Specification From Examples

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Specification From Examples

• Julia Johnson
• Dept. of Math Computer Science
• Laurentian University
• Sudbury, Ontario
• Canada

2
Problem

• To describe system characteristics by providing
  examples of systems that exhibit those
  characteristics.

3
Outline

• Problem Statement
• Criticism of Existing Solutions
• Suggested Solution
• 3.1 Rough Sets
• 3.2 Strength of a Rule
• 3.3 Rough Mereology
• 3.4 RM System Specification
• Conclusions

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ag

ag2
agm
ag1



ag21
ag20
ag11
ag1n
ag12
agm1
agmp
5
M
B
L
6
µB (B3,B1)
?B .25
B1
B3
µL (L3,L2)
?L .4
L2
L3
7
µM (C5,C1)
C1
C5
?M .14
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Rough Mereology
Mereology Theory of Part of relation,
Lesniewski Rough Mereology Theory of Relation
Part of to a degree, Polkowski
Skowron Applications of Rough Mereology
Control Skowron Polkowski 1994 Warsaw
Politecnica Building Poitr 1998-99 Polish
Academy of Science Scheduling
Johnson 1998-99 University of Regina/University
of Waterloo µ (x,y) is read
the degree in which x is a part of y -the rough
inclusion function
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For each construction of objects from
sub-objects, we form a vector, ?B
?L ?M Where if M1 O(B1L1) and
M2 O(B2L2) Then ?B
µB(B1,B2) ?L µL(L1,L2) ?M
µM(M1,M2)
M2 is constructed from B2 and L2
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The vector means If µB(B1,B2) gt ?B
(B1 is part of B2 to degree at least ?B) And
µL(L1,L2) gt ?L (L1 is part of L2 to degree
at least ?L) Then µM(M1,M2) gt ?M (M1 is part
of M2 to degree at least ?M)
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Rough Mereology
µB (B3,B1)
?B .25
B1
B3
µL (L2,L2)
?L 1
L2
L2
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µM (C4,C1)
C1
C4
?M .28
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Some Properties of µ

• (A) µ(x,y) ? 0,1
• (B) µ(x,x) 1
• If µ(x,y) 1 then µ(z,y) gt µ(z,x) for each
  object z
• A null object is any object n which satisfies
• (D) µ(n,y) 1 for every object y

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?L
?M
f
?B

• We wish to learn functions f from a set of
  vectors.

?B1 ?B2 ?B3 . . . ?Bn
?L1 ?L2 ?L3 . . . ?Ln
?M1 ?M2 ?M3 . . . ?Mn
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Assume

• C1 and C2 are cost specifications
• D1 and D2 are design specifications
• M1 and M2 provide maintenance requirements
• C1 O(D1,M1) , C2 O(D2,M2) e.g. C1
  results from design maintenance requirements
  specified by D1 and M1, respectively.
• ?Design µ(D1,D2) , ?Maint µ(M1,M2) ,
  ?Cost µ(C1,C2)

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Back to the Problem at Hand

• To describe system characteristics by providing
  examples of systems that exhibit those
  characteristics.
• To determine system cost by providing examples
  of systems whose design, maintenance and overall
  costs are known.

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• Suppose we know the following

Specs Designi and Designj similar to degree at
least ?Design , i,j not necessarily
distinct Maintenance requirements Maintk and
Maintq similar to degree at least ?Maint ,
possibly kq Cost1 and Cost2 , respectively, of
the two systems O(Design1, Maint1) and O(Design2,
Maint2) similar by at least ?Cost.
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?Design
?Cost
f
?Maint

• We wish to learn function f from a set of
  vectors.

?Design1 ?Design2 ?Design3 . .
. ?Designn
?Maint1 ?Maint2 ?Maint3 . .
. ?Maintn
?Cost1 ?Cost2 ?Cost3 . . .
?Costn
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Table 1 Criteria Inferred from Application Data
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• (ResponseTime, slow) and (Throughput, low)
• (Acceptable, no),
• (ResponseTime, fast) and (Memory, medium)
• (Acceptable, yes),
• (Throughput, high) and (Memory, large)
• (Acceptable, no).
• Uncertain (or possible) rules are
• (ResponseTime, fast) and (Throughput, high)
• (Acceptable, yes),
• (ResponseTime, fast) and (Throughput, high)
• (Acceptable, no).

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Table 3 Rough Inclusion for Table 1
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Table 2 Criteria Dictated by the Customer
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Table 4 Rough Inclusion for Table 2
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Table 5 Maintenance Criteria
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Table 6 Rough Inclusion for Table 5
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Table 7 Partial List of Arguments for µ
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Table 8 Partial List of Arguments for ?
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RM acceptable degree

• What? How?

?
µ (X, Y) threshold vectors
composition of objects agents message passing
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• What? How?

Simplicity user satisfaction
learnability ease of use
comprehensibility user - friendliness
µ (X, Y) threshold vectors
composition of objects agents message passing
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Summary Conclusions

• Our objective is to describe system
  characteristics such as user friendliness by
  providing examples of systems that exhibit such
  characteristics.
• The computer recognizes a pattern and generates
  rules for what a user friendly system, for
  example, would be.
• This is possible because computers are able to
  provide imprecise solutions to problems.
• We have demonstrated the feasibility of applying
  rough sets/rough mereology to the problem of
  systems requirements systems.

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?L
?M
f
?B

• We wish to learn functions f from a set of
  vectors.

?B1 ?B2 ?B3 . . . ?Bn
?L1 ?L2 ?L3 . . . ?Ln
?M1 ?M2 ?M3 . . . ?Mn