ch. 51 specification. ch. 52 outline discussion of the term "specification" types of...
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Ch. 5 1
Specification
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Ch. 5 2
Outline• Discussion of the term "specification"• Types of specifications
– operational • Data Flow Diagrams• (Some) UML diagrams• Finite State Machines• Petri Nets
– descriptive• Entity Relationship Diagrams• Logic-based notations• Algebraic notations
• Languages for modular specifications – Statecharts– Z
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Ch. 5 3
Specification• A broad term that means definition• Used at different stages of software
development for different purposes• Generally, a statement of agreement
(contract) between– producer and consumer of a service– implementer and user
• All desirable qualities must be specified
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Ch. 5 4
Uses of specification
• Statement of user requirements– major failures occur because of
misunderstandings between the producer and the user
– "The hardest single part of building a softwarem system is deciding precisely what to build" (F. Brooks)
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Ch. 5 5
Uses of specification (cont.)
• Statement of the interface between the machine and the controlled environment – serious undesirable effects can result
due to misunderstandings between software engineers and domain experts about the phenomena affecting the control function to be implemented by software
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Ch. 5 6
Uses of specification (cont.)
• Statement of requirements for implementation– design process is a chain of specification
(i.e., definition)–implementation–verification steps
• requirements specification refers to definition of external behavior
– design specification must be verified against it
• design specification refers to definition of the software architecture
– code must be verified against it
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Ch. 5 7
Uses of specification (cont.)
• A reference point during maintenance– corrective maintenance only changes
implementation– adaptive and perfective maintenance
occur because of requirements changes• requirements specification must change
accordingly
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Ch. 5 8
Specification qualities
• Precise, clear, unambiguous• Consistent• Complete
– internal completeness– external completeness
• Incremental
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Ch. 5 9
Clear, unambiguous, understandable
• Example: specification fragment for a word-processorSelecting is the process of designating areas of the document that you want to work on. Most editing and formatting actions require two steps: first you select what you want to work on, such as text or graphics; then you initiate the appropriate action.
can an area be scattered?
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Ch. 5 10
Precise, unambiguous, clear
• Another example (from a real safety-critical system)
The message must be triplicated. The threecopies must be forwarded through three different physical channels. The receiver accepts the message on the basis of a two-out-of-three voting policy.
can a message be accepted as soon as we receive 2 out of 3 identical copies of message or do we need to wait for receipt of the 3rd?
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Ch. 5 11
Consistent• Example: specification fragment for a
word-processor
The whole text should be kept in lines of equal length. The length is specified by the user. Unless the user gives an explicit hyphenation command, a carriage return should occur only at the end of a word.
What if the length of a word exceeds the length of the line?
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Ch. 5 12
Complete
• Internal completeness– the specification must define any new
concept or terminology that it uses• glossary helpful for this purpose
– the specification must document all the needed requirements• difficulty: when should one stop?
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Ch. 5 13
Incremental
• Referring to the specification process– start from a sketchy document and
progressively add details
• Referring to the specification document– document is structured and can be
understood in increments
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Ch. 5 14
Classification of specification styles
• Informal, semi-formal, formal• Operational
– Behavior specification in terms of some abstract machine
• Descriptive– Behavior described in terms of
properties
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Ch. 5 15
Example 1
• Specification of a geometric figure E:
E can be drawn as follows:1. Select two points P1 and P2 on a plane2. Get a string of a certain length and fix its ends
to P1 and P23. Position a pencil as shown in next figure4. Move the pen clockwise, keeping the string
tightly stretched, until you reach the point where you started drawing
this is an operational specification
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Ch. 5 16
P P 1 2
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Ch. 5 17
A descriptive specification
• Geometric figure E is describe by the following equation
ax2 + by2 + c = 0where a, b, and c are suitable constants
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Ch. 5 18
Another example
“Let a be an array of n elements. The result of its sorting is an array b of n elements such that the first element of b is the minimum of a (if several elements of a have the same value, any one of them is acceptable); the second element of b is the minimum of the array of n-1 elements obtained from a by removing its minimum element; and so on until all n elements of a have been removed.”
“The result of sorting array a is an array b which is a permutation of a and is sorted.”
OP
DES
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Ch. 5 19
How to verify a specification?
• “Observe” dynamic behavior of specified system (simulation, prototyping, “testing” specs)
• Analyze properties of the specified system
• Analogy with traditional engineering– physical model of a bridge– mathematical model of a bridge
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Ch. 5 20
Data Flow Diagrams (DFDs)
• A semi-formal operational specification
• System viewed as collection of data manipulated by “functions”
• Data can be persistent– they are stored in data repositories
• Data can flow – they are represented by data flows
• DFDs have a graphical notation
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Ch. 5 21
Graphical notation– bubbles represent functions– arcs represent data flows– open boxes represent persistent
store– closed boxes represent I/O
interactionThe function symbol
The data flow symbol
The data store symbol
The input device symbol
The output device symbol
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Ch. 5 22
Example
+ * +
*
a d c
b
specifies evaluation of(a + b) * (c + a * d)
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Ch. 5 23
A construction “method” (1)
... ...
