# Domain testing examples

This chapter follows a problem-based approach. We first show a program requirement, and then, show how we would apply equivalent class analysis and boundary testing.

We will follow a common strategy when applying domain testing, highly influenced by how Kaner et al. do:

2. We identify the input and output variables in play, together with their types, and their ranges.
3. We identify the dependencies (or independence) among input variables, and how input variables influence the output variable.
4. We perform equivalent class analysis (valid and invalid classes).
5. We explore the boundaries of these classes.
6. We think of a strategy to derive test cases, focusing on minimizing the costs while maximizing fault detection capability.
7. We generate a set of test cases that should be executed against the system under test.

See the videos for detailed explanations. See also the JUnit test cases we implemented for them in https://github.com/sttp-book/code-examples/tree/master/src/test/java/tudelft/domain.

## Exercise 1: The Sum Of Integers

A program receives two numbers and returns the sum of these two integers. Numbers are between 1 inclusive and 99 inclusive.

Variables Types Ranges
X - First number Integer [1, 99]
Y - Second number Integer [1, 99]
Sum - Output Integer [1, inf]

### Dependency among variables:

• X and Y are independent (X doesn't influence the range of Y, and vice-versa).
• X and Y are used to calculate Sum.

### Equivalence partitioning / Boundary analysis

Variable Equivalence classes Invalid classes Boundaries
X [1, 99] < 1 valid 1 | 0 invalid
> 99 99 | 100
Y [1, 99] < 1 1 | 0
> 99 99 | 100

### Strategy

• Variables are independent (X does not affect the range of Y, and vice-versa).
• 7 tests for X (3 partitions + 4 boundaries), in point for Y.
• 7 tests for Y (3 partitions + 4 boundaries), in points for X.
• Total of 14 tests

In-point X = 50, In-point Y = 50.

Test cases X Y Sum Remark
T1 50 50 100 x ∈ [1, 99], y remains fixed for T1-T7
T2 -100 50 invalid x < 1
T3 250 50 invalid x > 99
T4 0 50 invalid x = 0
T5 1 50 51 x = 1
T6 99 50 149 x = 99
T7 100 50 invalid x = 100
T8 50 50 100 y ∈ [1, 99], x remains fixed for T8-T14
T9 50 -100 invalid y < 1
T10 50 250 invalid y > 99
T11 50 0 invalid y = 0
T12 50 1 51 y = 1
T13 50 99 149 y = 99
T14 50 100 invalid y = 100

Questions:

• Do we need T3 and T7? Or are they testing the same thing (i.e. > 99)? (Same applies to T10 and T14.) If we remove one of them, we end up with 12 tests.
• What would change if the requirement had something like sum < 167 ?

## Exercise 2: The Sum Of Integers, part 2

A program receives two numbers and returns the sum of these two integers. Numbers are between 1 inclusive and 99 inclusive. Final sum should be <= 165.

Variables Types Ranges
X - First number Integer [1, 99]
Y - Second number Integer [1, 99]
Sum - Output Integer [0, 165]

### Dependency among variables:

• X and Y are independent (X doesn't influence the range of Y, and vice-versa).
• X and Y are used to calculate Sum.
• X + Y <= 165

### Equivalence partitioning / Boundary analysis

Variable Equivalence classes Invalid classes Boundaries
X [1, 99] < 1 valid 1 | 0 invalid
> 99 99 | 100
Y [1, 99] < 1 1 | 0
> 99 99 | 100
Sum [0, 165] > 165 165 | 166

### Strategy

• Variables are independent (X does not affect the range of Y, and vice-versa).
• 7 tests for X (3 partitions + 4 boundaries), in point for Y,
• 7 tests for Y (3 partitions + 4 boundaries), in points for X.
• 2 tests for the boundary on Sum.
• Total of 16 tests

In-point X = 50 (taking into consideration that X <= 165 - Y), In-point Y = 50 (taking into consideration that Y <= 165 - X).

