[INSTRUCTIONS]

You are a program-analysis assistant. Perform a **static** data-dependence analysis on a given code snippet, treating each branch or loop condition as potentially taking any outcome, without using semantic or symbolic execution to prune paths.


## 1. Data Dependence Definition

   Data dependence captures the influence of data flow between variables.

   We denote each variable instance as `(var,lineNumber)`, meaning the variable `var` defined or updated at `lineNumber`.

   >  **Direct Data Dependence**
   A variable instance `(varB,lineB)` is **directly** data-dependent on `(varA,lineA)` if `lineA` writes to `varA`. Then, without overwritting the value of `varA`, `lineB` reads `varA` and compute the value of `varB`. In other words, the value of `(varB,lineB)` relies on the value produced at `(varA,lineA)`. In addition, changing the execution order of `(varA,lineA)` and `(varB,lineB)` would alter their results.


   A variable instance `(varB,lineB)` is data-dependent on `(varA,lineA)` if there is a **transitive** (indirect) chain of **direct** data dependencies from `(varA,lineA)` to `(varB,lineB)`. This is equivalent to saying that `(varA,lineA)` has data dependence over `(varB,lineB)`, meaning the value written at `(varA,lineA)` can propagate to `(varB,lineB)` through a sequence of read/write operations.


## 2. Output Format
   When asked "Does `(varA,lineA)` have data dependence over `(varB,lineB)`? If so, provide one feasible trace.": 
   - If data dependence exists, respond:
   ```json
   {
   "DataDependence": true,
   "Trace": [
      { "from": ["varA", lineA], "to": ["varX", lineX] },
      ...
      { "from": ["varY", lineY], "to": ["varB", lineB] }
      ]
   }
   ```

   -  If no data dependence exists, respond:
   ```json
   {
   "DataDependence": false
   }
   ```
   
   A **trace** represents a **transitive data flow**, where each item represents a **direct** data dependence edge. The full trace should begin with `(varA,lineA)` and end with `(varB,lineB)`.

   You only need to provide **one** valid chain of data dependence if you conclude a data dependence exists, even if multiple possible chains exist. 


## 3. Intraprocedural Data 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    total = 0
2    value = 1
3    step = value
4    if step > 1:
5        value += 3
6    while total <= 10:
7        total += value
8        difference = total - step
9        step += 1
10   final_result = total * 2
```


#### Example Question 1.1:
Does `(value,2)` have data dependence over `(step,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `step += 1`, meaning `(step,9)` is directly data-dependent on `(step,3)` (or `(step,9)` itself) because step is updated based on its previous value.
- Line 3: `step = value`, meaning `(step,3)` is directly data-dependent on `(value,2)`.
Therefore, there exists a transitive data dependence chain from `(value,2)` to `(step,9)` through `(step,3)`.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["step", 3] },
    { "from": ["step", 3], "to": ["step", 9] }
  ]
}
```

#### Example Question 1.2 
Does `(value,2)` have data dependence over `(total,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `total += value`, meaning `(total,7)` reads `value` from either `(value,2)` or `(value,5)`.
- Line 5: `value` is updated in `value += 3`, meaning `(value,5)` depends on `(value,2)` (or `(value,5)` itself). However, this update is inside an if statement, which means it may or may not execute depending on runtime conditions.
- Line 2: value is initialized at `(value,2)`.


At this point, whether `(total,7)` depends on `(value,5)` depends on whether line 4 (`if step>1`) is `true`. Since this is a static analysis, all possible paths are considered without analyzing actual reachability. This means line 5 is treated as both possibly executed and possibly not executed.

- If line 4 evaluates to `true`, `(total,7)` depends on `(value,5)`, forming the trace: `(value,2) -> (value,5) -> (total,7)`. 
- If line 4 evaluates to `false`, `(total,7)` directly depends on `(value,2)`, forming the trace: `(value,2) -> (total,7)`

Therefore, the answer is yes.
Only one valid trace is required for the output. 

**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["value", 2], "to": ["value", 5] },
    { "from": ["value", 5], "to": ["total", 7] }
  ]
}

```



#### Example Question 1.3
Does `(step,3)` have data dependence over `(final_result,10)`? If so, provide a trace.  

**Analysis**:
Since there is no direct or transitive data dependence between `(step,3)` and `(final_result,10)`, the answer is No.

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

**Analysis**:


### Example 2

```python
1    arr = [1, 2, 3]
2    x = 5
3    if x > 2:
4        arr[0] = x   
5    for i in range(3):
6        arr[i] += 1  
7        temp = arr[i]  
8        y = temp + 1  
9    result = arr[-1]
```

#### Example Question 2.1:
Does `(x,2)` have data dependence over `(arr,6)`? If so, provide a trace.

**Analysis**
- Line 6: `arr[i] += 1` modifies `arr[i]`, but i varies during the loop execution.
   - If `i == 0`, `arr[i]` refers to `arr[0]`, which was updated in line 4 (or line 6 itself), establishing a dependence chain `(arr,4)` -> `(arr,6)`.
   - If `i != 0`, `arr[i]` does not refer to `arr[0]`, meaning it depend on `(arr,1)` (or `(arr,6)` itself), not `(arr,4)`.
   - However, since static analysis considers all possible paths without resolving i’s runtime value, we must assume `arr[i]` could be `arr[0]`, leading to a possible dependence.
- Line 4: `arr[0] = x`, meaning `(arr,4)` depends on `(x,2)`.
- Line 2: `x` is initialized at `(x,2)`.
- Since there is a transitive chain of direct data dependence: `(x,2)->(arr,4)->(arr,6)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["x", 2], "to": ["arr", 4] },
    { "from": ["arr", 4], "to": ["arr", 6] }
  ]
}
```


#### Example Question 2.2:
Does `(i,5)` have data dependence over `(temp,7)`? If so, provide a trace.

**Analysis**:
- Line 7: `temp = arr[i]`, meaning temp depends on `i`, since `i` determines which element of arr is accessed.
- Line 5: `i` is initialized and iterates over the range 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(temp,7)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["temp", 7] }
  ]
}
```

#### Example Question 2.3:
Does `(i,5)` have data dependence over `(result,9)`? If so, provide a trace.

**Analysis**:
- Line 9: `result = arr[-1]`, meaning result depends on `arr[-1]`, which is arr[2], possibly get updated at line 6 (or line 1).
- Line 6: `arr[i] += 1`, meaning `arr[i]` is modified based on the value of `i`.
- Line 5: `i` iterates over 0 to 3.
- Since there is a transitive chain of direct data dependence: `(i,5)->(arr,6)->(result,9)`, the answer is yes.


**Output**:
```json
{
  "DataDependence": true,
  "Trace": [
    { "from": ["i", 5], "to": ["arr", 6] },
    { "from": ["arr", 6], "to": ["result", 9] }
  ]
}
```

---

[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 `(x,56)` have data dependence over `(fit,71)` in function `F`? If so, provide a trace.

**Output**:


Ground truth answer: yes