tivdm1modelling unordered collections1 peter gorm larsen
TRANSCRIPT
TIVDM1 Modelling unordered collections 1
Modelling unordered collections
Peter Gorm Larsen
TIVDM1 Modelling unordered collections 2
Agenda
Set Characteristics and Primitives• The Minimum Safety Altitude Warning System• The Robot Controller
TIVDM1 Modelling unordered collections 3
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
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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”
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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 “”
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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}
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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}
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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}
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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}
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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}
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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}
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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}
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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},{}}
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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}
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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
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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}}
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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}
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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}
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Agenda
Set Characteristics and Primitives The Minimum Safety Altitude Warning System• The Robot Controller
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MSAW General Monitoring
500´ Threshold
Terrain Clearance Altitude
Minimum Safe Altitude (MSA)
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MSAW Approach Path Monitoring
Runway
Glideslope Path
1 nm
Alarm Trigger Area
(100´ below glideslope path)
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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.
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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)
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UML Class Diagram
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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 FOinstance variablesid : Id;coord : Coordinates;alt : Altitude; end FO
class Airspaceinstance variables airspace : set of FO;inv forall x,y in set airspace & x <> y => x.getId() <> y.getId()end Airspace
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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}
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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
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Get Hold of a Particular FOUsing the let-be-such-that expression we get
public getFO : Id ==> FOgetFO(id) == let fo in set airspace be st fo.getId() = id in return fopre FOExists(id,airspace);
and
functions
FOExists: Id * set of FO -> boolFOExists(id,space) == exists fo in set space & fo.getId() = id
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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
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Complete AirSpace Class
• This completes the AirSpace class• Visibility shown with icons• Stereotypes used to seperate operations and functions• Signatures can be listed
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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;
);
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What Instance Variables in Radar?
• What information is needed for each radar?
instance variables
location : Coordinates;
range : nat1;
detected : set of FO
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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);
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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;
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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 typespublic
ObstacleType = <Natural>|<Artificial>|<Airport>|<Military_Area>;
public FOWarning = ObstacleType;
public RadarWarning = <Saturated>;
public MinimumSafetyAltitude = nat | <NotAllowed>;
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The AirTrafficController Classclass 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;
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Finding Treats for FOspublic 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)
);
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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 = <NotAllowed>
then false
else msa < fo.getAltitude();
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Saturating a radarThere 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 : () ==> boolsaturatedRadar() == return card detected > range / maxFOs;
end Radar
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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 FOdetectedByTwoRadars(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 FOdetectedByAllRadars(radars) == dinter {x.getDetected()
| x in set radars};
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The World Classclass Worldinstance variables public static env : [Environment] := nil;
public static timerRef : Timer := new Timer(); operations
public World : () ==> WorldWorld() == (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
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The Environment Class (1)class Environment is subclass of GLOBALoperations public Environment : String ==> EnvironmentEnvironment(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
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The Environment Class (2)class Environment is subclass of GLOBALoperations 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
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Updating a Flying ObjectsSince 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
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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
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Agenda
Set Characteristics and Primitives The Minimum Safety Altitude Warning System The Robot Controller
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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.
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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.
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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 90o. 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.
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Class Diagram for Robot Controller
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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 Pointinstance variablesx: nat;y: nat;index: natend Point
class Controllerinstance variablesjourneyPlan : set of Point;end Controller
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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)}
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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
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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;
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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}
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Arriving at a Waypoint
• journeyPlan desirable index properties1. 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};
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Taking a Step on a Journey
• Inside the Point class:public TakeStep: () ==> PointTakeStep() ==( index := index - 1; return self)pre index > 1;
• Inside Route:static public TakeStep: set of Point -> set of PointTakeStep(pts) == let laterPoints = {pt | pt in set pts & pt.GetIndex() <> 1} in {p.TakeStep() | p in set laterPoints};
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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.
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The Update OperationUpdate: () ==> ()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) ));
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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
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Plotting a Courseclass 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
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Avoiding Obstacles
class Route…staticpublic AvoidanceRoutes( obstacles: set of (nat * nat), currentPosition: Point, nextWaypoint: Point) routes:set of Routepost 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?
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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
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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
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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.