Input 1
Input2
Input n
Output1
Output2
Outputm
information
system
1. Start from the “context” diagram
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Ch. 5 24
A construction “method” (2)
A
A1
A3
A2
A4
A5
A6
A7
B1
B2
B3B4
Ag
IO
I
O
H
K
J
M
N
P Q
R
S
K
T
K1
K2
K3
K4
M
N
2. Proceed by refinements until you reach “elementary” functions (preserve balancing)
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Ch. 5 25
A library exampleShelves
List of Authors
List of titles
List of topics
Title and author of requested book; name of the user
Get a book
Book
List of books borrowed
Book title; user name
Topic request by the user
Search by topics
Book request by the user
Book reception
TopicList of titles referring to the topic
Book
Author
Title
Display of the list of titles
Topic
Title
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Ch. 5 26
Refinement of“Get a book”
Shelves
List of Authors
List of titles
Title and author of requested book; name of the user
Book
List of books borrowed
Book title; user name
Book request by the user
Book reception
Book
Author
TitleFind book position
<shelf#, book#>
Get the book
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Ch. 5 27
Patient monitoring systems
The purpose is to monitor the patients’ vital factors--blood,pressure, temperature, …--reading them at specified frequenciesfrom analog devices and storing readings in a DB. If readings fall outside the range specified for patient or device fails an alarm must be sent to a nurse. The system also provides reports.
Patient
Nurse
PatientMonitoring
Nurse
Persistent data
Report
AlarmDataClinical
ReportRequest
Recent data
Data for report
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Ch. 5 28
A refinementNurse
Nurse
Patient archive
ReportRequest
Limits for patient
MonitoringCentral
Limits
Updatearchive
GenerateReport
Data forReport
RecentData
Formatted data
Alarm
PatientClinicalDataMonitoring
Local
Patient data
Report
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Ch. 5 29
More refinement
Limits
Formatted data alarm
dataPatient
decode
Check
violations limit
Temperature
Pulse
Pressure
Result
Pressure, pulse…
Format
data clockDate Time
producemessage
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Ch. 5 30
An evaluation of DFDs (1)• Easy to read, but …• Informal semantics
– How to define leaf functions?– Inherent ambiguities
A
C
E
B
F
D
• Outputs from A, B, C are all needed?
• Outputs for E and F areproduced at the same time?
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Ch. 5 31
An evaluation of DFDs (2)
• Control information is absent
BA
Possible interpretations:(a) A produces datum, waits until B consumes it(b)B can read the datum many times without
consuming it(c) a pipe is inserted between A and B
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Ch. 5 32
Formalization/extensions
• There have been attempts to formalize DFDs
• There have been attempts to extend DFDs (e.g., for real-time systems)
.
.
.
Trigger
d1
d2
dn
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Ch. 5 33
UML use-case diagrams
• Define functions on basis of actors and actions
borrow book
return book
library update
librarian
customer
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Ch. 5 34
UML sequence diagrams• Describe how objects interact by
exchanging messages• Provide a dynamic view
Librarian Catalogue
member card + book request membership
OK
book request
book available
book borrowed
time
Customer
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Ch. 5 35
UML collaboration diagrams
• Give object interactions and their order• Equivalent to sequence diagrams
Customer Librarian Catalogue
1: member card + book request
2: membership OK
3: book request
4: book available 5: book borrowed
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Ch. 5 36
Finite state machines (FSMs)
• Can specify control flow aspects• Defined as
a finite set of states, Q;a finite set of inputs, I;a transition function d : Q x I Q(d can be a partial function) a a
b
bc
q
q
q
q
1
20
3
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Ch. 5 37
Example: a lamp
On Off
Push switch
Push switch
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Ch. 5 38
Another example:a plant control system
On Off
High-pressure alarm
High-temperature alarm
Restart
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Ch. 5 39
A refinementPressure signal Temperature signal
Successful recovery
Unsuccessful recovery
OffNormal
Pressure action
OffNormal
Pressure action
Temperature signalTemperature action
Successful recovery
Unsuccessful recovery
Pressure signal
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Ch. 5 40
Classes of FSMs
• Deterministic/nondeterministic• FSMs as recognizers
– introduce final states
• FSMs as transducers– introduce set of outputs
• . . .
