
AutoToS prompt:

The task is a 5x5 mini crosswords. A state is a 2D array representing the current puzzle state, where the initial grid is all "None". Note that some of the possible answers are not exactly 5 character long. Given an input of possible answers to horizontal clues and vertical clues, how is the successor state function defined? Please first think step by step. Then provide the successor state function in Python code. 

The possible clues for each row and each column are given separately. Here is an example of possible horizontal and vertical clues:

horizontal_answers = [
        ["tasks", "goals", "plans", "agend", "chores", "works", "deeds", "items", "lists", "brief"],
        ["motor", "power", "drive", "diesel", "steam", "pumps", "crank", "gears", "turbn", "motor"],
        ["grand", "artsy", "showy", "ornate", "fancy", "vain", "proud", "vogue", "swank", "luxus"], 
        ["venue", "salle", "forum", "atria", "lobby", "parls", "court", "malls", "mall", "lobby"], 
        ["jeer", "scoff", "sleer", "deris", "sneer", "scorn", "derid", "gibes", "gibed", "flout"] 
]
vertical_answers = [
        ["amass", "stack", "hoard", "pile", "store", "heaps", "massy", "gathe", "lumps", "mound"],
        ["nilga", "goral", "eland", "lepus", "gazal", "kudu", "oryx", "gnu", "imps", "carb"],
        ["scheme", "design", "ettle", "nettle", "sting", "wiles", "plans", "ideas", "plots", "cocks"], 
        ["spout", "nosle", "snout", "mouth", "nostr", "ports", "inlet", "vents", "outlt", "beaks"], 
        ["drier", "arid", "sere", "parch", "dryer", "wring", "drear", "sear", "pall", "lack"]
]

where horizontal_answers is a list where element i is a list of possible answers to clue in row i, and vertical_answers is a list where element i is a list of possible answers to clue in column i.

Model response:
To define the successor state function for a 5x5 mini crossword puzzle, we need to consider how to update the puzzle state based on the given possible answers for horizontal and vertical clues. The goal is to fill in the puzzle grid one cell at a time, ensuring that each cell is filled correctly according to the constraints provided by the possible answers.

### Step-by-Step Approach:

1. **Initialization**: Start with an empty 5x5 grid (all cells set to `None`).
2. **Selection**: Choose a row or column to fill next. This choice can be made based on the current state of the puzzle, aiming to minimize conflicts or maximize progress.
3. **Validation**: For the chosen row or column, check each possible answer to see if it fits within the current state of the puzzle without conflicts.
4. **Update**: If a valid answer is found, update the corresponding row or column in the puzzle grid.
5. **Iteration**: Repeat the process until the puzzle is fully solved or no further progress can be made.

### Successor State Function in Python:

The following Python function `get_successor_state` implements the successor state logic:

python
```
def get_successor_state(current_state, horizontal_answers, vertical_answers):
    def is_valid_placement(state, answer, row_or_col, is_row):
        if is_row:
            for i in range(len(answer)):
                if state[row_or_col][i] is not None and state[row_or_col][i] != answer[i]:
                    return False
        else:
            for i in range(len(answer)):
                if state[i][row_or_col] is not None and state[i][row_or_col] != answer[i]:
                    return False
        return True

    def place_answer(state, answer, row_or_col, is_row):
        if is_row:
            for i in range(len(answer)):
                state[row_or_col][i] = answer[i]
        else:
            for i in range(len(answer)):
                state[i][row_or_col] = answer[i]

    for row in range(5):
        for answer in horizontal_answers[row]:
            if len(answer) <= 5 and is_valid_placement(current_state, answer, row, True):
                new_state = [row[:] for row in current_state]
                place_answer(new_state, answer, row, True)
                yield new_state

    for col in range(5):
        for answer in vertical_answers[col]:
            if len(answer) <= 5 and is_valid_placement(current_state, answer, col, False):
                new_state = [row[:] for row in current_state]
                place_answer(new_state, answer, col, False)
                yield new_state

# Example usage:
initial_state = [[None] * 5 for _ in range(5)]
for state in get_successor_state(initial_state, horizontal_answers, vertical_answers):
    for row in state:
        print(row)
    print("-----")
```

### Explanation:

1. **is_valid_placement**: This helper function checks if a given answer can be placed in a specified row or column without conflicts.
2. **place_answer**: This helper function places a valid answer into the specified row or column of the puzzle grid.
3. **get_successor_state**: This function iterates over each row and column, trying to place each possible answer. If a valid placement is found, it yields the updated state.

