Reaching definitions analysis in Compiler Design - Step-by-Step Execution

Concept Flow - Reaching definitions analysis

Start: Entry block

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Initialize IN and OUT sets

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For each block

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Calculate GEN and KILL sets

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Iterate until IN and OUT stabilize

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Output reaching definitions for each block

The analysis starts by initializing sets, then repeatedly updates IN and OUT sets for each block until no changes occur, showing which definitions reach each point.

Execution Sample

Compiler Design

Block1: d1: x=1
Block2: d2: y=2
Block3: d3: x=3

Compute IN and OUT sets for each block.

This example shows how definitions d1, d2, d3 reach different blocks in a simple control flow.

Analysis Table

Step	Block	IN Set	GEN Set	KILL Set	OUT Set	Change?
1	Block1	{}	{d1}	{}	{d1}	Yes
2	Block2	{d1}	{d2}	{d3}	{d1,d2}	Yes
3	Block3	{d1,d2}	{d3}	{d1}	{d2,d3}	Yes
4	Block1	{}	{d1}	{}	{d1}	No
5	Block2	{d1}	{d2}	{d3}	{d1,d2}	No
6	Block3	{d1,d2}	{d3}	{d1}	{d2,d3}	No

💡 No changes in IN and OUT sets after step 6, analysis converged.

State Tracker

Variable	Start	After Step 1	After Step 2	After Step 3	After Step 6
IN(Block1)	{}	{}	{}	{}	{}
OUT(Block1)	{}	{d1}	{d1}	{d1}	{d1}
IN(Block2)	{}	{d1}	{d1}	{d1}	{d1}
OUT(Block2)	{}	{}	{d1,d2}	{d1,d2}	{d1,d2}
IN(Block3)	{}	{}	{d1,d2}	{d1,d2}	{d1,d2}
OUT(Block3)	{}	{}	{d2,d3}	{d2,d3}	{d2,d3}

Key Insights - 3 Insights

Why do we keep updating IN and OUT sets multiple times?

Because definitions can flow through multiple blocks, we must iterate until no changes occur to ensure all reaching definitions are found, as shown by changes in steps 1-3 and no changes after step 6.

What does the KILL set represent in this analysis?

The KILL set contains definitions that are overwritten by new definitions in the block, so they no longer reach beyond this block, as seen in Block2 killing d3 and Block3 killing d1.

Why is the OUT set of a block different from its GEN set?

OUT includes GEN plus any reaching definitions from predecessors that are not killed, showing all definitions reaching after the block, e.g., Block2's OUT is {d1,d2} not just {d2}.

Visual Quiz - 3 Questions

Look at the execution_table at step 3. What is the OUT set of Block3?

A {d1,d2}

B {d2,d3}

C {d3}

D {d1,d3}

💡 Hint

Check the OUT Set column for Block3 at step 3 in the execution_table.

At which step does the analysis stop changing the IN and OUT sets?

A Step 3

B Step 4

C Step 6

D Step 2

💡 Hint

Look for the first step where Change? is 'No' for all blocks in the execution_table.

If Block2 did not kill d3, how would OUT(Block2) change at step 2?

A {d1,d2,d3}

B {d1,d2}

C {d2}

D {d3}

💡 Hint

Consider that without killing d3, it remains in the OUT set along with d1 and d2.

Concept Snapshot

Reaching Definitions Analysis:
- Tracks which variable assignments (definitions) reach each program point.
- Uses IN, OUT, GEN, KILL sets per block.
- Iterates until IN and OUT sets stabilize (no changes).
- Helps optimize by knowing where variables are defined and used.
- Key rule: OUT = GEN ∪ (IN - KILL).

Full Transcript

Reaching definitions analysis is a method used in compilers to find out which assignments to variables can reach a certain point in the program. It works by assigning sets of definitions to each block: GEN for definitions made in the block, KILL for definitions overwritten, IN for definitions reaching the block, and OUT for definitions leaving the block. The analysis starts with empty sets and repeatedly updates IN and OUT for each block until no changes occur. This process ensures all possible definitions reaching each point are found. For example, in a simple program with three blocks defining variables x and y, the analysis shows how definitions flow through blocks and which are killed or preserved. This information is useful for optimizations like removing unnecessary assignments or detecting variable usage. The key formula is OUT = GEN union (IN minus KILL). Iteration continues until IN and OUT sets stabilize, meaning no new information is found. Understanding this helps in compiler design and program analysis.

Practice

1. What is the main purpose of reaching definitions analysis in compiler design?

easy

A. To find all places where a variable's value could have been assigned before a given point

B. To check syntax errors in the source code

C. To optimize memory allocation for variables

D. To generate machine code from source code

Solution

1. Step 1: Understand the goal of reaching definitions

   Reaching definitions analysis tracks where each variable's value might come from at different points in the program.

