[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<stdlib.h>
3   #include<stdbool.h>
4   #include<ctype.h>
5   int sol[151];
6   int queryList[150];
7   char solStr[151];
8   int generateQueries(int n)
9   {
10      int beginning = 1, end = n, i = 0;
11      while(beginning <= end)
12      {
13          queryList[i] = beginning;
14          beginning++;
15          i++;
16          if(beginning < end)
17          {
18              queryList[i] = end;
19              end--;
20              i++;
21          }
22      }
23      while(i < 150)
24      {
25          queryList[i] = -1;
26          i++;
27      }
28  }
29  int findFlipBit(int n)
30  {
31      int i = 1;
32      while(i <= n && sol[i] != 2)
33      {
34          if(sol[i] == sol[n-i+1])
35          {
36              return i;
37          }
38          else i++;
39      }
40      return -1;
41  }
42  int findSwapBit(int n)
43  {
44      int i = 1;
45      while(i <= n && sol[i] != 2 && sol[n-i+1] != 2)
46      {
47          if(sol[i] != sol[n-i+1])
48          {
49              return i;
50          }
51          else i++;
52      }
53      return -1;
54  }
55  int query(int i)
56  {
57      int ret;
58      printf("%d\n", i);
59      fflush(stdout);
60      scanf("%d", &ret);
61      if(ret == 'N')
62      {
63          exit(1);
64      }
65      else
66      {
67          return ret;
68      }
69  }
70  void flipSol(int n)
71  {
72      for(int i = 1; i <= n; i++)
73      {
74          if(sol[i] == 1) sol[i] = 0;
75          else if(sol[i] == 0) sol[i] = 1;
76      }
77  }
78  void swapSol(int n)
79  {
80      int beginning = 1, end = n;
81      while (beginning < end)
82      {
83          int aux = sol[beginning];
84          sol[beginning] = sol[end];
85          sol[end] = aux;
86          beginning++;
87          end--;
88      }
89  }
90  void resolve(int n)
91  {
92      for(int i = 1; i <= n; i++)
93      {
94          sol[i] = 2;
95      }
96      generateQueries(n);
97      int queries = 0;
98      int known = 0;
99      int flipBit = -1, swapBit = -1;
100      int flipRes, swapRes, doubleRes;
101      bool flipped = false, swapped = false;
102      while(true)
103      {
104          if(queries % 10 == 0 && queries != 0)
105          {
106              if(flipBit == -1) flipBit = findFlipBit(n);
107              if(flipBit != -1)
108              {
109                  flipRes = query(flipBit);
110                  if(flipRes != sol[flipBit]) flipSol(n);
111                  else flipped = false;
112                  queries++;
113              }
114              else
115              {
116                  query(1);
117                  queries++;
118              }
119              if(swapBit == -1)
120              {
121                  swapBit = findSwapBit(n);
122              }
123              if(swapBit != -1)
124              {
125                  swapRes = query(swapBit);
126                  if(swapRes != sol[swapBit]) swapSol(n);
127                  else swapped = false;
128                  queries++;
129              }
130              else
131              {
132                  query(1);
133                  queries++;
134              }
135          }
136          else
137          {
138              sol[queryList[known]] = query(queryList[known]);
139              known++;
140              queries++;
141              if(queryList[known] == -1)
142              {
143                  for(int i = 1; i < n; i++)
144                  {
145                      printf("%d", sol[i]);
146                  }
147                  printf("%d\n", sol[n]);
148                  fflush(stdout);
149                  char answer  = 10;
150                  while (answer == 10)
151                  {
152                      answer = getchar();
153                  }
154                  if(answer == 'Y')
155                  {
156                      return;
157                  }
158                  else
159                  {
160                      exit(7);
161                  }
162              }
163          }
164      }
165  }
166  int main(void)
167  {
168      int t, n;
169      scanf("%d %d", &t, &n);
170      for(int test = 0; test < t; test++)
171      {
172          resolve(n);
173      }
174      return 0;
175  }
```


**Question**: Does `(n,8)` have data dependence over `(end,10)` in function `generateQueries`? If so, provide a trace.

**Output**:


Ground truth answer: yes