[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**. 

```Python
1   def areInConflict(x, y):
2       bx = "{0:b}".format(x)[::-1]
3       by = "{0:b}".format(y)[::-1]
4       lx = len(bx)
5       ly = len(by)
6       for i in range(min(lx, ly)):
7           if bx[i]=="1" and by[i]=="1":
8               return True
9       return False
10  def solve(x,y):
11      origX = x
12      origY = y
13      x = max(x, -x)
14      y = max(y, -y)
15      if x%2==y%2:
16          return "IMPOSSIBLE"
17      x1 = 1
18      while x1 < x:
19          x1 *= 2
20      x2 = x1 - x
21      y1 = 1
22      while y1 < y:
23          y1 *= 2
24      y2 = y1 - y
25      if not areInConflict(x, y):
26          bx = "{0:b}".format(x)[::-1]
27          by = "{0:b}".format(y)[::-1]
28          result = [None] * max(len(bx),len(by))
29          for i in range(len(bx)):
30              if bx[i] == "1":
31                  result[i] = "E"
32          for i in range(len(by)):
33              if by[i] == "1":
34                  result[i] = "N"
35      elif not areInConflict(x2, x1) and not areInConflict(x2, y) and not areInConflict(x1, y):
36          bx1 = "{0:b}".format(x1)[::-1]
37          bx2 = "{0:b}".format(x2)[::-1]
38          by = "{0:b}".format(y)[::-1]
39          result = [None] * max(len(bx1), len(bx2), len(by))
40          for i in range(len(bx1)):
41              if bx1[i] == "1":
42                  result[i] = "E"
43          for i in range(len(bx2)):
44              if bx2[i] == "1":
45                  result[i] = "W"
46          for i in range(len(by)):
47              if by[i] == "1":
48                  result[i] = "N"
49      elif not areInConflict(y2, y1) and not areInConflict(y2, x) and not areInConflict(y1, x):
50          by1 = "{0:b}".format(y1)[::-1]
51          by2 = "{0:b}".format(y2)[::-1]
52          bx = "{0:b}".format(x)[::-1]
53          result = [None] * max(len(by1), len(by2), len(bx))
54          for i in range(len(bx)):
55              if bx[i] == "1":
56                  result[i] = "E"
57          for i in range(len(by1)):
58              if by1[i] == "1":
59                  result[i] = "N"
60          for i in range(len(by2)):
61              if by2[i] == "1":
62                  result[i] = "S"
63      elif not areInConflict(y2, y1) \
64          and not areInConflict(y2, x1) \
65          and not areInConflict(y2, x2) \
66          and not areInConflict(y1, x1) \
67          and not areInConflict(y1, x2) \
68          and not areInConflict(x2, x1):
69          bx1 = "{0:b}".format(x1)[::-1]
70          bx2 = "{0:b}".format(x2)[::-1]
71          by1 = "{0:b}".format(y1)[::-1]
72          by2 = "{0:b}".format(y2)[::-1]
73          result = [None] * max(len(by1), len(by2), len(bx))
74          for i in range(len(bx1)):
75              if bx1[i] == "1":
76                  result[i] = "E"
77          for i in range(len(bx2)):
78              if bx2[i] == "1":
79                  result[i] = "W"
80          for i in range(len(by1)):
81              if by1[i] == "1":
82                  result[i] = "N"
83          for i in range(len(by2)):
84              if by2[i] == "1":
85                  result[i] = "S"
86      else:
87          return "IMPOSSIBLE"
88      if x != origX:
89          for i in xrange(len(result)):
90              if result[i]=="W":
91                  result[i] = "E"
92              elif result[i]=="E":
93                  result[i] = "W"
94      if y != origY:
95          for i in xrange(len(result)):
96              if result[i]=="N":
97                  result[i] = "S"
98              elif result[i]=="S":
99                  result[i] = "N"
100      return "".join(result)
101  T = int(raw_input())
102  for t in range(T):
103      x,y = map(int,raw_input().split())
104      print "Case #{}: {}".format(t+1, solve(x, y))
```


**Question**: Does `(origY,12)` have data dependence over `(x,18)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: no