Test cases X Y Sum Remark
T1 50 50 100 x ∈ [1, 99], y remains fixed for T1-T7
T2 -100 50 invalid x < 1
T3 250 50 invalid x > 99
T4 0 50 invalid x = 0
T5 1 50 51 x = 1
T6 99 50 149 x = 99
T7 100 50 invalid x = 100
T8 50 50 100 y ∈ [1, 99], x remains fixed for T8-T14
T9 50 -100 invalid y < 1
T10 50 250 invalid y > 99
T11 50 0 invalid y = 0
T12 50 1 51 y = 1
T13 50 99 149 y = 99
T14 50 100 invalid y = 100
T15 82 83 165
T16 83 83 invalid

A student passes an exam if s/he gets a grade >= 5.0. Grades below that are a fail.

Grades range from [1.0, 10.0]. Assume the system doesn't allow for invalid grades (e.g., 0.9, 10.5).

### Variables

Variables Types Ranges Notes
grade float [1, 10] (no one gets a 0...)
pass boolean true/false output variable

### Dependencies among variables

• No dependencies among input variables (grade is the only one).
• Grade is used to decide the pass/fail.

### Equivalence partitioning / Boundary analysis

Variable Equivalence Classes Invalid Classes Boundaries
4.9
5
[5, 10] 4.9
5
10

We do not need to test 0.9 and 10.1 because we assume that the system doesn't allow for them.

### Strategy

Boundaries seem to be enough.

Test Case Grade (input) Pass (output) Notes
T1 1 false
T2 4.9 false
T3 5 true
T4 7.5 true extra in-point
T5 10 true

## Exercise 4: Passing Concepts

The final grade of a student is calculated as follows:

• 1 <= grade < 5 => F
• 5 <= grade < 6 => E
• 6 <= grade < 7 => D
• 7 <= grade < 8 => C
• 8 <= grade < 9 => B
• 9 <= grade <= 10 => A

The system does not allow for invalid grades (e.g. 0.9, 10.5)

### Variables

Variable type range remark
Grade float [1, 10] no one gets a 0
Concept Enumerate [F, E, D, B, C, A] output variable

### Dependency among variables

*There are no dependencies among the input variables, since we only have one variable.

• The grade is used to decide the concept.

### Equivalence Partitioning/Boundary Analysis

Variable Equivalence classes Invalid classes Boundaries
1
5
4.9
[5,6[
4.9
5
6
5.9
[6,7[
5.9
6
7
6.9
[7,8[
6.9
7
8
7.9
[8,9[
7.9
8
9
8.9
[9,10]
8.9
9
10

### Strategy

Test all boundaries, yielding 12 tests.

T1 1 F
T2 4.9 F
T3 5 E
T4 5.9 E
T5 6 D
T6 6.9 D
T7 7 C
T8 7.9 C
T9 8 B
T10 8.9 B
T11 9 A
T12 10 A

## Exercise 5: The MSc admission problem

A student can only join the MSc if :

• ACT = 36 and GPA >= 3.5
• ACT >= 35 and GPA >= 3.6
• ACT >= 34 and GPA >= 3.7
• ACT >= 33 and GPA >= 3.8
• ACT >= 32 and GPA >= 3.9
• ACT >= 31 and GPA = 4.0 ACT is an integer between 0 and 36 (inclusive). GPA are float variables between 0.0 and 4.0 (single decimal digit) (inclusive).

### Variables

Variables Types Ranges
ACT integer [0, 36]
GPA float [0.0, 4.0]
Decision boolean true/false

### Dependency among variables

• ACT and GPA have a joint effect
• ACT and GPA are used to calculate Decision.

### Equivalence partitioning / Boundary Analysis

The conditions from the requirements are really close with each other. Because of that, finding values that cross the boundary between partitions is challenging. We also note that the boundary that matters here is from (student being approved) -> (student not being approved). So, for each partition, we should find its boundary that would force the student not being approved.

These boundary points can found through the following process. We first find the on-point of the boundary, for instance (35, 3.6), then discover the closest inputs that would make the student to fail. In this example, (34, 3.6) and (35, 3.5). Note that there exists one special case (= 36, >= 3.5). Since it contains an equality condition, we have two off-points to explore.

Also note that we are using in-points that are on-points too. This is less common, but in the case of this problem choosing anohter in-point might make the final outcome to still not change between the boundary tests we devised. For example, if we choose a GPA in-point of 3.6, the 35 and 36 ACT will have the same outcome value (as the next rule will intervene).