This approach ensures that the puzzle is filled incrementally, respecting the constraints provided by the possible answers for each row and column

AutoToS prompt:

Please provide a goal test function in Python ensuring that all cells are not None, all words horizontally are matching at least one of the horizontal answers, and all words vertically match at least one of the vertical answers. The function should receive a state, horizontal_answers, and vertical_answers as before and return True for a goal state and False otherwise.

Model response:
Certainly! Below is the Python function `is_goal_state` that checks if a given state of the 5x5 mini crossword puzzle meets the criteria for being a goal state:

python
```
def is_goal_state(state, horizontal_answers, vertical_answers):
    # Check if all cells are not None
    for row in state:
        if None in row:
            return False

    # Check if all horizontal words match at least one of the horizontal answers
    for i, row in enumerate(state):
        word = ''.join(row)
        if word not in horizontal_answers[i]:
            return False

    # Check if all vertical words match at least one of the vertical answers
    for j in range(5):
        word = ''.join(state[i][j] for i in range(5))
        if word not in vertical_answers[j]:
            return False

    return True

# Example usage:
goal_state = [
    ['t', 'a', 's', 'k', 's'],
    ['m', 'o', 't', 'o', 'r'],
    ['g', 'r', 'a', 'n', 'd'],
    ['v', 'e', 'n', 'u', 'e'],
    ['j', 'e', 'e', 'r', 's']
]

print(is_goal_state(goal_state, horizontal_answers, vertical_answers))  # Should return True
```

### Explanation:

1. **Check for None Cells**: The function first iterates through each row of the state to ensure that no cell contains `None`. If any cell is `None`, it returns `False`.

2. **Check Horizontal Words**: For each row, the function constructs the word by joining the characters in the row and checks if this word is present in the corresponding list of horizontal answers. If any word does not match, it returns `False`.

3. **Check Vertical Words**: For each column, the function constructs the word by joining the characters in the column and checks if this word is present in the corresponding list of vertical answers. If any word does not match, it returns `False`.

4. **Return True**: If all checks pass, the function returns `True`, indicating that the state is a goal state.

This function ensures that the puzzle is fully solved and that all words match the given clues for both horizontal and vertical directions

System message:
Goal Soundness Test Passed
Soundness test
Successor Iteration 1

AutoToS prompt:
Successor state has the same number of filled cells as the parent state.
Let's think step by step. First, think what you did wrong.
Then, think of in what ways successor state should be different from the parent state.
Then, provide the complete Python code for the revised successor function that returns a list of successor states.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.

Input state: [[None, None, None, None, 'l'], [None, None, None, None, 'a'], [None, None, None, None, 'c'], [None, None, None, None, 'k'], [None, None, None, None, None]]
Example wrong successor state: [[None, None, None, None, 'l'], [None, None, None, None, 'a'], [None, None, None, None, 'c'], [None, None, None, None, 'k'], [None, None, None, None, None]]
Horizontal clues: [['tasks', 'goals', 'plans', 'agend', 'chores', 'works', 'deeds', 'items', 'lists', 'brief'], ['motor', 'power', 'drive', 'diesel', 'steam', 'pumps', 'crank', 'gears', 'turbn', 'motor'], ['grand', 'artsy', 'showy', 'ornate', 'fancy', 'vain', 'proud', 'vogue', 'swank', 'luxus'], ['venue', 'salle', 'forum', 'atria', 'lobby', 'parls', 'court', 'malls', 'mall', 'lobby'], ['jeer', 'scoff', 'sleer', 'deris', 'sneer', 'scorn', 'derid', 'gibes', 'gibed', 'flout']]
Vertical clues: [['amass', 'stack', 'hoard', 'pile', 'store', 'heaps', 'massy', 'gathe', 'lumps', 'mound'], ['nilga', 'goral', 'eland', 'lepus', 'gazal', 'kudu', 'oryx', 'gnu', 'imps', 'carb'], ['scheme', 'design', 'ettle', 'nettle', 'sting', 'wiles', 'plans', 'ideas', 'plots', 'cocks'], ['spout', 'nosle', 'snout', 'mouth', 'nostr', 'ports', 'inlet', 'vents', 'outlt', 'beaks'], ['drier', 'arid', 'sere', 'parch', 'dryer', 'wring', 'drear', 'sear', 'pall', 'lack']]

