[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<string.h>
3   int F(int p,int q,int r,char ft[100],int fc){
4   	int fit[100],ic,x,t;
5   	ic=1;
6   	for(t=0;t<100;t++){
7   		fit[t]=0;
8   	}
9   	while(ic<fc){
10  		if(ft[ic]=='P'){
11  			fit[ic]=p;
12  		}
13  		if(ft[ic]=='Q'){
14  			fit[ic]=q;
15  		}
16  		if(ft[ic]=='R'){
17  			fit[ic]=r;
18  		}
19  		if(ft[ic]=='0'){
20  			fit[ic]=0;
21  		}
22  		if(ft[ic]=='1'){
23  			fit[ic]=1;
24  		}
25  		if(ft[ic]=='2'){
26  			fit[ic]=2;
27  		}
28  		if(ft[ic]=='+'){
29  			fit[ic]=3;
30  		}
31  		if(ft[ic]=='-'){
32  			fit[ic]=4;
33  		}
34  		if(ft[ic]=='*'){
35  			fit[ic]=5;
36  		}
37  		ic++;
38  	}
39  	ic--;
40  	x=1;
41  	while(x<ic){
42  		if(fit[x]==3){
43  			if(fit[x-1]==0&&fit[x-2]==0){
44  				fit[x-2]=0;
45  			}
46  			else if(fit[x-1]==2||fit[x-2]==2){
47  				fit[x-2]=2;
48  			}
49  			else{
50  				fit[x-2]=1;
51  			}
52  			ic-=2;
53  			for(t=x-1;t<=ic;t++){
54  				fit[t]=fit[t+2];
55  			}
56  			x-=2;
57  		}
58  		if(fit[x]==4){
59  			fit[x-1]=2-fit[x-1];
60  			ic--;
61  			for(t=x;t<=ic;t++){
62  				fit[t]=fit[t+1];
63  			}
64  			x--;
65  		}
66  		if(fit[x]==5){
67  			if(fit[x-1]==0||fit[x-2]==0){
68  				fit[x-2]=0;
69  			}
70  			else if(fit[x-1]==2&&fit[x-2]==2){
71  				fit[x-2]=2;
72  			}
73  			else{
74  				fit[x-2]=1;
75  			}
76  			ic-=2;
77  			for(t=x-1;t<=ic;t++){
78  				fit[t]=fit[t+2];
79  			}
80  			x-=2;
81  		}
82  		x++;
83  	}
84  	return fit[1];
85  }
86  int main(void){
87  	int c,nc,oc,fc;
88  	char txt[100],ng[100],op[100],ft[100];
89  	while(1){
90  		scanf("%s",&txt);
91  		if(txt[0]=='.'){
92  			break;
93  		}
94  		nc=oc=fc=0;
95  		for(c=0;c<strlen(txt);c++){
96  			if(txt[c]==0||txt[c]==1||txt[c]==2||txt[c]=='P'||txt[c]=='Q'||txt[c]=='R'){
97  				fc++;
98  				ft[fc]=txt[c];
99  				if(nc!=0){
100  					while(nc!=0){
101  						fc++;
102  						ft[fc]=ng[nc];
103  						nc--;
104  					}
105  				}
106  			}
107  			if(txt[c]=='+'){
108  				if(oc==0||op[oc]=='('){
109  					oc++;
110  					op[oc]=txt[c];
111  				}
112  				else{
113  					while(oc!=0){
114  						fc++;
115  						ft[fc]=op[oc];
116  						oc--;
117  					}
118  					oc++;
119  					op[oc]=txt[c];
120  				}
121  			}
122  			if(txt[c]=='*'){
123  				if(oc==0||op[oc]=='+'||op[oc]=='('){
124  					oc++;
125  					op[oc]=txt[c];
126  				}
127  				else{
128  					while(oc!=0){
129  						fc++;
130  						ft[fc]=op[oc];
131  						oc--;
132  					}
133  					oc++;
134  					op[oc]=txt[c];
135  				}
136  			}
137  			if(txt[c]=='('){
138  				oc++;
139  				op[oc]=txt[c];
140  			}
141  			if(txt[c]==')'){
142  				while(op[oc]!='('){
143  					fc++;
144  					ft[fc]=op[oc];
145  					oc--;
146  				}
147  				op[oc]=0;
148  				oc--;
149  			}
150  			if(txt[c]=='-'){
151  				nc++;
152  				ng[nc]=txt[c];
153  			}
154  		}
155  		if(oc!=0){
156  			while(oc!=0){
157  				fc++;
158  				ft[fc]=op[oc];
159  				oc--;
160  			}
161  		}
162  		for(c=1;c<=fc;c++){
163  			printf("%c",ft[c]);
164  		}
165  		printf("\n");
166  		for(c=0;c<100;c++){
167  			ng[c]=op[c]=0;
168  		}
169  		int p,q,r,ans;
170  		ans=0;
171  		for(p=0;p<3;p++){
172  			for(q=0;q<3;q++){
173  				for(r=0;r<3;r++){
174  					if(F(p,q,r,ft,fc)==2){
175  						ans++;
176  					}
177  				}
178  			}
179  		}
180  		printf("%d\n",ans);
181  	}
182  	return 0;
183  }
```


**Question**: Does line `41` have control dependence over line `78` in function `F`? If so, provide a trace.

**Output**:


Ground truth answer: yes