Variable Equivalence classes Boundaries Remark
(ACT, GPA) (= 36, >= 3.5) (35, in) in-point GPA = 3.5
(36, in) on-point
(37, in)
(in, 3.5) in-point ACT = 36
(in, 3.4)
(>= 35, >= 3.6) (35, in) in-point GPA = 3.6
(34, in)
(in, 3.6) in-point ACT = 35
(in, 3.5)
(>= 34, >= 3.7) (34, in) in-point GPA = 3.7
(33, in)
(in, 3.7) in-point ACT = 34
(in, 3.6)
(>= 33, >= 3.8) (33, in) in-point GPA = 3.8
(32, in)
(in, 3.8) in-point ACT = 33
(in, 3.7)
(>= 32, >= 3.9) (32, in) in-point GPA = 3.9
(31, in)
(in, 3.9) in-point ACT = 32
(in, 3.8)
(>= 31, = 4.0) (31, in) in-point GPA = 4.0
(30, in)
(in, 4.0) in-point ACT = 31
(in, 3.9)
(< 0, < 0.0) (-1, in) in-point GPA = 0.1
(in, -0.1) in-point ACT = 1
(> 36, > 4.0) (37, in) in-point GPA = 4
(in, 4.1) in-point ACT = 36

### Strategy

• There are 24 boundaries (for conditions on valid inputs), but some are repeated. 14 boundary tests.
• (37, 3.5) is an invalid path, so we can ignore. Therefore 13 test cases.
Test cases ACT GPA Decision
T1 35 3.5 False
T2 36 3.5 True
T3 36 3.4 False
T4 35 3.6 True
T5 34 3.7 True
T6 34 3.6 False
T7 33 3.8 True
T8 32 3.8 False
T9 33 3.7 False
T10 32 3.9 True
T11 31 4.0 True
T12 30 4.0 False
T13 31 3.9 False
T14 -1 4.0 Invalid
T15 37 3.5 Invalid
T16 36 -0.1 Invalid
T17 36 4.1 Invalid

## Exercise 6: The printing label

A printer prints mailing labels. The first line is the name of the person.

The program builds the name from three fields: first name, middle name, and last name. Each field can hold up to 30 characters. The label can be up to 70 characters wide.

### Variables

Variables Types Ranges
Len. of FN integer [1, 30]
Len. of MN integer [0, 30]
Len. of LN integer [0, 30]
Output length integer [1, 70]

### Dependencies among variables

FN + MN + LN <= 68

(The difference of 2 (to 70) happens as the system needs to add an empty space in between names)

### Equivalence partitioning / Boundary analysis

Variable Equivalence Classes Invalid Classes Boundaries Notes
FN [1, 30] invalid string 0 everybody has a first name
1
30
31 same as >30
MN [0, 30] invalid string 0 not everybody has a middle name
-1
30
31
LN [0, 30] invalid string 0 not everybody has a last name
-1
30
31
(FN, MN, LN) FM + MN + LN <= 68 68
69

### Strategy

• Each variable has 2 partitions, plus a restriction.

• Let's not combine "invalid strings" with them all. So: 3 tests for exceptional cases + 5 5 5 = 125 + 3 = 128.

• If we focus on the on-points and off-points, and in-points for others, we'd have 4 + 4 + 4 = 12 tests plus invalid cases: 15 tests.

• In-points always taking the FN + MN + LN <= 68 restriction into account.

• Two tests for the extra restriction: 17 tests.

Test Case FN MN LN (length) output
T1 0 15 7 22 invalid FN boundaries
T2 1 7 2 10 valid
T3 30 3 9 42 valid
T4 31 21 12 64 invalid
T5 15 0 15 30 valid MN boundaries
T6 20 -1 7 26 invalid
T7 21 30 6 57 valid
T8 22 31 3 56 invalid
T9 7 3 0 10 valid LN boundaries
T10 2 6 -1 7 invalid
T11 9 18 30 57 valid
T12 12 27 31 70 invalid
T13 invalid string 14 20 invalid invalid classes
T14 11 invalid string 23 invalid
T15 9 19 invalid string invalid
T16 23 23 22 68 valid FN + MN + LN restriction
T17 23 23 23 69 invalid

### Notes:

• We simplified the output by basically returning "valid" or "invalid". You might also wanna check the final name that was generated.
• Test cases with strings of length -1 might then not be possible.