Model response:
### Step-by-Step Analysis and Revision

#### What Went Wrong:
The initial successor state function did not ensure that each successor state had a different number of filled cells compared to the parent state. This means that the function was not progressing the state towards a solution but rather generating states that were identical to the parent state.

#### How to Make Successor State Different:
To ensure that each successor state is different from the parent state, we need to:
1. Only attempt to place words in rows or columns that have empty cells.
2. Ensure that placing a word in a row or column changes the state by filling in at least one `None` cell.
3. Avoid generating states that are identical to the parent state.

#### Revised Successor State Function:
The revised function will focus on filling in the next available `None` cell by trying to place valid words from the horizontal and vertical clues.

python
```
def get_successor_state(current_state, horizontal_answers, vertical_answers):
    def is_valid_placement(state, answer, row_or_col, is_row):
        if is_row:
            for i in range(len(answer)):
                if state[row_or_col][i] is not None and state[row_or_col][i] != answer[i]:
                    return False
        else:
            for i in range(len(answer)):
                if state[i][row_or_col] is not None and state[i][row_or_col] != answer[i]:
                    return False
        return True

    def place_answer(state, answer, row_or_col, is_row):
        if is_row:
            for i in range(len(answer)):
                state[row_or_col][i] = answer[i]
        else:
            for i in range(len(answer)):
                state[i][row_or_col] = answer[i]

    successor_states = []

    # Try to place horizontal answers
    for row in range(5):
        if None in current_state[row]:  # Only attempt to place if there are None cells
            for answer in horizontal_answers[row]:
                if len(answer) <= 5 and is_valid_placement(current_state, answer, row, True):
                    new_state = [row[:] for row in current_state]
                    place_answer(new_state, answer, row, True)
                    if new_state != current_state:  # Ensure the state has changed
                        successor_states.append(new_state)

    # Try to place vertical answers
    for col in range(5):
        if any(current_state[row][col] is None for row in range(5)):  # Only attempt to place if there are None cells
            for answer in vertical_answers[col]:
                if len(answer) <= 5 and is_valid_placement(current_state, answer, col, False):
                    new_state = [row[:] for row in current_state]
                    place_answer(new_state, answer, col, False)
                    if new_state != current_state:  # Ensure the state has changed
                        successor_states.append(new_state)

    return successor_states

# Example usage:
initial_state = [
    [None, None, None, None, 'l'],
    [None, None, None, None, 'a'],
    [None, None, None, None, 'c'],
    [None, None, None, None, 'k'],
    [None, None, None, None, None]
]

successor_states = get_successor_state(initial_state, horizontal_answers, vertical_answers)
for state in successor_states:
    for row in state:
        print(row)
    print("-----")
```

### Explanation:
1. **is_valid_placement**: Checks if an answer can be placed in a specified row or column without conflicts.
2. **place_answer**: Places a valid answer into the specified row or column of the puzzle grid.
3. **get_successor_state**: Iterates over each row and column, trying to place each possible answer. It ensures that only rows or columns with `None` cells are attempted, and it only adds states to the successor list if they are different from the current state.

This revised function ensures that each successor state has a different number of filled cells compared to the parent state, progressing the puzzle towards a solution

System message:
Successor States Soundness Test Passed
Completeness test
Successor Completeness Test Passed
Evaluating the final outcome
Success: 20, failure: 0, total expansions 1587973, total generated 5939765