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Ch. 5 41
FSMs as recognizers
q
q q q q
q q
q
b
e g i
n
e
n
d
0
1 2 3 4
5 6
f
qf is a final state
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Ch. 5 42
FSMs as recognizers
q <letter>
<letter>
_
<digit>
q q
<letter> Legend: is an abbreviation for a set of arrows
labeled a, b,..., z, A,..., Z, respectively
is an abbreviation for a set of arrows labeled 0, 1,..., 9, respectively <digit>
0 1 2
<letter>
<digit>
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Ch. 5 43
Limitations
• Finite memory• State explosion
– Given a number of FSMs with k1, k2, … kn states, their composition is a FSM with k1 * k2 *… * kn. This growth is exponential with the number of FSMs, not linear (we would like it to be k1 + k2 +… + kn )
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Ch. 5 44
State explosion: an example
Producer
p1
c2
Storage
1
produce
deposit
get
consume
deposit
get get
deposit
p2
Consumer
c1
20
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Ch. 5 45
The resulting FSM
<0, p ,c >
<0, p ,c >
consume
produce
consume
produce
consume
produce
consume
produce
produce produce
consume consume
write
read
write
read
read
write read
write
1
1 2
<0, p , c >
1
2 2
<1, p ,c >
<0, p ,c >
1 1
<1, p ,c>
<1, p ,c >
<1, p ,c >
2 1
1 2
1
2
2 2 <2, p ,c > 2 2
<2, p ,c > 1 2
<2, p ,c > 2 1
<2, p ,c > 1 1
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Ch. 5 46
Petri netsA quadruple (P,T,F,W)
P: places T: transitions (P, T are finite) F: flow relation (F {PT} {TP} ) W: weight function (W: F N – {0} )Properties:(1) P T = Ø(2) P T Ø(3)F (P T) (T P)(4) W: F N-{0}
Default value of W is 1State defined by marking: M: P N
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Ch. 5 47
places
transitions flows
marking
3 weight
P
P
P
t
t
P
t
1
3
1
3
4
6
5
P
P
P
2
5
7
t
t
t
2
4
6
Graphical representation
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Ch. 5 48
Semantics: dynamic evolution
• Transition t is enabled iff pt's input places, M(p) W(<p,t>)
• t fires: produces a new marking M’ in places that are either t's input or output places or both– if p is an input place: M'(p) = M(p) - W(<p,t>)– if p is an output place: M'(p) = M(p) +
W(<t,p>)– if p is both an input and an output place:
M'(p) = M(p) - W(<p,t>) + W(<t,p>)
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Ch. 5 49
Nondeterminism
• Any of the enabled transitions may fire
• Model does not specify which fires, nor when it fires
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Ch. 5 50
Modeling with Petri nets
• Places represent distributed states• Transitions represent actions or
events that may occur when system is in a certain state
• They can occur as certain conditions hold on the states
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Ch. 5 51
P
P
P
t
tP
t
1
3
1
3
4
6
5
P
P
P
2
5
7
t
t
t
2
4
6
P
P
P
t
tP
t
1
3
1
3
4
6
5
P
P
P
2
5
7
t
t
t
2
4
6
P
P
P
t
tP
t
1
3
1
3
4
6
5
P
P
P
2
5
7
t
t
t
2
4
6
P
P
P
t
tP
t
1
3
1
3
4
6
5
P
P
2
5
7
t
t
t
2
4
6
P
(a) (b)
(c) (d)
after (a) either (b) or (c) may occur, and then (d)
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Ch. 5 52
Common cases• Concurrency
– two transitions are enabled to fire in a given state, and the firing of one does nor prevent the other from firing
• see t1 and t2 in case (a)
• Conflict– two transitions are enabled to fire in a given state,
but the firing of one prevents the other from firing• see t3 and t4 in case (d)• place P3 models a shared resource between two processes
– no policy exists to resolve conflicts (known as unfair scheduling)
– a process may never get a resource (starvation)
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Ch. 5 53
How to avoid starvation
P P
P
P P
t t
t t
P P
t t
1
1 2
3
4
5
6
7
4
2
3
6
5
imposes alternation
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Ch. 5 54
A conflict-free net
R
P P
t t
t'
t"
t
t'
t"
t
1
1
3
3
2
2
4
4
56
2 2
this net candeadlock!consider
'42
'31 t,t , t,t
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Ch. 5 55
A deadlock-free net
R
P P
t t
t'
t"
t
t'
t"
t
1
1
3
3
2
2
4
4
56
2 2
2 2
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Ch. 5 56
A case of partial starvation
t
t
1
3
t
t
2
4
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Ch. 5 57
Producer-consumer example (1)
P P
write
produce
C
C
consume
0 1 2
read read
write write
read
1
1
2
2
separate netsfor the subsystems
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Ch. 5 58
Producer-consumer example (2)
C1C2
consume
0 1 2
read
writewrite
read
P1 P2produce
one net for theentire system
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Ch. 5 59
Limitations and extensions
P
channel1 channel2
Token represents a message.You wish to say that the delivery channel dependson contents.How?Petri nets cannot specify selection policies.