## Exercise 7: Tax Income

Your income is taxed as follows:*

• 0 <= Income < 22100Tax = 0.15 x Income
• 22100 <= Income < 53500Tax = 3315 + 0.28 * (Income - 22100)
• 53500 <= Income < 115000Tax = 12107 + 0.31 * (Income - 53500)
• 115000 <= Income < 250000Tax = 31172 + 0.36 * (Income - 115000)
• 250000 <= IncomeTax = 79772 + 0.396 * (Income - 250000)

### Variables, Types, Ranges

Variable Type Range Remark
Income double [0, infinite] input
Tax double [0, infinite] output

### Dependency between variables

• Income is used to calculate Tax

### Equivalence partitioning / Boundary analysis

Variable Equivalence classes Invalid classes Boundaries Remark
Income [0, 22100[ -1 negative number
0
22099 22099.99 is better!
22100
[22100,53500[ 22099
22100
53499
53500
[53500,115000[ 53499
53500
114999
115000
[115000,250000[ 114999
115000
250000
250001
[250000, infinite[ 249999
250000

### Strategy

Test the boundaries (removing duplicates) → 10 tests

### Test cases

# Income Tax
T1 -1 CANNOT CALC TAX
T2 0 0
T3 22099 3314.85
T4 22100 3315
T5 53499 12106.72
T6 53500 12107
T7 114999 31171.69
T8 115000 31172
T9 249999 79771.64
T10 250000 79772

### Question

• Would you consider 22100 and 22099 a duplicate?

### Another approach

One may argue that this functions is continuous in all of its boundaries, which makes the result the same for all (on, off) pairs.

As, a consequence when looking just at the results at the boundaries we are not exercising them in the most efficient way. The whole purpose of boundary testing is to minimize the costs while maximizing fault detection capability.

What can we do to make our test cases more efficient? We can observe that what is really changing when making the transition from one partition to another is not the result of the function itself, but its derivative. Thus, we can focus on looking at the derivative for on and off points.

Let's recall the definition of the derivative of function f at point a: f'(a) = lim h -> 0 ((f(a+h) - f(a)) / h) To determine the derivative numerically we have to substitute h with sufficiently small number. For our case let h = 0.01.

We can now look at the boundary between first and second partition. off = 22099.99 on = 22100

By substituting those values into the definition of the derivative, we can determine the derivative of f at off on:

1. f'(off) = (f(off+h) - f(off)) / h = (f(22099.99 + 0.01) - f(22099.99)) / 0.01 = (f(22100) - f (22099.99)) / 0.01
2. f'(on) = (f(on+h) - f(a)) / h = (f(22100 + 0.01) - f(22100)) / 0.01 = (f(22100.01) - f (22100)) / 0.01

Now we find the expected derivatives for the corresponding off and on points from the specification:

1. f(off) = 0.15 * off thus f'(off) = 0.15
2. f(on) = 3315 + 0.28 * (on - 22100) thus f'(a) = 0.28

Now in our tests we can check whether actual derivatives for on and off points are equal to the expectation. What is left to do is to repeat the whole process for all other boundaries.

Note that if we decide to exercise the boundaries in this way, we have to remember to write the tests for the results too, as correct derivative does not imply correct function. Maybe the developer, during implementation, instead of 3315 + 0.28 * (on - 22100) wrote 315 + 0.28 * (on - 22100), which by testing only for derivatives we will not spot.

This example shows that as a software testers we, not only have to identify the boundaries between partitions, but also think about what is really changing when crossing those boundaries, in order to maximize the efficiency of our test cases.

## Exercise 8: The ATM

An ATM allows you to withdraw 20 to 200 euros (inclusive) in increments of 20. For the example purposes, the program returns true or false, depending whether the amount required is valid.

### Variables

Variable Type Range
Amount integer {20, 40, 60, 80, 100, 120, 140, 160, 180, 200}
Valid boolean true, false

### Dependency among variables

Valid depends on the Amount

### Equivalence Partitioning/Boundary Analysis

Variable Equivalence classes Invalid classes Boundaries
Amount 20 19
21
40 39
41
60 59
61
80 79
81
100 99
101
120 119
121
140 139
141
160 159
161
180 179
181
200 199
201
-1

Note that -1 may be impossible to test.

