Peter Gorm Larsen

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Modelling unordered collections

• Peter Gorm Larsen

2
Agenda

• Set Characteristics and Primitives
• The Robot Controller

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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 -gt
bool e not in set s1 Not membership (?) A set
of A -gt bool s1 union s2 Union (?) set of A
set of A -gt set of A s1 inter s2 Intersection
(?) set of A set of A -gt set of A s1 \
s2 Difference (\) set of A set of A -gt set of
A s1 subset s2 Subset (?) set of A set of A
-gt bool s1 psubset s2 Proper subset (?) set of
A set of A -gt bool s1 s2 Equality () set
of A set of A -gt bool s1 ltgt s2 Inequality (?)
set of A set of A -gt bool card s1 Cardinality
set of A -gt nat dunion s1 Distr. Union (?) set
of set of A -gt set of A dinter s1 Distr.
Intersection (?) set of set of A -gt set of
A power s1 Finite power set (P) set of A -gt
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 lt 3 or 3 x x in set
  0,,2
• x x nat x lt 5 or x x in set 0,,4

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Questions

• What are the set enumerations for
• xx nat x lt 3
• xx nat x gt 3 and x lt 6
• y y in set 3,1,7,3
• xy 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
  gt y
• yy in set 0,1,2 exists x in set 0,,3 x
  2 y
• x 7 x in set 1,,10 x lt 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 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

• The robot's current position is always available
  to the controller from a position sensor.
• The robot has a predetermined journey plan based
  on a collection of waypoints.
• The robot must navigate from waypoint to waypoint
  without missing any.
• 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.
• If the robot is off-course, i.e., it cannot find
  a route to the next waypoint, it should stop in
  its current position.
• 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
class Point instance variables x nat y
nat index nat end Point

• What instance variables should the Point class
  have?
• How should the journeyPlan association between
  the Controller and Point be made?

class Controller instance variables journeyPlan
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
  -gt
• 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
  -gt
• 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 () gt 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 properties
• Next waypoint has index 1
• Final waypoint has index equal to number of
  waypoints
• 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 () gt Point
• TakeStep()
• ( index index - 1
• return self
• )
• pre index gt 1
• Inside Route
• static public TakeStep set of Point -gt set of
  Point
• TakeStep(pts)
• let laterPoints pt pt in set pts
• pt.GetIndex() ltgt 1
• in
• p.TakeStep() p in set laterPoints

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Controlling the Robot

• Find out the robot's current position.
• Find out the next waypoint that the robot must
  visit.
• 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.
• Calculate the commands needed by the steering
  controller to get the robot to this next
  waypoint.
• Give these commands to the steering controller.

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The Update Operation

• Update () gt ()
• Update()
• let currentPosition ins.GetPosition()
• in
• ( if RouteGetPointAtIndex(journeyPlan,1).GetCoord
  ()
• currentPosition.GetCoord()
• then
• if card journeyPlan 1
  
• then CompleteJourney()
• else
• ( journeyPlan RouteTakeStep(journeyPlan)
• let obstacles obs.GetData(),
• route PlotCourse(obstacles)
• in
• if route nil
• then emergencyBrake.Enable()
• else
• def dfps ComputeDesiredSteerPosition(
• ins.GetDirection(),

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Neighbours of a Journey Point

• class Point
• public Neighbour () gt set of Point
• Neighbour ()
• return new Point(x, y1, index 1)
• y1 in set y-1,y1
• y1 gt 0 union
• new Point(x1, y, index 1)
• x1 in set x-1,x1
• x1 gt 0
• end Point

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Plotting a Course

• class Controller
• PlotCourse set of (nat nat) gt Route
• PlotCourse(obstacles)
• let nextWaypoint RouteGetPointAtIndex(journeyP
  lan, 1),
• posRoutes RouteAvoidanceRoutes(obstacles,
  
• 
  ins.GetPosition(),
• 
  nextWaypoint)
• in
• if posRoutes
• then return nil
• else ShortestFeasibleRoute(posRoutes)
• end Controller

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Avoiding Obstacles

• class Route
• static
• public AvoidanceRoutes(
• obstacles set of (nat nat),
• currentPosition Point,
• nextWaypoint Point) routesset
  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?
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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() gt p1 p2
  and
• forall p in set points
• p.GetIndex() ltgt card points
• gt 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 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
Do not worry about your difficulties in
Mathematics. I can assure you mine are still
greater.
By Albert Einstein (1879 - 1955)