Logic Coverage Exercise

1
Logic Coverage Exercise

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
2
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,2,3,5gt
3
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,2,3,5gt

4
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,2,3,5gt (Agt0) V (Blt0) ?

5
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,2,3,5gt (Agt0) V (Blt0) ?

6
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,2,3,5gt (Agt0) V (Blt0) ?

(A5B0) V ((A5)B0)
7
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,3,4,5gt
8
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,3,4,5gt

9
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,3,4,5gt (A0) ? (B0) ?

10
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,3,4,5gt (A0) ? (B0) ?

11
Logic Coverage Exercise (a)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

1
2
3
4
5
Path condition for lt1,3,4,5gt (A0) ? (B0) ?
(ABgt0)
? (ABlt0)
12
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition
comp. cond. path
1
2
3
4
5
test case 1 A1, B3
13
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition
comp. cond. path
1
2
3
4
5
test case 1 A1, B3
14
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

T
F
statement branch condition
comp. cond. path
1
2
3
4
5
test case 1 A1, B3
15
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

T
F
statement branch condition
comp. cond. path
1
2
3
4
5
test case 1 A1, B3
16
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

T
F
statement branch condition
comp. cond. path
1
2
3
T
F
4
5
test case 1 A1, B3
17
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

T
F
statement branch condition
comp. cond. path
1
2
3
T
F
4
5
test case 1 A1, B3
18
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition
comp. cond. path
T
F
1
2
3
T
F
4
5
test case 1 A1, B3 test case 2 A-1, B-5
19
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition
comp. cond. path
T, F
F, T
1
2
3
T
F
4
5
test case 1 A1, B3 test case 2 A-1, B-5
20
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition
comp. cond. path
T, F
F, T
1
2
3
T, F
F, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
21
Logic Coverage Exercise (b)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition ? comp.
cond. path
T, F
F, T
1
2
3
T, F
F, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
22
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition ? comp. cond.
path
T, F
F, T
1
2
3
T, F
F, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1
23
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition ? comp. cond.
path
T, F, T
F, T, T
1
2
3
T, F
F, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1
24
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition ? comp. cond.
path
T, F, T
F, T, T
1
2
3
T, F, T
F, T, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1
25
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement branch condition ? comp. cond.
path
T, F, T
F, T, T
1
2
3
T, F, T
F, T, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1
26
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement ? branch condition ? comp. cond.
path
T, F, T
F, T, T
1
2
3
T, F, T
F, T, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1
27
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement ? branch condition ? comp. cond.
path
T, F, T
F, T, T
1
2
3
T, F, T
F, T, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1 test case 4 A0,
B0
28
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement ? branch condition ? comp. cond.
path
T, F, T, F
F, T, T, F
1
2
3
T, F, T
F, T, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1 test case 4 A0,
B0
29
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement ? branch condition ? comp. cond.
path
T, F, T, F
F, T, T, F
1
2
3
T, F, T
F, T, T
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1 test case 4 A0,
B0
30
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

statement ? branch condition ? comp. cond.
path
T, F, T, F
F, T, T, F
1
2
3
T, F, T, F
F, T, T, F
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1 test case 4 A0,
B0
31
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

T, F, T, F
F, T, T, F
statement ? branch ? condition ? comp.
cond. path
1
2
3
T, F, T, F
F, T, T, F
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1 test case 4 A0,
B0
32
Logic Coverage Exercise (c)

• 1. input(A,B)
• if (Agt0) OR (Blt0) then
• A A5
• end_if
• 3. if (ABgt0) AND (ABlt0) then
• 4. B AB
• end_if
• 5. output(A,B)

T, F, T, F
F, T, T, F
statement ? branch ? condition ? comp.
cond. ? path
1
2
3
T, F, T, F
F, T, T, F
4
5
test case 1 A1, B3 test case 2 A-1, B-5
test case 3 A3, B-1 test case 4 A0,
B0
33
While Loop Verification Exercise

• y gt 0
• z x
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

34
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

35
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• initialization PgtI

36
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• initialization PgtI
• Does (ygt0 ? zx ? ny) ? (zx(y1-n) ? n1)?

