Peter Gorm Larsen

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Modelling ordered collections

• Peter Gorm Larsen

2
Agenda

• Sequence Characteristics and Primitives
• The Congestion Warning System

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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 of Type
• seq1 of Type (for non-empty sequences)

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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

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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

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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

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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

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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

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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

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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)

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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 -gt
• 2,4,3,1,1, 2 -gt 7,5,1 -gt 8
• true,false, 3 -gt true,false

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Sequence Concatenation

• Two sequences A and B can be concatenated
  together to form a new sequence where As
  elements are followed by Bs elements
• Sequence concatenation is written as A B
• Quick examples
• 1,2,2 1,6,5
• true
• 3,2,3,1 4

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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

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Sequence Operators
hd l Head seq1 of A -gt A tl l Tail seq1 of A -gt
seq of A len l Length seq of A -gt nat elems
l Elements seq of A -gt set of A inds
l Indexes seq of A -gt set of nat1 l1
l2 Concatenation seq of A seq of A -gt seq of
A conc ll Distr. conc. seq of seq of A -gt seq of
A l(i) Seq. application seq1 of A nat1 -gt A l
m Seq. modification seq1 of A map nat1 to A
-gt seq1 of A l1 l2 Equality seq of A seq
of A -gt bool l1 ltgt l2 Inequality seq of A seq
of A -gt bool
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Sequence Comprehensions

• Using predicates to define sequences implicitly
• In VDM formulated like
• element numeric set binding 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 gt 2

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Questions

• What are the sequence enumerations for
• xx in set 8,,1 x lt 3
• xx in set 1,,10 x gt 3 and x lt 6
• y y in set 3,1,7,3
• x6 x in set 1,2
• mk_(x,8) x in set 1,2,7 x gt 4
• 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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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)

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Agenda

• Sequence Characteristics and Primitives
• The Congestion Warning System

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The Congestion Warning System

• A system for warning drivers of upcoming
  congestion on highways with lower speed limits to
  reduce the likelihood of collisions.

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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.

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Overview of the CWS System
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UML Class Diagram for CWS
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Example Journey Plan

• class CWS
• instance variables
• roadNetwork seq of CongestionMonitor
• sensors seq of PassageSensor
• inv len roadNetwork len sensors
• am ActuatorManager new ActuatorManager()
• op OperatorControl new OperatorControl()
• types
• Location nat1
• end CWS

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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
• )

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The AddCongestionMonitor Operation

• public AddCongestionMonitor Location gt ()
• 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(loc1,...,
  
• 
  numberOfWarners)
• sensors sensors(1,...,loc)
  sensor
• sensors(loc1,...,numberOfW
  arners)
• )
• am.AddActuator(loc)
• )

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Different kinds of Sensors
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Sensors and PassageSensors

• class Sensor
• instance variables
• protected location CWSLocation
• end Sensor
• class PassageSensor is subclass of Sensor
• instance variables
• passages seq of CWSSpeed
• operations
• public PassageSensor CWSLocation gt
  PassageSensor
• PassageSensor(loc)
• location loc
• end PassageSensor

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Finding the Average Speed

• class PassageSensor is subclass of Sensor
• public AverageSpeed nat1 gt CWSSpeed
• AverageSpeed(numberOfPassages)
• ( dcl accSpeed CWSSpeed 0
• let passInAccount passages(1,...,numberOfPassa
  ges)
• in
• ( for speed in passInAccount do
• accSpeed accSpeed speed
• return (accSpeed/numberOfPassages)
• )
• )
• pre len passages gt numberOfPassages
• end PassageSensor

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The Congestion Sensor

• class CongestionSensor is subclass of Sensor
• types
• public CongestionStatus ltCongestiongtltNoCongesti
  ongt
• ltDoubtgt
• operations
• public CongestionSensor PassageSensor gt
  CongestionSensor
• CongestionSensor(sensor)
• passageSensor sensor
• public IssueCongestionStatus () gt
  CongestionStatus
• IssueCongestionStatus()
• def averageSpeed passageSensor.AverageSpeed(no
  Passages)
• in
• if averageSpeed lt congestionThreshold
• then return ltCongestiongt
• elseif averageSpeed gt noCongestionThreshold
• then return ltNoCongestiongt
• else return ltDoubtgt
• end CongestionSensor

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Actuator Structure
public Signal ltNoWarninggt
ltPreAnnouncementgt
ltCongestionWarninggt
as seq of Actuator
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Show Signal in Actuation Manager

• class ActuationManager
• public ShowSignal CWSLocation
• CongestionMonitorSignal gt
  ()
• 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 ltNoWarninggt or
• signal ltCongestionWarninggt)
• end ActuationManager

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Show Signal at a given Location

• class ActuationManager
• ShowSignalAtLoc CongestionMonitorSignal
  Actuator
• Actuator gt ()
• ShowSignalAtLoc(signal,downstream,actuator)
• if signal ltNoWarninggt
• then def downstreamsignal downstream.GetSignal
  ()
• in
• if downstreamsignal ltCongestionWarninggt
  
• then actuator.SetSignal(ltPreAnnouncementgt
  )
• else actuator.SetSignal(ltNoWarninggt)
• else def currentsignal actuator.GetSignal()
• in
• let safest MostRestrictive(currentsigna
  l,
• signal)
• in
• actuator.SetSignal(safest)
• end ActuationManager

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Most Restrictive Signal

• class ActuationManager
• functions
• MostRestrictive CongestionMonitorSignal
• CongestionMonitorSignal -gt
• CongestionMonitorSignal
• MostRestrictive(s1, s2)
• if s1 ltCongestionWarninggt or s2
  ltCongestionWarninggt
• then ltCongestionWarninggt
• elseif s1 ltPreAnnouncementgt or s2
  ltPreAnnouncementgt
• then ltPreAnnouncementgt
• else ltNoWarninggt
• end ActuationManager

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Adding and Replacing Actuators

• class ActuationManager
• public AddActuator CWSLocation gt ()
• AddActuator(loc)
• def act new Actuator()
• in
• as as(1,...,loc) act as(loc1,...,
  len as)
• pre loc in set inds as
• public ReplaceActuator CWSLocation gt ()
• ReplaceActuator(loc)
• def act new Actuator()
• in
• as as loc -gt act
• pre loc in set inds as
• end ActuationManager

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Operator Control

• class OperatorControl
• instance variables
• messageLog seq of seq1 of char
• locations seq of CWSLocation
• inv len messageLog len locations
• end OperatorControl

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Manipulating Log Messages

• class OperatorControl
• operations
• public ResetLog () gt ()
• ResetLog()
• atomic
• (messageLog
• locations
• )
• public WriteLog seq1 of char CWSLocation gt
  ()
• WriteLog(message, location)
• atomic
• (messageLog messageLog
• message ConvertNum2String(locati
  on)
• locations locations location
• )
• end OperatorControl

Notice that WriteLog has an error in the book.
This is the right version.
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Operator Utilities

• class OperatorControl
• operations
• public CongestionSpots () gt set of
  CWSLocation
• CongestionSpots()
• return elems locations
• ConvertLog2File () gt seq of char
• ConvertLog2File()
• return conc messageLog
• end OperatorControl

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Summary

• What have I presented today?
• The notion of sequences as ordered collections
• The basic operations in VDM for manipulating
  sequences
• The congestion warning system example
• What do you need to do now?
• Continue with your project
• Present your status to all of us
• Read chapter 8 before next lecture

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Quote of the day
In science the credit goes to the man who
convinces the world, not the man to whom the idea
first occurs.
By Sir Francis Darwin (1848 - 1925)