Boundary Value Analysis

Boundary Value Analysis (BVA) focuses testing on the edges of equivalence partitions, where faults are most likely to occur. Off-by-one errors and incorrect boundary operators are among the most common programming mistakes.

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Why Boundaries Matter

Empirical observation: Faults cluster at partition boundaries.

Common boundary bugs:

• Loop termination (< n vs <= n)
• Array indexing (0-based vs 1-based)
• Range checks (x < max vs x <= max)
• Comparison operators (> vs >=)

“Boundary problems are most likely to happen at boundaries.” [1]

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Boundary Fault Types

Fault Type	Description	Example
Closure	Wrong operator (< vs ≤)	if (x < 10) should be if (x <= 10)
Shift	Boundary at wrong value	if (x < 10) should be if (x < 11)
Tilt	Coefficient wrong (2D+)	ax + by = K vs ax + cy = K
Missing	Boundary not implemented	No upper bound check
Extra	Spurious boundary	Unnecessary constraint added

Example: Off-by-One Error

# Bug: Processes items 1-9, misses item 10
for i in range(1, 10):
    process(items[i])

# Correct: Processes items 1-10
for i in range(1, 11):
    process(items[i])

BVA would test values 9, 10, and 11 to catch this.

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ON/OFF Point Formalism

A rigorous approach to boundary testing from Tian [1]:

Definitions

Concept	Definition
ON point	Exactly on the boundary (satisfies f(x) = K)
OFF point	Just off boundary, within ε distance
ε-neighborhood	Small distance distinguishing ON from OFF

Key Rule

The OFF point must be in the opposite processing region from the ON point.

Closed Boundaries (≤, ≥)

The boundary point is included in the valid region.

flowchart LR
    subgraph "Closed: x ≤ 100"
        OUT["x = 101<br>OFF point<br>(outside)"]
        ON["x = 100<br>ON point<br>(boundary)"]
        IN["x = 50<br>Interior"]
    end

    style OUT fill:#ffcccc,stroke:#cc0000
    style ON fill:#fff3cd,stroke:#ffc107
    style IN fill:#ccffcc,stroke:#009900

Test points for x ≤ 100:

• ON point: x = 100 (valid, on boundary)
• OFF point: x = 101 (invalid, just outside)

Open Boundaries (<, >)

The boundary point is excluded from the valid region.

flowchart LR
    subgraph "Open: x < 100"
        ON["x = 100<br>ON point<br>(excluded)"]
        OFF["x = 99<br>OFF point<br>(inside)"]
        IN2["x = 50<br>Interior"]
    end

    style ON fill:#ffcccc,stroke:#cc0000
    style OFF fill:#fff3cd,stroke:#ffc107
    style IN2 fill:#ccffcc,stroke:#009900

Test points for x < 100:

• ON point: x = 100 (invalid, exactly on boundary)
• OFF point: x = 99 (valid, just inside)

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BVA Testing Strategies

Basic BVA (4k + 1)

For each of k input variables, test:

• Minimum value
• Just above minimum (min + ε)
• Nominal (middle) value
• Just below maximum (max - ε)
• Maximum value

Plus one test with all nominal values.

Formula: 4k + 1 tests for k variables

Example: k = 3 variables → 13 tests

Robust BVA (6k + 1)

Adds values beyond valid boundaries:

• Below minimum (min - ε)
• Above maximum (max + ε)

Formula: 6k + 1 tests for k variables

Example: k = 3 variables → 19 tests

Strategy Comparison

Strategy	Points per Boundary	Detects	Formula
Weak 1×1	1 ON + 1 OFF	Closure, shift	2 per condition
Weak N×1	N ON + 1 OFF	+ tilt detection	(n+1) × b + 1
Basic BVA	4 per variable	Standard faults	4k + 1
Robust BVA	6 per variable	+ beyond-boundary	6k + 1

Practical minimum: Weak 1×1 (one ON + one OFF per boundary condition).