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Ch. 5 60
Extension 1assigning values to tokens• Transitions have associated
predicates and functions• Predicate refers to values of tokens
in input places selected for firing• Functions define values of tokens
produced in output places
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Ch. 5 61
Example
PP
P
PP
34
71 4
t t1 2
45
12
3
Predicate P2 > P1 and function P4 := P2 + P1 associated with t1
Predicate P3 = P2 and functions P4 := P3 P2 and P5 := P2 + P3 are associated with t2
The firing of t1 by using <3,7> would produce the value 10 in P4. t2 can then fire using <4, 4>
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Ch. 5 62
Extension 2specifying priorities
• A priority function pri from transitions to natural numbers:
• pri: T N• When several transitions are
enabled, only the ones with maximum priority are allowed to fire
• Among them, the one to fire is chosen nondeterministically
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Ch. 5 63
Extension 3Timed Petri nets
• A pair of constants <tmin, tmax> is associated with each transition
• Once a transition is enabled, it must wait for at least tmin to elapse before it can fire
• If enabled, it must fire before tmax has elapsed, unless it is disabled by the firing of another transition before tmax
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Ch. 5 64
Examplecombining priorities and
time
P P P
t t t
tmin = 1 tmax = 4
tmin = 2 tmax = 3
tmin = 0 tmax = 5
priority = priority = priority = 1 3 2
P
1
1 2
2
3
3
4
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Ch. 5 65
Original message
Message triplication
Message copies
Message copies transmission
tmin = tmax =
tmin = tmax =
tmin = tmax = 0
0
for all three transitions
PC1
PC2
PC3
c1 k1
c2k2
Forwarded message
tvoting1 tvoting2 tvoting3
{
{
{
Precise specificationof message triplication problemCase (1)
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Ch. 5 66
Original message
Message triplication
Message copies
Message copies transmission
tmin = tmax =
tmin = tmax =
tmin = 0 tmax = 0
PC1
PC2
PC3
c1 k1
c2k2
tvoting
Forwarded message
Precise specificationof message triplication problemCase (2)
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Ch. 5 67
Case study
• An n elevator system to be installed in a building with m floors
• Natural language specs contain several ambiguities
• Formal specification using PNs removes ambiguities
• Specification will be provided in a stepwise fashion
• Will use modules, each encapsulating fragments of PNs which describe certain system components
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Ch. 5 68
From informal specs…“The illumination is cancelled when the
elevator visits the floor and is either moving in the desired direction, or ...”
2 different interpretations (case of up call)– switch off as the elevator arrives at the floor
from below (obvious restrictions for 1st and last floor)
– switch off after the elevators starts moving up• in practice you may observe the two cases!
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Ch. 5 69
…more analysis of informal specs
“The algorithm to decide which to service first should minimize the waiting time for both requests.”what does this mean?
• in no other way can you satisfy either request in a shorter time
– but minimizing for one may require longer for the other
• the sum of both is minimal– why the sum?
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Ch. 5 70
Initial sketch of movement
Elevator at floor j
Elevator at floor j + 1
Pushing internal button for floor j + 1
Button illumination
Transfer from floor j to j+1
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Ch. 5 71
Button moduleC P P u s h
S e t
O f f
R e s e t
O n
0 . 1 . . 0 . 0 5 . . 0 . 0 5
0 . . 0
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Ch. 5 72
Elevator position (sketch)F
F DFUF
UF F DF
UF FDF
F
m
m-1
3
2
1
m-1 m-1
3 3
2 2
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Ch. 5 73
Assume j+1hm
Fj+1 UFj+1
Fj"
t Fj'
UPh
On
On
DOWNh
ILBh
DOWNj+1
On
On
On
On
ILBj+1
Fj
t1 t2 t3 t4 t5 t6
t7
t8
t9 t10
t11 t12
UPj+1
More precise description of elevator position
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Ch. 5 74
Switch internal button off
I L B j
F j
S e t
R e s e t
O f fO n0 . . 0
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Ch. 5 75
Switch external button off
Set
x..x Reset
ti'
Fj
Fj'
On Off
UPj
x time needed bya person to enter + pushing button
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Ch. 5 76
Specifying policy
U
U _ D
D
D _ U
D KU K
A “fair” solution:Keep the direction unchanged as long as there are calls that require elevator to go in that direction
[0,0][0,0]
[x,x]
[x,x]
t7, t8, t9 have higher priority than t10, t11, t12
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Ch. 5 77
A general schedulerSCHEDULER
...all transitions
• Each transition has predicate OK(Scheduler)• Token in SCHEDULER stores information about
the state of the system that is useful for scheduling transitions to fire
• The token is “permanent” (it is always reproduced after the firing of any transition)