### Strategy

• Black-box testing: maybe all the on-points, off-points and invalid points.
• White-box testing: maybe just on-points, few-off points, invalid points.
Why? If you know the implementation, you know how hard you need to test it.
Test case Amount Valid (output)
T1 20 true
T2 40 true
T3 60 true
T4 80 true
T5 100 true
T6 120 true
T7 140 true
T8 160 true
T9 180 true
T10 200 true
T11 19 false
T12 41 false
T13 61 false
T14 -1 false

## Exercise 9: Piecewise

The input domain of a function is a set of all points (x, y) that meet the criteria:

• 1 < x <= 10
• 1 <= y <= 10
• y <= 14 - x

### Variables

Variable Type Range
x integer 1 < x <= 10
y integer 1 <= y <= 10, y <= 14 - x

### Dependency among variables

• x and y are dependent, since the range in y varies according to x.

### Equivalence Partitioning/Boundary Analysis

Variable Equivalence classes Invalid classes Boundaries
x 1 < x <= 10
(1, in)
(2, in)
(10, in)
(11, in)
y 1 <= y <= 10
(in, 1)
(in, 0)
(in, 10)
(in, 11)
y <= 14 - x
(4, 10)
(5, 10)
(10, 4)
(11, 4)

### Strategy

Make tests for all 12 boundaries

Test case x y output
T1 1 5 false
T2 2 5 true
T3 10 2 true
T4 11 2 false
T5 3 1 true
T6 3 0 false
T7 3 10 true
T8 3 11 false
T9 4 10 true
T10 5 10 false
T11 10 4 true
T12 11 4 false

### Another approach

We might look at the plot of the function. In the plot, we identify 5 boundaries (one at each of the extremes of the figure). As a tester, we can exercise these boundaries.

Test case x y output
T1 1 1 false
T2 2 1 true
T3 10 1 true
T4 11 1 false
T5 2 0 false
T6 2 10 true
T7 2 11 false
T8 4 10 true
T9 4 11 false
T10 10 4 true
T11 10 5 false
T12 11 4 false
T13 1 10 false
T14 5 10 false

## Exercise 10: Chocolate bars

A package should store a total number of kilograms. There are small bars (1 kg each) and big bars (5 kg each).

• The input of the program is the number of small bars and big bars available and the total number of kilos to store.
• We should calculate the number of small bars to use, assuming we always use big bars before small bars. Output -1, if it can't be done.
Variables Types Ranges
Small bars integer [0, inf]
Big bars integer [0, inf]
Total kilos integer [0, inf]
Used small bars (output) integer [-1, inf]

### Dependency between variables

• Input variables are independent (they don't affect each other's range).
• Output variable depends on the three input values.
• Constraint: Use big bars before small bars.

Analyse it from the perspective of the output variable: how can the input variables influence the result?

### Equivalence partitioning / Boundary analysis

Variable Equivalence classes Boundaries
(small, big, total weight) only big bars only big bars -> small and big bars
only big bars -> not enough bars
only small bars only small bars -> small and big bars
only small bars -> not enough bars
small and big bars small and big bars -> only big bars
small and big bars -> only small bars
small and big bars -> not enough bars
not enough bars not enough bars -> only big bars
not enough bars -> only small bars
not enough bars -> small and big bars

### Strategy

• There are 4 equivalence classes and 10 boundaries, but many of these boundaries are actually the same.
• For example, boundary 1 and 5 are in fact the same boundary. Boundaries are not directional
Test cases (Small bars, Big bars, Total weight) Used small bars (output) Remark
T2 (10, 2, 10) 0 small and big bars ->
T1 (10, 1, 10) 5 only big bars
T3 (10, 1, 10) 5 small and big bars ->
T4 (10, 0, 10) 10 only small bars
T5 (5, 0, 5) 5 only small bars ->
T6 (4, 0, 5) -1 not enough bars
T7 (4, 2, 10) 0 only big bars ->
T8 (4, 1, 10) -1 not enough bars
T9 (3, 1, 8) 3 small and big bars ->
T10 (2, 1, 8) -1 not enough bars (needed more small bars)
T11 (3, 1, 8) 3 small and big bars ->
T12 (3, 0, 8) -1 not enough bars (needed more big bars)