Abstract Domain Explorer

Overview

This skill applies abstract interpretation to statically analyze programs using various abstract domains. It infers invariants, value ranges, and relationships between variables without executing the code. Different domains offer different trade-offs between precision and efficiency.

Analysis Workflow

Follow these steps to analyze programs with abstract domains:

1. Select Appropriate Domain(s)

Choose based on analysis goals:

Interval Domain:

• Use for: Range analysis, bounds checking, array indexing
• Precision: Low to medium
• Cost: Very efficient
• Example: Determine if x ∈ [0, 100]

Sign Domain:

• Use for: Sign analysis, division by zero detection
• Precision: Low
• Cost: Very efficient
• Example: Determine if x is positive, negative, or zero

Congruence Domain:

• Use for: Modular arithmetic, alignment analysis
• Precision: Medium (for specific patterns)
• Cost: Efficient
• Example: Determine if x ≡ 0 (mod 4)

Octagon Domain:

• Use for: Relational analysis, loop invariants with simple relationships
• Precision: Medium to high
• Cost: Moderate (O(n³) operations)
• Example: Infer x ≤ y + 5, x + y ≤ 10

Polyhedra Domain:

• Use for: Complex linear relationships, precise invariants
• Precision: High
• Cost: Expensive (exponential worst-case)
• Example: Infer 2x + 3y ≤ z + 10

Reduced Product:

• Use for: Combining strengths of multiple domains
• Precision: Higher than individual domains
• Cost: Sum of component costs
• Example: Intervals × Congruence for precise range with modular constraints

2. Initialize Abstract State

Set initial values for program entry:

• Constants: Exact values (e.g., x = 5 → x ∈ [5, 5])
• Inputs: Top element (e.g., user input → x ∈ [-∞, +∞])
• Uninitialized: Bottom element (unreachable)

Example:

int x = 0;        // x ∈ [0, 0]
int y = input();  // y ∈ [-∞, +∞]

3. Apply Transfer Functions

For each statement, compute abstract semantics:

Assignment (x = e):

• Evaluate expression e in abstract domain
• Update abstract state for x

Condition (assume c):

• Refine abstract state based on condition
• Intersect with constraint

Join (control flow merge):

• Compute least upper bound of incoming states
• Used after if-else, at loop headers

Example (Intervals):

x = y + 5;        // If y ∈ [a, b], then x ∈ [a+5, b+5]
assume(x > 10);   // If x ∈ [a, b], then x ∈ [max(a, 11), b]

4. Handle Loops with Widening

For loops, apply widening to ensure termination:

1. Compute first iteration
2. Compute second iteration
3. Apply widening operator (∇)
4. Check for convergence
5. Continue until fixpoint reached

Example:

int x = 0;
while (x < 100) {
    x = x + 1;
}

• Iteration 0: x ∈ [0, 0]
• Iteration 1: x ∈ [0, 1]
• Widening: x ∈ [0, +∞]
• Refine with condition: x ∈ [0, 99] in loop
• Exit: x ∈ [100, 100]

5. Refine with Narrowing (Optional)

Apply narrowing to improve precision:

• Iterate a few more times (typically 1-3)
• Use narrowing operator (△)
• Refine over-approximations from widening

6. Extract Invariants

Identify inferred properties:

• Value ranges for each variable
• Relationships between variables
• Loop invariants
• Preconditions for safe operations

Report format:

• Variable: abstract value
• Invariants: logical formulas
• Safety properties: bounds checks, division by zero, etc.

Analysis Examples

Example 1: Range Analysis with Intervals

Code:

int x = 0;
int y = 100;
while (x < 10) {
    x = x + 1;
    y = y - 1;
}
// What are the values of x and y here?

Analysis (Interval Domain):

• Entry: x ∈ [0, 0], y ∈ [100, 100]
• Loop iterations with widening:
  • Iter 0: x ∈ [0, 0], y ∈ [100, 100]
  • Iter 1: x ∈ [0, 1], y ∈ [99, 100]
  • Widening: x ∈ [0, +∞], y ∈ [-∞, 100]
  • Refine with x < 10: x ∈ [0, 9], y ∈ [-∞, 100]
• Narrowing: y ∈ [91, 100]
• Exit: x ∈ [10, 10], y ∈ [90, 90]

Inferred Invariants:

• Loop: x ∈ [0, 9], y ∈ [91, 100]
• Exit: x = 10, y = 90
• Implicit: x + y = 100 (not captured by intervals alone)

Example 2: Relational Analysis with Octagons

Code:

int x = 0;
int y = 0;
while (x < 10) {
    x = x + 1;
    y = y + 1;
}

Analysis (Octagon Domain):

• Entry: x = 0, y = 0, x - y = 0
• Loop iterations:
  • Maintains x - y = 0 throughout
  • x ∈ [0, 10], y ∈ [0, 10]
• Exit: x = 10, y = 10, x = y

Inferred Invariants:

• x = y (captured by octagon)
• x ∈ [0, 10], y ∈ [0, 10]

Advantage over Intervals: Intervals would only infer x ∈ [0, 10], y ∈ [0, 10] but miss the relationship x = y.