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Ch. 5 78
Declarative specifications
ER diagrams: semiformal specs
Logic specificationsAlgebraic specifications
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Ch. 5 79
ER diagrams
• Often used as a complement to DFD to describe conceptual data models
• Based on entities, relationships, attributes
• They are the ancestors of class diagrams in UML
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Ch. 5 80
Example
STUDENT
CLASS
ENROLLED_IN
NAME
SEX
AGE
SUBJECT
COURSE_ID
MAX_ENROLLMENT
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Ch. 5 81
Relations
• Relations can be partial• They can be annotated to define
– one to one
– one to many
– many to one
– many to many
A R B
A R B
A R B
A R B
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Ch. 5 82
Non binary relations
D i r e c t o r
P r o j e c t
E m p l o y e e
D e p a r t m e n t
A s s i g n e dP a r t i c i p a t e
H e a d O f
D a t a D u r a t i o n
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Ch. 5 83
Logic specifications
Examples of first-order theory (FOT) formulas:• x > y and y > z implies x > z• x = y y = x• for all x, y, z (x > y and y > z implies x >
z)• x + 1 < x – 1• for all x (exists y (y = x + z))• x > 3 or x < -6
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Ch. 5 84
Specifying complete programs
A property, or requirement, for P is specified as a formula of the type
{Pre (i1, i2,..., in) }P{Post (o1, o2,..., om, i1, i2,..., in)}
Pre: preconditionPost: postcondition
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Ch. 5 85
Example
• Program to compute greatest common divisor{i1 > 0 and i2 > 0}P{(exists z1, z2 (i1 = o * z1 and i2 = o * z2)and not (exists h (exists z1, z2 (i1 = h * z1 and i2 = h * z2) and h > o))}
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Ch. 5 86
Specifying procedures
{n > 0} -- n is a constant valueprocedure search (table: in integer_array; n: in integer;
element: in integer; found: out Boolean);{found (exists i (1 i n and table (i) = element))}
{n > 0 } procedure reverse (a: in out integer_array; n: in integer);{for all i (1 i n) implies (a (i) = old–a (n - i +1))}
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Ch. 5 87
Specifying classes
• Invariant predicates and pre/post conditions for each method
• Example of invariant specifying an array implementing ADT set
for all i, j (1 i length and 1 j length and ij) implies IMPL[i]IMPL[j](no duplicates are stored)
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Ch. 5 88
Specifying non-terminating behaviors
• Example: producer+consumer+buffer• Invariant specifies that whatever has
been produced is the concatenation of what has been taken from the buffer and what is kept in the buffer
input_sequence = append (output_sequence,contents(CHAR_BUFFER))
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Ch. 5 89
A case-study using logic specifications
• We outline the elevator example• Elementary predicates
– at (E, F, T) • E is at floor F at time T
– start (E, F, T, up) • E left floor F at time T moving up
• Rules– (at (E, F, T) and on (EB, F1, T) and F1 > F)
implies start (E, F, T, up)
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Ch. 5 90
States and events
• Elementary predicates are partitioned into– states, having non-null duration
– standing(E, F, T1, T2)» assumption: closed at left, open at right
– events• instantaneous (caused state change occurs at same time)• represented by predicates that hold only at a particular
time instant– arrived (E, F, T)
• For simplicity, we assume• zero decision time• no simultaneous events
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Ch. 5 91
Events (1)
• arrival (E, F, T) – E in [1..n], F in [1..m], T t0, (t0 initial time)
• does not say if it will stop or will proceed, nor where it comes from
• departure(E, F, D, T) – E in [1..n], F in [1..m], D in {up, down}, T t0
• stop (E, F, T) – E in [1..n], F in [1.. m], T t0
• specifies stop to serve an internal or external request
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Ch. 5 92
Events (2)
• new_list (E, L, T) – E in [1..n], L in [1.. m]*, T t0
• L is the list of floors to visit associated with elevator (scheduling is performed by the control component of the system)
• call(F, D, T) – external call (with restriction for 1, N)
• request(E, F, T) – internal reservation
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Ch. 5 93
States
• moving (E, F, D, T1, T2)• standing (E, F, T1, T2)• list (E, L, T1, T2)
– We implicitly assume that state predicates hold for any sub- interval (i.e., the rules that describe this are assumed to be automatically added)• Nothing prevents that it holds for larger
interval
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Ch. 5 94
Rules relating events and states
R1:When E arrives at floor F, it continues to move if there isno request for service from F and the list is empty.If the floor to serve is higher, it moves upward; otherwise it moves downward.
arrival (E, F, Ta) andlist (E, L, T, Ta) and first (L) > F
impliesdeparture (E, F, up, Ta)
A similar rule describes downward movement.