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Test Count Examples

Variables (k)	Basic BVA (4k+1)	Robust BVA (6k+1)
1	5	7
2	9	13
3	13	19
4	17	25
5	21	31

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Closed vs Open Boundary Summary

Boundary Type	Operator	ON Point	OFF Point Location
Closed	≤	On boundary (valid)	Outside (invalid)
Closed	≥	On boundary (valid)	Outside (invalid)
Open	<	On boundary (invalid)	Inside (valid)
Open	>	On boundary (invalid)	Inside (valid)

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Multidimensional BVA

When testing functions with multiple numeric inputs:

Single Variable at Extreme

Hold all variables at nominal, vary one to extreme:

Test	X	Y	Z
T1	min	nom	nom
T2	max	nom	nom
T3	nom	min	nom
T4	nom	max	nom
T5	nom	nom	min
T6	nom	nom	max
T7	nom	nom	nom

Combining Extremes

For higher assurance, combine extreme values:

Test	Latitude	Longitude
T1	-90 (min)	-180 (min)
T2	-90 (min)	+180 (max)
T3	+90 (max)	-180 (min)
T4	+90 (max)	+180 (max)
T5	0 (nom)	0 (nom)

Ordered Pairs

When relationship between variables matters:

Relationship	Test Values	Purpose
x < y	(1, 5)	Normal order
x = y	(3, 3)	Equal values
x > y	(5, 1)	Inverted order

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Practical Example: Date Validation

Function: isValidDate(year, month, day) -> Boolean

Boundary Identification

Variable	Min	Max	Special
Year	1	9999	Leap years
Month	1	12	-
Day	1	28/29/30/31	Month-dependent

Test Cases

Test	Year	Month	Day	Expected	Boundary
T1	2024	1	0	Invalid	Day below min
T2	2024	1	1	Valid	Day at min
T3	2024	1	31	Valid	Day at max (Jan)
T4	2024	1	32	Invalid	Day above max
T5	2024	0	15	Invalid	Month below min
T6	2024	13	15	Invalid	Month above max
T7	2024	2	29	Valid	Leap year Feb
T8	2023	2	29	Invalid	Non-leap Feb
T9	0	6	15	Invalid	Year below min

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BVA Limitations

1. Case-like processing assumption: BVA assumes inputs map directly to processing regions without loops that modify the input.

2. Coincidental correctness: A bug may exist but the test happens to pass because of specific values chosen.

3. ε-limit problems: Floating-point precision varies across platforms. What is “just above” 1.0?

4. Closed domain issues: For some domains, the OFF point may be undefined (e.g., minimum of unsigned integer).

5. Specification dependency: Boundaries must be documented; implicit boundaries are easily missed.

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Combining BVA with EP

BVA complements Equivalence Partitioning:

flowchart TD
    EP[Equivalence Partitioning] --> |"Identifies partitions"| P1[Child: 0-12]
    EP --> P2[Teen: 13-19]
    EP --> P3[Adult: 20-64]
    EP --> P4[Senior: 65+]

    P1 --> |"BVA adds boundaries"| B1["0, 12, 13"]
    P2 --> B2["13, 19, 20"]
    P3 --> B3["20, 64, 65"]
    P4 --> B4["65, 120"]

    style EP fill:#e1f5fe
    style B1 fill:#fff3cd
    style B2 fill:#fff3cd
    style B3 fill:#fff3cd
    style B4 fill:#fff3cd

EP alone: Tests 5, 15, 30, 70 (one per partition) EP + BVA: Adds 0, 12, 13, 19, 20, 64, 65 (boundary points)

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Key Takeaways

1. Faults cluster at boundaries — test edges of every partition
2. ON/OFF formalism provides rigorous boundary point selection
3. Closed boundaries: OFF point outside; Open boundaries: OFF point inside
4. 4k+1 formula for basic BVA; 6k+1 for robust BVA
5. Combine with EP — EP identifies what to partition, BVA tests the edges
6. Weak 1×1 is practical minimum when resources are limited

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References

1. J. Tian, Software Quality Engineering: Testing, Quality Assurance, and Quantifiable Improvement. Wiley-IEEE Computer Society Press, 2005.

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