2. Step 2: Identify the correct purpose

   This analysis helps find all assignments (definitions) that can reach a particular point, which is useful for detecting variable usage before assignment and optimization.

3. Final Answer:

   To find all places where a variable's value could have been assigned before a given point -> Option A

4. Quick Check:

   Reaching definitions = find variable assignments [OK]

Hint: Reaching definitions track where variables get their values [OK]

Common Mistakes:

• Confusing reaching definitions with syntax checking
• Thinking it manages memory allocation
• Assuming it generates machine code

2. Which of the following is a correct statement about the sets used in reaching definitions analysis?

easy

A. The IN set contains definitions that reach the start of a basic block

B. The OUT set contains definitions that are killed before the block starts

C. The GEN set contains definitions that are never generated in the block

D. The KILL set contains definitions that are generated in the block

Solution

1. Step 1: Recall the meaning of IN, OUT, GEN, and KILL sets

   IN set: definitions reaching the start of a block; OUT set: definitions reaching the end; GEN set: definitions generated inside the block; KILL set: definitions invalidated inside the block.

2. Step 2: Match the correct statement

   Only the statement about IN set containing definitions reaching the start of a block is correct.

3. Final Answer:

   The IN set contains definitions that reach the start of a basic block -> Option A

4. Quick Check:

   IN set = definitions at block start [OK]

Hint: IN set means definitions reaching block start [OK]

Common Mistakes:

• Mixing up GEN and KILL sets
• Thinking OUT set contains killed definitions
• Confusing KILL with GEN

3. Consider the following code snippet and its control flow graph with blocks B1 and B2:

B1: x = 1
y = 2
B2: x = 3
z = x + y

Which definitions reach the start of block B2?

medium

A. {y = 2}

B. {x = 3, y = 2}

C. {x = 1, y = 2}

D. {} (empty set)

Solution

1. Step 1: Identify definitions in B1

   Block B1 defines x = 1 and y = 2. These definitions reach the end of B1.

2. Step 2: Determine what reaches B2

   Since B2 follows B1, the definitions from B1 reach the start of B2. The new definition x = 3 is generated inside B2, so it does not reach the start of B2.

3. Final Answer:

   {x = 1, y = 2} -> Option C

4. Quick Check:

   Definitions from previous block reach next block start [OK]

Hint: Definitions inside a block don't reach its own start [OK]

Common Mistakes:

• Including definitions generated inside the block itself
• Ignoring definitions from previous blocks
• Assuming empty set at block start

4. In a reaching definitions analysis, a student wrote the following incorrect equation for the OUT set of a block:

OUT[B] = IN[B] ∩ GEN[B]

What is the error in this equation?

medium

A. It should use only GEN[B] without IN[B]

B. It should use difference (\) instead of intersection (∩)

C. It should use intersection (∩) but with KILL[B]

D. It should use union (∪) instead of intersection (∩)

Solution

1. Step 1: Recall the correct OUT set formula

   The correct formula is OUT[B] = GEN[B] ∪ (IN[B] \ KILL[B]), meaning definitions generated in B plus those reaching B and not killed.

2. Step 2: Identify the error in the student's formula

   The student's formula uses intersection (∩) instead of union (∪), which incorrectly limits the OUT set.

3. Final Answer:

   It should use union (∪) instead of intersection (∩) -> Option D

4. Quick Check:

   OUT set uses union, not intersection [OK]

Hint: OUT = GEN union (IN minus KILL) [OK]

Common Mistakes:

• Using intersection instead of union
• Ignoring the KILL set
• Using only GEN set

5. Given a program with a loop where variable x is assigned inside the loop and also before the loop, how does reaching definitions analysis help optimize the code?

hard

A. By duplicating the assignment inside the loop to improve speed

B. By identifying that the initial assignment to x before the loop is overwritten and can be removed

C. By removing all assignments to x inside the loop

D. By ignoring the assignments before the loop since loops always override variables

Solution

1. Step 1: Understand variable assignments in loops

   If x is assigned before the loop and also inside the loop, the assignment before the loop might be redundant if the loop always assigns x.

2. Step 2: Apply reaching definitions analysis

   This analysis shows which definitions reach each point. If the initial assignment never reaches uses inside the loop, it can be safely removed to optimize code.

3. Final Answer:

   By identifying that the initial assignment to x before the loop is overwritten and can be removed -> Option B

4. Quick Check:

   Reaching definitions find redundant assignments [OK]

Hint: Remove overwritten assignments before loops [OK]

Common Mistakes:

• Thinking all assignments inside loops are removable
• Duplicating assignments unnecessarily
• Ignoring the effect of loops on variable values