Example 3: Linear Relationships with Polyhedra

Code:

int x = 0, y = 0, z = 0;
while (x < 10) {
    x = x + 1;
    y = y + 2;
    z = x + y;
}

Analysis (Polyhedra Domain):

• Entry: x = 0, y = 0, z = 0
• Loop iterations:
  • Infers: y = 2x, z = x + y, z = 3x
  • x ∈ [0, 10]
• Exit: x = 10, y = 20, z = 30

Inferred Invariants:

• y = 2x
• z = x + y
• z = 3x
• x ∈ [0, 10]

Advantage over Octagons: Polyhedra can express y = 2x, which octagons cannot.

Example 4: Modular Arithmetic with Congruence

Code:

int sum = 0;
for (int i = 0; i < 100; i++) {
    sum = sum + 3;
}

Analysis (Congruence Domain):

• Entry: sum ≡ 0 (mod 3), i ≡ 0 (mod 1)
• Loop: sum ≡ 0 (mod 3) maintained
• Exit: sum ≡ 0 (mod 3)

Analysis (Interval Domain):

• Exit: sum ∈ [0, 300]

Combined (Reduced Product):

• sum ∈ [0, 300] and sum ≡ 0 (mod 3)
• Refined: sum ∈ {0, 3, 6, 9, ..., 297, 300}
• Exact: sum = 300

Inferred Invariants:

• sum is always divisible by 3
• sum ∈ [0, 300]

Example 5: Division by Zero Detection with Sign

Code:

int x = read_input();
int y = x * x;
int z = 100 / y;  // Safe?

Analysis (Sign Domain):

• x: ⊤ (unknown sign)
• y = x * x: ≥0 (non-negative)
• Problem: y could be 0 if x = 0
• Division 100 / y: potential division by zero

Analysis (Interval Domain):

• x: [-∞, +∞]
• y: [0, +∞]
• Problem: 0 ∈ [0, +∞]
• Division 100 / y: potential division by zero

Inferred Property:

• Unsafe: Division by zero possible when x = 0

Example 6: Array Bounds Checking

Code:

int arr[10];
int i = 0;
while (i < 10) {
    arr[i] = 0;  // Safe?
    i = i + 1;
}

Analysis (Interval Domain):

• Loop: i ∈ [0, 9]
• Array access: arr[i] where i ∈ [0, 9]
• Array bounds: [0, 9]
• Safe: i always within bounds

Inferred Property:

• No array bounds violation

Example 7: Nested Loops with Octagons

Code:

int i = 0, j = 0;
while (i < 10) {
    j = 0;
    while (j < i) {
        j = j + 1;
    }
    i = i + 1;
}

Analysis (Octagon Domain):

• Outer loop: i ∈ [0, 10]
• Inner loop: j ∈ [0, i], j ≤ i
• Exit: i = 10, j = 9, j ≤ i

Inferred Invariants:

• j ≤ i (relational invariant)
• i ∈ [0, 10]
• j ∈ [0, 9]

Domain Comparison

Domain	Precision	Cost	Relationships	Best For
Sign	Very Low	O(1)	None	Sign errors, division by zero
Interval	Low-Medium	O(1)	None	Range analysis, bounds checking
Congruence	Medium	O(1)	None	Modular patterns, alignment
Octagon	Medium-High	O(n³)	±x ± y ≤ c	Simple relational invariants
Polyhedra	High	Exponential	Linear	Complex linear relationships
Reduced Product	Higher	Sum of components	Combined	Precise analysis with multiple aspects

Choosing the Right Domain

Start with Intervals if:

• You need range information
• Efficiency is critical
• No relationships needed

Use Octagons if:

• You need simple relationships (x ≤ y + c)
• Loop invariants involve variable pairs
• Moderate cost acceptable

Use Polyhedra if:

• Complex linear relationships needed
• Precision is critical
• Small programs or specific code sections
• Cost is acceptable

Use Reduced Products if:

• Multiple aspects needed (range + modular)
• Willing to pay combined cost
• Need precision from multiple domains

Use Sign if:

• Only sign information needed
• Very large programs
• Quick analysis required

Use Congruence if:

• Modular arithmetic patterns
• Alignment analysis
• Complement to intervals

Constraints

MUST:

• Select appropriate domain(s) for analysis goals
• Apply transfer functions correctly
• Use widening for loops to ensure termination
• Report inferred invariants clearly
• Indicate precision limitations of chosen domain

MUST NOT:

• Claim exact values when domain gives approximations
• Ignore widening (may not terminate)
• Use expensive domains unnecessarily
• Report unsound results

Resources

references/abstract_domains.md

Comprehensive reference covering:

• Abstract interpretation fundamentals
• Detailed domain specifications (intervals, octagons, polyhedra, sign, congruence)
• Domain operations and transfer functions
• Widening and narrowing techniques
• Reduced product domains
• Fixpoint computation
• Complete analysis examples