1. TIVDM1Modelling unordered collections1 Peter Gorm Larsen

2. TIVDM1Modelling unordered collections2 Agenda  Set Characteristics and Primitives 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 Robot Controller

20. TIVDM1Modelling unordered collections20 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.

21. TIVDM1Modelling unordered collections21 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.

22. TIVDM1Modelling unordered collections22 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.

23. TIVDM1Modelling unordered collections23 Class Diagram for Robot Controller

24. TIVDM1Modelling unordered collections24 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

25. TIVDM1Modelling unordered collections25 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)}

26. TIVDM1Modelling unordered collections26 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

27. TIVDM1Modelling unordered collections27 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;

28. TIVDM1Modelling unordered collections28 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}

29. TIVDM1Modelling unordered collections29 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};

30. TIVDM1Modelling unordered collections30 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};

31. TIVDM1Modelling unordered collections31 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.

32. TIVDM1Modelling unordered collections32 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) ) );

33. TIVDM1Modelling unordered collections33 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

34. TIVDM1Modelling unordered collections34 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

35. TIVDM1Modelling unordered collections35 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?

36. TIVDM1Modelling unordered collections36 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

37. TIVDM1Modelling unordered collections37 Summary What have I presented today? The notion of sets as unordered collections The basic operations in VDM++ for manipulating sets 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

38. TIVDM1Modelling unordered collections38 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.