37
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• initialization PgtI
• Does (ygt0 ? zx ? ny) ? (zx(y1-n) ? n1)?
• Yes. Clearly, if zx and ny, it follows that
• zx(y1-n)
• Similarly, if nygt0, it follows that n1.

38
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• preservation I?b s I

39
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• preservation I?b s I
• zx(y1-n) ? n1 ? ngt1

40
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• preservation I?b s I
• zx(y1-n) ? n1 ? ngt1 zx(y1-n) ? ngt1

41
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• preservation I?b s I
• zx(y1-n) ? n1 ? ngt1 zx(y1-n) ? ngt1
• z zx

42
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• preservation I?b s I
• zx(y1-n) ? n1 ? ngt1 zx(y1-n) ? ngt1
• z zx
• zx(y2-n) ? ngt1

43
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• preservation I?b s I
• zx(y1-n) ? n1 ? ngt1 zx(y1-n) ? ngt1
• z zx
• zx(y2-n) ? ngt1
• n n-1

44
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• preservation I?b s I
• zx(y1-n) ? n1 ? ngt1 zx(y1-n) ? ngt1
• z zx
• zx(y2-n) ? ngt1
• n n-1
• zx(y2-(n1)) ? (n1)gt1

45
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• preservation I?b s I
• zx(y1-n) ? n1 ? ngt1 zx(y1-n) ? ngt1
• z zx
• zx(y2-n) ? ngt1
• n n-1
• zx(y2-(n1)) ? (n1)gt1
• zx(y1-n) ? n1 I

46
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• finalization (I?b) gt Q

47
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• finalization (I?b) gt Q
• Does (zx(y1-n) ? n1 ? n1) ? zxy?

48
While Loop Verification Exercise

• y gt 0 I z x(y1-n)
• z x ? n 1
• n y
• while ngt1 do
• z zx
• n n-1
• end_while
• z xy

• finalization (I?b) gt Q
• Does (zx(y1-n) ? n1 ? n1) ? zxy?
• Yes, since (zx(y1-n) ? n1 ? n1) ? (n1?
  zx(y1-1)) ? zxy

49
While Loop Verification Exercise

• n 0
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

50
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

51
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• initialization PgtI

52
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• initialization PgtI
• Does (n0 ? p1 ? k0) ? (p2k? kn)?

53
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• initialization PgtI
• Does (n0 ? p1 ? k0) ? (p2k? kn)?
• Yes. Clearly, if p1 and k0, it follows that
• p2k
• Similarly, if n0 and k0 it follows that kn.

54
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• preservation I?b s I
• p2k? kn ? kltn

55
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• preservation I?b s I
• p2k? kn ? kltn p2k? kltn

56
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• preservation I?b s I
• p2k? kn ? kltn p2k? kltn
• p 2p

57
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• preservation I?b s I
• p2k? kn ? kltn p2k? kltn
• p 2p
• p2k1? kltn

58
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• preservation I?b s I
• p2k? kn ? kltn p2k? kltn
• p 2p
• p2k1? kltn
• k k1

59
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• preservation I?b s I
• p2k? kn ? kltn p2k? kltn
• p 2p
• p2k1? kltn
• k k1
• p2k? kn I

60
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• finalization (I?b) gt Q

61
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• finalization (I?b) gt Q
• Does (p2k? kn ? kn) ? p2n?

62
While Loop Verification Exercise

• n 0 I p 2k? k n
• p 1
• k 0
• while kltn do
• p 2p
• k k1
• end_while
• p 2n

• finalization (I?b) gt Q
• Does (p2k? kn ? kn) ? p2n?
• Yes, since (p2k? kn ? kn) ? (kn ? p2k) ?
  p 2n