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Ch. 5 95
R2: Upon arrival at F, E stops if F must be serviced (Fappears as first of the list)
arrival (E, F, Ta) andlist (E, L, T, Ta) andfirst (L) = F
impliesstop (E, F,Ta)
R3: E stops at F if it gets there with an empty list
arrival (E, F, Ta) andlist (E, empty, T, Ta)
impliesstop (E, F, Ta)
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Ch. 5 96
R4: Assume that elevators have a fixed time to service a floor. If the list is not empty at the end of such interval, the elevator leaves the floor immediately.
stop (E, F, Ta) andlist (E, L, T, Ta + Dts) andfirst (L) > F,
impliesdeparture (E, F, up, Ta + Dts
R5: If the elevator has no floors to service, it stops until its list becomes nonempty.
stop (E, F, Ta) and list (E, L, Tp, T) andTp > Ta + Dts and list (E, empty, Ta + Dts, Tp) andfirst (L) > F
impliesdeparture (E, F, up, Tp)
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Ch. 5 97
R6: Assume that the time to move from on floor to the next is known and fixed. The rule describes movement.
departure (E, F, up, T)implies
arrival (E, F + 1, T + Dt)
R7: The event of stopping initiates standing for at least Dts.
stop (E, F, T)implies
standing (E, F, T, T + Dts)
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Ch. 5 98
R8: At the end of the minimum stop interval Dts, E remainsstanding if there are no floors to service.
stop (E, F, Ts) andlist (E, empty, Ts + Dts, T)
impliesstanding (E, F, Ts, T)
R9: Departure causes moving.
departure (E, F, D, T)implies
moving (E, F, D, T, T + Dt)
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Ch. 5 99
Control rulesExpress the scheduling strategy (by describing “new_list” events and “list” states)
Internal requests are inserted in the list from current floor to top if the elevator is moving up
External calls are inserted in the list of the closest elevatorthat is moving in the correct direction, or in a standing elevator
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Ch. 5 100
R10: Reserving F from inside E, which is not standing at F,causes immediate update of L according to previous policy
request (E, F, TR) and not (standing (E, F, Ta, TR)) andlist (E, L, Ta, TR) and LF = insert_in_order(L, F, E)
impliesnew_list (E, LF, TR)
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Ch. 5 101
R11: Effect of arrival of E at floor F
arrival (E, F, Ta) and list (E, L, T, Ta) andF = first (L) and Lt = tail (L)
impliesnew_list (E, Lt, Ta)
R12: How list changes
new_list (E, L, T1) and not (new_list (E, L, T2) and T1 < T2 < T3)
implieslist (E, L, T1, T3)
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Ch. 5 102
Verifying specifications
• The system can be simulated by providing a state (set of facts) and using rules to make deductions
standing (2, 3, 5, 7) elevator 2 at floor 3 at least from instant 5 to 7
list(2, empty, 5, 7)
request(2, 8, 7)
new_list(2, {8}, 7)
(excluding other events) departure (2, up, 7 + Dts) arrival (2, 8, 7 + Dts + Dta *(8-3))
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Ch. 5 103
Verifying specifications
• Properties can be stated and proved via deductionsnew_list (E, L, T) and F L
impliesnew_list (E, L1, T1) and F L1 and T1 > T2
(all requests are served eventually)
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Ch. 5 104
Descriptive specs
• The system and its properties are described in the same language
• Proving properties, however, cannot be fully mechanized for most languages
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Ch. 5 105
Algebraic specifications
• Define a heterogeneous algebra • Heterogeneous = more than 1 set• Especially useful to specify ADTs
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Ch. 5 106
Example
• A system for strings, with operations for– creating new, empty strings (operation new)– concatenating strings (operation append)– adding a new character at the end of a
string (operation add)– checking the length of a given string
(operation length)– checking whether a string is empty
(operation isEmpty)– checking whether two strings are equal
(operation equal)
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Ch. 5 107
Specification: syntax algebra StringSpec;introduces
sorts String, Char, Nat, Bool;operations
new: () String;append: String, String String;add: String, Char String;length: String Nat;isEmpty: String Bool;equal: String, String Bool
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Ch. 5 108
Specification: propertiesconstrains new, append, add, length, isEmpty, equal so thatfor all [s, s1, s2: String; c: Char]
isEmpty (new ()) = true;isEmpty (add (s, c)) = false;length (new ()) = 0;length (add (s, c)) = length (s) + 1;append (s, new ()) = s;append (s1, add (s2,c)) = add (append (s1,s2),c);equal (new (),new ()) = true;equal (new (), add (s, c)) = false;equal (add (s, c), new ()) = false; equal (add (s1, c), add (s2, c) = equal (s1,s2);
end StringSpec.
