
AutoToS prompt:

Logical deduction is performed by applying rules of the form if X then Y to our current knowledge about a world, extending our knowledge with each single rule application. 
A rule if X then Y is applicable if we know X. If a state is defined as our current knowledge of the world, how are successor states defined, given a collection of rules? Please think step by step. Then, provide only the Python implementation for the successor function that receives a state and rules and returns a list of successor states.

Example initial state: {"integer"}
Example rules: [("integer", "real number"), ("integer", "float")]

Model response:
To define successor states given a collection of rules, we need to follow these steps:

1. **Identify Applicable Rules**: For each rule `if X then Y`, check if `X` is present in the current state.
2. **Generate Successor States**: For each applicable rule, create a new state by adding `Y` to the current state.
3. **Ensure Uniqueness**: Each successor state should be unique to avoid redundant states.

Let's break down the process step-by-step:

1. **Identify Applicable Rules**:
   - Iterate through the list of rules.
   - For each rule `(X, Y)`, check if `X` is in the current state.

2. **Generate Successor States**:
   - If `X` is in the current state, create a new state by copying the current state and adding `Y` to it.

3. **Ensure Uniqueness**:
   - Use a set or a similar data structure to ensure that each successor state is unique.

Now, let's implement this in Python:

python
```
def successor(state, rules):
    successor_states = []
    for rule in rules:
        X, Y = rule
        if X in state:
            new_state = state.copy()
            new_state.add(Y)
            if new_state not in successor_states:
                successor_states.append(new_state)
    return successor_states

# Example usage:
initial_state = {"integer"}
rules = [("integer", "real number"), ("integer", "float")]
print(successor(initial_state, rules))
```

### Explanation:
- **state**: A set representing the current knowledge of the world.
- **rules**: A list of tuples where each tuple `(X, Y)` represents a rule `if X then Y`.
- **successor_states**: A list to store unique successor states.
- **for rule in rules**: Iterate through each rule.
- **if X in state**: Check if `X` is in the current state.
- **new_state = state.copy()**: Create a copy of the current state.
- **new_state.add(Y)**: Add `Y` to the new state.
- **if new_state not in successor_states**: Ensure the new state is unique before adding it to the list of successor states.

This function will return a list of unique successor states based on the given rules and the current state

AutoToS prompt:

Provide only the Python function for the goal test function that tests, given a target fact, whether a state is a goal state. 

Model response:
python
```
def goal_test(state, target_fact):
    return target_fact in state
``

System message:
Goal Soundness Test Passed
Soundness test

System message:
Successor States Soundness Test Passed
Completeness test
Successor Completeness Test Passed
Success: 4000, failure: 0, total expansions 85523, total evaluations 244776
