[INSTRUCTIONS]

You are a program-analysis assistant. Your task is to statically analyze the control dependence of a given code snippet. 

## 1. Control Dependence Definition
  Control dependence captures the influence of control-flow decisions on the execution of statements.

  A statement `S2` is control-dependent on `S1` if there is a **transitive** (indirect) chain of **direct** control dependencies from `S1` to `S2`. This is equivalent to saying that `S1` has control dependence over `S2`, meaning `S1`’s condition influences whether `S2` executes.


  **Direct Control Dependence**:
  A statement `S2` is **directly** control-dependent on a statement `S1` if:
  
  1.	`S1` is a conditional control statement (e.g., an `if`, `while`, `for`, `switch`, etc.).
  
  2.	`S1` directly determines whether `S2` executes. That is, `S1` has multiple successor branches, 

  - there exists **at least one branch** in which `S2` **always executes**, and
  - there exists **at least one other branch** in which `S2` does **not necessarily execute**.

    “Not necessarily execute” means that `S2` might execute or might not, but it is **not guaranteed** to execute in that branch.

  `S2` is **control-dependent** on `S1` if there exists a **transitive chain** of control dependencies from `S1` to `S2`, where each intermediate step in the chain represents a **direct** control dependence between two statements.



## 2. Output Format
When asked, “Does line `S1` have control dependence over line `S2`? If so, provide a trace.” You should respond in JSON format as follows:

- If there is a control dependence:

```json
{
  "ControlDependence": true,
  "Trace": [S1, ..., S2]
}
```

- If there is no control dependence:
```json
{
  "ControlDependence": false
}
```

A **trace** represents a **transitive control flow**, where each adjacent pair in the list reflects a **direct** control dependence. The trace must starts with `S1` and ends with `S2`.


    

## 3. Interprocedural Control Dependence
  All dependence analysis is performed within **a single function**. We do not track dependencies across function boundaries. The analysis only applies to variables and control structures inside the **specified function**.


## 4. Example Code Snippet

### Example 1

```python
1  if x > 0:
2      y = 10       # this line is directly control-dependent on line 1 (x>0)
3      if y > 5:
4          z = 20   # this line is directly control-dependent on line 3 (y>5)
5          w = 30   # this line is directly control-dependent on line 3 (y>5)
6  v = 40           # this line is NOT control-dependent on any line
```


**Analysis**:
- Line 2 is directly control-dependent on line 1 (`x>0`)
- Lines 4 and 5 are directly control-dependent on line 3 (`y>5`).
- Lines 4 and 5 are **indirectly** control-dependent on line 1 (`x>0`) becasue even if line 1 (`x>0`) is `true`, lines 4 and 5 may not execute if line 3’s condition (`y>5`) is `false`. However, since control dependence is transitive, lines 4 and 5 are **indirectly** control-dependent on line 1. 
- Line 6 is not control-dependent on any lines because Line 6 always executes, regardless of whether line 1 (`x>0`) or line 3 (`y>5`) is true or false.



#### Example Question 1.1: 
Does line `1` have control dependence over line `5`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": true,
  "Trace": [1, 3, 5]
}
```

#### Example Question 1.2:
Does line `3` have control dependence over line `6`? If so, provide a trace.

**Output**: 
```json
{
  "ControlDependence": false
}
```


### Example 2

```python
1  count = 0
2  if count < 5:
3      count += 1
4  print("Step 1 done")
5  while count < 10:
6      if count == 7:
7          break
8      if count % 2 == 0:
9          continue
10     count += 2
11     if count > 9:
12         count = 9
13     print("Iteration done")
14 print("End of program")
```

#### Example Question 2.1: 
Does line `5` have control dependence over line `13`? If so, provide a trace.

**Analysis**:
  - Line 13 is directly control-dependent on line 8 because if line 8 evaluates to `true`, continue skips line 13 for that iteration.
  - Line 8 is directly control-dependent on line 6 because if line 6 evaluates to `true`, the break statement terminates the loop, preventing line 8 from executing.
  - Line 6 is directly control-dependent on line 5 because the loop condition at line 5 determines whether line 6 executes. If line 5 evaluates to `false`, execution skips the loop entirely.


Since control dependence is transitive, line 5 indirectly controls line 13 through lines 6 and 8. The trace is `5->6->8->13`.

**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [5, 6, 8, 13]
}
```

#### Example Question 2.2: 
Does line `8` have control dependence over line `10`? If so, provide a trace.


**Analysis**: Similar to the explanation above, line 8 is an `if` condition inside the `while` loop with a `continue`, so it can make the execution skip the rest of the loop entirely, including line 10. Therefore, line 10 directly control-dependent on 8.