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Ch. 5 109
Example: editor• newF
– creates a new, empty file
• isEmptyF– states whether a file is empty
• addF– adds a string of characters to the end of a file
• insertF– inserts a string at a given position of a file (the rest of
the file will be rewritten just after the inserted string)
• appendF– concatenates two files
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Ch. 5 110
algebra TextEditor;introduces
sorts Text, String, Char, Bool, Nat;operations
newF: () Text;isEmptyF: Text Bool;addF: Text, String Text;insertF: Text, Nat, String Text;appendF: Text, Text Text;deleteF: Text Text;lengthF : Text Nat;equalF : Text, Text Bool;addFC: Text, Char Text;
{This is an auxiliary operation that will be needed to define addF and other operations on files.}
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Ch. 5 111
constrains newF, isEmptyF, addF, appendF, insertF, deleteF so that TextEditor generated by [newF, addFC]for all [f, f1,f2: Text; s: String; c: Char; cursor: Nat]
isEmptyF (newF ()) = true;isEmptyF (addFC (f, c)) = false;addF (f, newS ()) = f;addF (f, addS (s, c)) = addFC (addF (f, s), c);lengthF (newF ()) = 0;lengthF (addFC (f, c)) = lengthF (f) + 1;appendF (f, newF ()) = f;appendF (f1, addFC (f2, c)) =
addFC (appendF (f1, f2), c);equalF (newF (),newF ()) = true;equalF (newF (), addFC (f, c)) = false;equalF (addFC (f, c), new ()) = false; equalF (addFC (f1, c1), addFC (f2, c2) = equalF (f1, f2) and equalC (c1, c2);insertF (f, cursor, newS ()) = f;((equalF (f, appendF (f1, f2)) and (lengthF (f1) = cursor - 1))
implies equalF (insertF (f, cursor, s), appendF (addF (f1, s), f2))) = true;
end TextEditor.
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Ch. 5 112
Requirements for a notation
• Ability to support separation of concerns– e.g., separate functional specs from
• performance specs • user-interface specs• …
• Support different views
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Ch. 5 113
Example of viewsdocument production
User
Formatting options
Predefined Text skeletons
Customers
Customer data (name, type of document)
Print Document
Predefined Formats
Document production
data flow view (1)
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Ch. 5 114
Control flow view (2)
Search in Customers
Get user name
Get other data from the data base
Get other relevant data from user interaction
Get appropriate text skeletons from predefined text library
Print document
Compose the document by choosing formatting options (this involves interaction with the user and access to the Formats data base)
(b)
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Ch. 5 115
UML notations
• Class diagrams– describe static architecture in terms
of classes and associations– dynamic evolution can be described
via Statecharts (see later)• Activity diagrams
– describe sequential and parallel composition of method executions, and synchronization
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Ch. 5 116
An activity diagram
G
C
D
F
B
[c1] [c3]
[c2] [c4]
(2)
(1)
end
E A
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Ch. 5 117
Building modular specifications
• The case of algebraic specifications– How to combine algebras taken from
a library– How to organize them in a hierarchy
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Ch. 5 118
Algebras used by StringSpecalgebra BoolAlg;
introducessorts Bool;operations
true () Bool;false () Bool;not : Bool Bool;and: Bool, Bool Bool;or: Bool, Bool Bool;implies: Bool, Bool Bool; : Bool, Bool Bool;
constrains true, false, not, and, or, implies, so thatBool generated by [true, false] for all [a, b: Bool]
not (true) = false;not (false) = true;a and b = not (not (a) or not (b));a implies b = not (a) or b;
…end BoolAlg.
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Ch. 5 119
Algebras used by StringSpec (cont.)
algebra NatNumb;introduces
sorts Nat, Bool;operations
0: () Nat;Succ: Nat Nat;+ : Nat, Nat Nat;- : Nat, Nat Nat;= : Nat, Nat Bool;
…constrains 0, Succ, +, -, =,..., so thatNatNumb generated by [0, Succ] …end NatNumb.
algebra CharAlg;introduces
sorts Char, Bool;operations
‘a’: ()® Char;‘b’ : ()® Char;…
end CharAlg.
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Ch. 5 120
StringSpec revisited
algebra StringSpec;imports BoolAlg, NatNumb, CharAlg;introduces
sorts String, Char, Nat, Bool;operations
new: () String;…
end StringSpec.
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Ch. 5 121
Incremental specification of an ADT
• We want to target stacks, queues, sets
• We start from "container" and then progressively specialize it
• We introduce another structuring clause– assumes
• defines inheritance relation among algebras
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Ch. 5 122
Container algebraalgebra Container;imports DataType, BoolAlg, NatNumb;introduces
sorts Cont;operations
new: () Cont;insert: Cont, Data Cont;
{Data is the sort of algebra DataType, to which elements to be stored in Cont belong}
isEmpty: Cont Bool;size: Cont Nat;
constrains new, insert, isEmpty, size so thatCont generated by [new, insert]for all [d: Data; c: Cont]
isEmpty (new ()) = true;isEmpty (insert (c, d)) = false;size (new ()) = 0;
end Container.
