set of (nat * nat) ==> [Route] PlotCourse(obstacles) == let nextWaypoint = Route`GetPointAtIndex(journeyPlan, 1), posRoutes = Route`AvoidanceRoutes(obstacles, ins.GetPosition(), nextWaypoint) in if posRoutes = {} then return nil else ShortestFeasibleRoute(posRoutes); end Controller

62. TIVDM1Modelling unordered collections62 Avoiding Obstacles class Route … static public AvoidanceRoutes( obstacles: set of (nat * nat), currentPosition: Point, nextWaypoint: Point) routes:set of Route post forall r in set routes & r.GetFirst().GetCoord() = currentPosition.GetCoord() and r.GetLast().GetCoord() = nextWaypoint.GetCoord() and r.GetCoords() inter obstacles = {}; end Route Does this work?

63. TIVDM1Modelling unordered collections63 An Invariant for the Route Class class Route … instance variables points: set of Point; inv forall p1, p2 in set points & p1.GetCoord() = p2.GetCoord() => p1 = p2 and forall p in set points & p.GetIndex() <> card points => GetNext(p).GetCoord() in set {n.GetCoord() | n in set p.Neighbour()} end Route

64. TIVDM1Modelling unordered collections64 Summary What have I presented today? The notion of sets as unordered collections The basic operations in VDM++ for manipulating sets The MSAW system The robot controller example What do you need to do now? Continue with your project Present your status to all of us Read chapter 7 before next lecture

65. TIVDM1Modelling unordered collections65 Quote of the day By Albert Einstein (1879 - 1955) Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.