**Output**:
```json
{
  "ControlDependence": true,
  "Trace": [8, 10]
}
```

#### Example Question 2.3: 
Does line `5` have control dependence over line `14`? If so, provide a trace.

**Analysis**: Line 5 doesn't affect whether line 14 is reached. Line 14 is outside the while loop (lines 5 -- 13) and will be executed regardless of the condition. 

**Output**:
```json
{
  "ControlDependence": false
}
```


#### Example Question 2.4: 
Does line `2` have control dependence over line `12`? If so, provide a trace.

**Analysis**: Line 2 doesn't affect whether line 12 is reached. The condition in line 2 only controls whether line 3 is executed. Line 12 is in while loop (line 5 -- 13) and will or will not be executed regardless of the condition in line 2. 

**Output**:
```json
{
  "ControlDependence": false
}
```

---

[YOUR TURN]

Below is **your target snippet**. 

```C
1   #include <stdio.h>
2   #include <stdlib.h>
3   #include <math.h>
4   void findSol(char** C, int A, int i);
5   int main(int argc, char *argv[]) {
6   	int T, A;
7   	scanf("%d", &T);
8   	for (int i=0; i<T; i++) {
9   		scanf("%d", &A);
10  		char **C;
11  		C = (char**) calloc(A, sizeof(char*));
12  		for (int j=0; j<A; j++) {
13  			 C[j] = (char *) calloc(501, sizeof(char));
14  			 scanf("%s", C[j]);
15  		}
16  		findSol(C, A, i+1);
17  	}
18  	return 0;
19  }
20  void findSol(char** C, int A, int T) {
21  	printf("Case #%d: ", T);
22  	char S[501], L;
23  	int X[3] = {0, 0, 0}, noSol=0, n=0, haveWon = 1;
24  	int *count;
25  	count = (int*) calloc(A, sizeof(int));
26  	while (1) {
27  		for (int i=0; i<A; i++) {
28  			if ( count[i] == -1 ) {
29  				continue;
30  			}
31  			haveWon=0;
32  			if ( C[i][count[i]] == '\0' ) {
33  				count[i] = 0;
34  			}
35  			if ( C[i][count[i]] == 'R' ) {
36  				X[0] = 1;
37  			} else if ( C[i][count[i]] == 'P' ) {
38  				X[1] = 1;
39  			} else {
40  				X[2] = 1;
41  			}
42  			if ( X[0] == 1 && X[1] == 1 && X[2] == 1 ) {
43  				noSol = 1;
44  				break;
45  			}
46  		}
47  		if (noSol || n > 500) {
48  			printf("IMPOSSIBLE\n");
49  			return;
50  		}
51  		if (haveWon) {
52  			S[n] = '\0';
53  			printf("%s\n", S);
54  			return;
55  		}
56  		if (X[0]) {
57  			if (X[2]) {
58  				S[n] = 'R';
59  				L = 'S';
60  			} else {
61  				S[n] = 'P';
62  				L = 'R';
63  			}
64  		} else {
65  			if (X[1]) {
66  				S[n] = 'S';
67  				L = 'P';
68  			} else {
69  				S[n] = 'R';
70  				L = 'S';
71  			}
72  		}
73  		for (int j=0; j<A; j++) {
74  			if ( count[j] == -1 || C[j][count[j]] == L) {
75  				count[j] = -1;
76  				continue;
77  			}
78  			count[j]++;
79  		}
80  		X[0] = 0; X[1] = 0; X[2] = 0;
81  		n++;
82  		haveWon = 1;
83  	}
84  }
```


**Question**: Does line `51` have control dependence over line `31` in function `findSol`? If so, provide a trace.

**Output**:


Ground truth answer: yes