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Ch. 5 123
Table specializes Containeralgebra TableAlg;
assumes Container;introduces
sorts Table;operations
last: Table Data;rest: Table Table;equalT : Table, Table Bool;delete: Table, Data Table;
constrains last, rest, equalT, delete, isEmpty, new, insert so thatfor all [d, d1, d2: Data; t, t1, t2: Table]
last (insert (t, d)) = d;rest (new ()) = new ();rest (insert (t, d)) = t;equalT (new (), new ()) = true;equalT (insert (t, d), new ()) = false;equalT (new (), insert (t,d)) = false;equalT (t1,t2) = equalD(last (t1), last (t2)) and equalT (rest (t1),rest (t2));delete (new (), d) = new ();delete (insert (t,d),d) = delete (t, d);if not equalD(d1, d2) thendelete (insert (t, d1), d2) = insert (delete (t, d2), d1);
end TableAlg.
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Ch. 5 124
Queue specializes Containeralgebra QueueAlg;
assumes Container;introduces
sort Queue;operations
last: Queue Data;first: Queue Data;equalQ : Queue , Queue Bool;delete:Queue Queue;
constrains last, first, equalQ, delete, isEmpty, new, insert so thatfor all [d: Data; q, q1, q2: Queue]
last (insert (q, d)) = d;first (insert (new(), d) = dfirst (insert (q, d)) = if not isEmpty (q) then first (q);equalQ (new (), new ()) = true;equalQ (insert (q, d), new ()) = false;equalQ (new (), insert (q, d)) = false;equalQ (insert (q1, d1), insert (q2, d2)) = equalD (d1, d2) and equalQ (q1,q2);delete (new ()) = new ();delete (insert (new (), d)) = new ();if not equalQ (q, new ()) thendelete (insert (q,d)) = insert (delete (q), d);
end QueueAlg.
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Ch. 5 125
A graphical view
Table Algebra
Queue Algebra
Bool Algebra
Container Algebra
Nat Algebra DataType
Algebra
Legend: imports relation assumes relation
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Ch. 5 126
A richer hierarchy
Table Algebra
Queue Algebra
Bool Algebra
Container Algebra
Nat Algebra
DataType Algebra
Sorted DataTypeArray
Algebra
Sorted TableTreeMultiset
Set
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Ch. 5 127
From specs to an implementation
• Algebraic spec language described so far is based on the "Larch shared language"
• Several "interface languages" are available to help transitioning to an implementation– Larch/C++, Larch/Pascal
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Ch. 5 128
Using Larch/Pascal for StringSpec
type String exports isEmpty, add, append,...based on Boolean, integer, character
function isEmpty (s: String) : Booleanmodifies at most [ ] {i.e., it has no side effects};
procedure add (var s : String; c : char)modifies at most [s]{modifies only the string parameter s};
function length (s: String) : integermodifies at most [ ] ;
procedure append (var s1, s2, s3: string)modifies at most [s3]{only s3, the result of the operation, is modified};
end StringSpec.
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Ch. 5 129
Modularizing finite state machines
• Statecharts do that• They have been incorporated in
UML• They provide the notions of
– superstate– state decomposition
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Ch. 5 130
Sequential decomposition--chemical plant control
example--
RecoveryFailure
Normal
Recovery
Pressure Action
Temperature Action
Recovery Identification
AnomalyDetection RecoverySuccess
Press
Done Done Temp
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Ch. 5 131
Parallel decomposition
Idle
ConcurrentWork
P1
P2
produce
write
C1
C2
read
consume
1
2
0
write write
read
read
start stop
Consumer Producer Buffer
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Ch. 5 132
Object state diagram using Statecharts
Empty NotEmptyy
Top
Push(item)
Pop[stack contains more than 1 item]
Push(item)
Pop[stack contains 1 item]
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Ch. 5 133
Modularizing logic specifications: Z
• System specified by describing state space, using Z shemas
• Properties of state space described by invariant predicates– predicates written in first-order logic
• Operations define state transformations
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Ch. 5 134
The elevator example in Z
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Ch. 5 135
Complete state spaceattempt #1
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Ch. 5 136
Complete state spaceattempt #2
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Ch. 5 137
Complete state spacefinal
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Ch. 5 138
Operations(1)
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Ch. 5 139
Operations(2)
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Ch. 5 140
Specifications for the end-user
• Specs should be used as common reference for producer and user
• They help removing ambiguity, incompleteness, …
• Can they be understood by end-user?– They can be the starting point for a
prototype– They can support some form of
animation (e.g., see Petri nets)
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Ch. 5 141
Conclusions (1)• Specifications describe
– what the users need from a system (requirements specification)
– the design of a software system (design and architecture specification)
– the features offered by a system (functional specification)
– the performance characteristics of a system (performance specification)
– the external behavior of a module (module interface specification)
– the internal structure of a module (internal structural specification)
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Ch. 5 142
Conclusions (2)
• Descriptions are given via suitable notations– There is no “ideal” notation
• They must be modular• They support communication and
interaction between designers and users