Logical Coverage (ch. 3)
Course Software Testing & Verification
2013/14
Wishnu Prasetya
How to test this?
m(p) { if (!member(p)
||
(p.age>17 && !admin(p))
S0 ;
else S1 ;
}
m(p) { if (!member(p)) goto L0
if (!p.age17) goto L1
if (!admin(p)) goto L1
L0: S0 ;
L1: S1 ;
}
The code to the left is often transformed to the
“short-circuiting” version above by a compiler.
• Edge coverage or even full path coverage on the original code
may feel inadequate.
• The short circuit version do have more paths (4), but requires
us to know what the compiled code is.
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The obvious ones
Member(p)  (p.age>17  admin(p))
a  (b  c)
• (C3.12) Predicate coverage
• (C3.13) Clause coverage; Note that AO call “Member(p)” a
clause instead of e.g. proposition. |TR| = ... ?
– feels too weak
– strangely does not subsume predicate coverage, nor the
other way around.
• (C3.14) All combinations  explodes if the formula is large
3
Using active clauses
a  (b  c)
a
b
c
0
0
0
1
0
0
1
0
0
1
1
0
1
1
0
duplicates
1
1
1
• Major and minor clause.
• (Def 3.42) a major clause c of p determines p if it is possible to
make it active : to assign values to the minor clauses so that
changing the value of c changes the value of p.
• (Def3.43, Active Clause Covrg, ACC) For each major clause c
which determines p, TR includes the cases c=1 and c=0. Minor
clauses must be given values that would make p active.
• |T|  2*n, but duplicates can be removed, ending up with |T|
 n+1, if n is number of clauses. Sounds good, but....
4
You need to fix the def. of ACC!
a  b
a
b
ab
0
0
1
1
1
1
• Active Clause Covrg (ACC): for each major clause c which
determines p, TR includes the cases c=1 and c=0; etc  need
refinement!
– does not specify if you should use the same values of minor
clauses or not.
– does not require p to actually change when you change c
– does not subsume predicate covrg. See example above.
5
Correlated and Restricted ACC
a  b
a
b
ab
0
0
1
1
0
0
0
0
1
0
1
0
• (C3.15) General ACC: for each major clause c which determines p,
TR includes the cases c=1 and c=0. Minor clauses must be given
values that would make p active. The values of the minors in both
cases do not have to be the same.  actually this is just the
same as ACC.
• (C3.16) Correlated ACC: as above, but for those two cases of c,
one must cause p to be 1 and the other 0.
• (C3.17) Restricted ACC: as in ACC but the values of the minors
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chosen in both cases of c must be the same.
Correlated vs Restricted ACC
a  (b c)
a:
b:
c:
•
•
•
•
isStudent(p)  ( isMasterStudent(p)  c )
a
b
c
whole
a
b
c
whole
0
0
1
0
0
1
0
0
1
1
0
1
1
1
0
1
1
0
0
0
1
0
0
0
1
1
0
1
1
1
0
1
1
0
0
0
1
0
0
0
1
0
1
1
1
0
1
1
a:
b:
c:
Left shows tests that cover CACC; after removing duplicates we get |T| = 4.
Right covers RACC; after removing duplicates |T| = 4.
RACC subsumes CACC
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Keep in mind some combinations may actually be impossible
Inactive-clause approach
• ACC-variants makes sure that each situation where a clause
affects the whole predicate is tested. Inactive clause is the
opposite.
• A major clause c is in inactive if it is possible to assign values
to minor clauses so that changing the value of c won’t change
the value of p.
• (Def3.44, Inactive Clause Coverage, ICC) For each major clause
c, if it is possible to make it inactive, TR includes: (1) a test
where c=0 and p=1, (2) a test where c=1 and p=1, (3) a test
where c=0 and p=0, (4) a test where c=1 and p=0. Minor
clauses must be given values that make c inactive.
• (C3.18, GICC) if the values of minor clauses in 1,2,3,4 do not
have to be the same.
• (C3.19, RICC) if the values of minors in 1/2 and 3/4 have to be 8
te same.
Overview
Combinatorial
Clause Coverage
COC
Restricted Active
Clause Coverage
RACC
Restricted Inactive
Clause Coverage
RICC
Correlated Active
Clause Coverage
CACC
General Inactive
Clause Coverage
GICC
General Active
Clause Coverage
GACC
Clause
Coverage
CC
Predicate
Coverage
PC
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Finding values to set a clause
active/inactive
• In ACC-variants we need to figure out how to assign
values to minor clauses to make a major clause c
determines p. Is there an algorithm for this!? 
yes; simple, namely by solving (finding values of the
variables that make the formula true) :
pc=0  pc=1
• Equivalently, solve: pc=0 xor pc=1 = 1
• How to solve such a formula? You can imagine how;
but if p is large, use e.g. a SAT solver.
10
Constructing the TR
• We’ll take CACC as an example. Given a p, and a
constraint B over the clauses (e.g. isMasterStd(p) 
isStd(p)).
• For each clause c, solve these:
(1) Bc=1  (pc=1 = 1)  ( pc=0 xor pc=1)
(2) Bc=0  (pc=0 = 0)  ( pc=0 xor pc=1)
Add the solutions to TR.
• Remove duplicates from TR.
