[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   from math import sqrt, pow, log, ceil, log10
2   from sys import stdin, setrecursionlimit
3   from sets import Set
4   setrecursionlimit(100000)
5   debug = False
6   def isCycleCorrect(A, C, L, ans):
7       la = len(ans)
8       for i in range(A):
9           pos = 0
10          isThereAWinner = False
11          while (isThereAWinner == False) and (pos < L[i] * la):
12              if C[i][pos % L[i]] != ans[pos % la]:
13                  if debug:
14                      print "->", C[i][pos % L[i]], ans[pos % la], A, i
15                  if (C[i][pos % L[i]], ans[pos % la]) == ('R', 'S'):
16                      return False
17                  if (C[i][pos % L[i]], ans[pos % la]) == ('S', 'R'):
18                      isThereAWinner = True
19                      break
20                  if (C[i][pos % L[i]], ans[pos % la]) == ('R', 'P'):
21                      isThereAWinner = True
22                      break
23                  if (C[i][pos % L[i]], ans[pos % la]) == ('P', 'R'):
24                      return False
25                  if (C[i][pos % L[i]], ans[pos % la]) == ('P', 'S'):
26                      isThereAWinner = True
27                      break
28                  if (C[i][pos % L[i]], ans[pos % la]) == ('S', 'P'):
29                      return False
30              pos += 1
31          return False
32      return True
33  def solve(A, C, L):
34      ans = ""
35      pos = 0
36      isAlive = [True] * A
37      while True:
38          possibilities = {}
39          possibilities['R'] = 0
40          possibilities['S'] = 0
41          possibilities['P'] = 0
42          predR = []
43          predS = []
44          predP = []
45          if debug:
46              print "alive ans", isAlive, ans
47          if not any(isAlive):
48              print ans
49              return
50          for i in range(A):
51              if isAlive[i]:
52                  possibilities[C[i][pos % L[i]]] = 1
53                  if C[i][pos % L[i]] == 'R':
54                      predR += [i]
55                  if C[i][pos % L[i]] == 'S':
56                      predS += [i]
57                  if C[i][pos % L[i]] == 'P':
58                      predP += [i]
59                  if possibilities['R'] + possibilities['S'] + possibilities['P'] == 3:
60                      print "IMPOSSIBLE"
61                      return
62          (r, s, p) = (possibilities['R'], possibilities['S'], possibilities['P'])
63          if debug:
64              print "->", (r, s, p)
65          if (r, s, p) == (1, 0, 0):
66              ans += "P"
67              print ans
68              return
69          if (r, s, p) == (0, 1, 0):
70              ans += "R"
71              print ans
72              return
73          if (r, s, p) == (0, 0, 1):
74              ans += "S"
75              print ans
76              return
77          if len(ans) == 500:
78              print "IMPOSSIBLE"
79              if debug:
80                  print "LEN", ans
81              return
82          if (r, s, p) == (1, 1, 0):
83              for i in predS:
84                  isAlive[i] = False
85              ans += "R"
86          if (r, s, p) == (1, 0, 1):
87              for i in predR:
88                  isAlive[i] = False
89              ans += "P"
90          if (r, s, p) == (0, 1, 1):
91              for i in predP:
92                  isAlive[i] = False
93              ans += "S"
94          pos += 1
95  T = int(stdin.readline())
96  for i in range(1,T+1):
97      A = stdin.readline()
98      A = int(A)
99      C = [""] * A
100      L = [0] * A
101      for j in range(A):
102          C[j] = stdin.readline().rstrip() 
103          L[j] = len(C[j])
104      print "Case #" + str(i) + ":", 
105      if debug:
106          print " -> ", A, C, L
107      solve(A, C, L)
```


**Question**: Does `(possibilities,41)` have data dependence over `(r,62)` in function `solve`? If so, provide a trace.

**Output**:


Ground truth answer: yes