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Going back to program
p.pt+1 < 1000
awardPt(p) {
p.pt ++ ;
if (birthday(p) && p.pt <1000 ) notify(p) ;
}
• Apply a logic-coverage at each decision point. This produces a
TR for each. But :
– we still have to calculate the corresponding program’s
inputs  calculate it by hand?
– Can we automate the calculation?  what to do with
loops/recursions?
12
Disjunctive Normal Form (DNF)
((a  c)  b)  (a  c)
(a  c)  (b  c)
• Rather than investigating each clause of p, we can
opt to investigate different “logical scenarios” that
affect p.
• Every proposition logic formula f can be rewritten to
an equivalent DNF. A DNF consists of implicants,
connected by . Each implicant consists of literals
connected by . A literal is a clause, or a negation of
a clause.
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Implicants of p and p
p = ab  bc
 p =  (ab  bc) = b  ac
• We’ll write p to abbreviate p
• Implicants of p  scenarios leading to p becomes
true
• Implicants of p  scenarios leading to p become
false.
• Idea: cover all those scenarios.
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Implicant Coverage
p = ab  bc
p =  (ab  bc) = b  ac
implicant
a
b
c
ab
1
1
0
bc
1
1
0
b
0
0
1
ac
0
0
1
• (C3.20, IC) Let p be a DNF, and p be a DNF of p; for
each implicant i in p and p TR includes the case that i
= 1.
• IC subsumes pred. cov (PC); but not necessarily
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subsumes any ACC variant.
Unique true point
p = abc  abc  bc
p = ab  ac  bc
• (p140) a unique true point of a DNF p wrt implicant i is values
assignment that makes i = 1 and other implicants (of p) = 0.
• (C3.21, Unique True Point Covrg, UTPC) Given minimal DNF p
and its negation p; for every implicant i in p and p the TR
includes a unique true point for i.
• (p140) Prime implicant, e.g. abc in red; abc is not.
• (p140) Redundant implicant, e.g ab in blue.
• A DNF is minimal if every implicant is prime and not
redundant. In such a DNF, there exists a test satisfying an
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implicant that does not satisfy the others.
IC vs UTPC
p = ab  bc
p = b  ac
implicant
a
b
c
ab
1
1
0
bc
1
1
0
b
0
0
1
ac
0
0
1
implicant
a
b
c
ab
1
1
1
bc
0
1
0
b
0
0
0
ac
0
1
1
|T| = 2
|T| = 4
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Converting to DNF
ab
a
b
0
1
0
1
1
1
1
0
c
00
01
11
10
0
0
1
1
1
1
1
0
1
1
• Rewriting, using Boolean algebra
• For small number of clauses we can also use Karnaugh map
(p144 -145). Important property: the combinations that
belong to two adjacent cells, differ only in one bit.
(“adjancent” in donut sense.) This allows simplication over
those cells.
• Karnaugh gives you minimal DNF.  if you make sure that
every group has at least one non-shared cell.
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Near False Point
• Not suprisingly, UTPC does not subsume GACC  it just
disregard clauses.
• Let p be a DNF with implicant i. Let c be a clause in i. A nearfalse point is a values assignment that makes p=0, but if we flip
the value of c, it will make i=1 (thus also p).
p = ab  bc
Near-false points for a in ab :
011
Flipping the value of a :
111  makes ab true (and p too)
• NFP gives us a concept of how a clause influences an implicant.
• Hey, this 111 is actually also a UTP of ab !
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Near False Point
p = ab  bc
Near-false points for a in ab :
011
Flipping the value of a :
111  makes ab true (and p too)
is actually also a UTP of ab
• (C3.22, Corresponding Unique True Point and Near False Point Pair
Covrg) Given a minimalDNF p, for every clause c and implicant i, TR
contains a unique true point for i and a near-false point wrt c and i,
such that these points only differ in the value of c.
• Does not subsume UTPC (simply because you dont have the DNF
for p) , but it subsumes RACC.
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UTPC vs CUTPNFP
• How strong are they compared to each other?
• Well, we just said they are unrelated.
• Still... let’s define a new metric: their ability to
reveal certain types of faults  let’s take
hypothetical faults you can make when you
manually write down a DNF
21
Some mistakes when coding down a DNF
•
•
•
•
•
•
•
•
•
ENF: Expression Negation Fault
TNF: Term/Implicant Neg. Fault
TOF: Term Omission Fault
LNF: Literal Negation Fault
LRF: Literal Reference Fault
LOF: Literal Omission Fault
LIF: Literal Insertion Fault
ORF+: Operator Reference Fault
ORF*: Operator Reference Fault
f = ab+c
f = ab+c
f = ab+c
f = ab+c
f = ab + bcd
f = ab + c
f = ab + c
f = ab + c
f = ab + c
f’ = ab+c
f’ = ab+c
f’ = ab
f’ = ab+c
f’ = ad + bcd
f’ = a + c
f’ = ab + bc
f’ = abc
f’ = a + b + c
Key idea: fault classes are related to testing. A test case “reveals”
a fault if it results f’ is different than the expected result on f.
A test-case that reveals certain faults may reveal additional faults.
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Literal Insertion
Fault
LIF
Term Omission
Fault
TOF
Operator
Reference Fault
ORF+
Literal Reference
Fault
LRF
Literal Negation
Fault
LNF
Literal Omission
Fault
LOF
Operator
Reference Fault
ORF*
Term Negation
Fault
TNF
Expression
Negation Fault
ENF
UTPC can detect TOF. CUTPNFP detects TOF as well, and detects LOF too.
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