
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:
```python
def successor(state, rules):
    def apply_rules(state, rules):
        successors = set()
        for x, y in rules:
            if x in state:
                successors.add(y)
        return successors

    return list(apply_rules(state, rules))
```

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 is_goal_state(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 Iteration 1

AutoToS prompt:
Successor function when run on the state {'painted lady'} failed to produce all successors.
Missing successors are: [{'painted lady', 'butterfly'}]
First think step by step why the successor function failed to produce all successors of the state.
Then, fix the successor function.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.
Rules: [('arthropod', 'protostome'), ('lepidopteran', 'insect'), ('painted lady', 'butterfly'), ('insect', 'arthropod'), ('invertebrate', 'animal'), ('arthropod', 'not-bony'), ('protostome', 'invertebrate'), ('whale', 'bony'), ('butterfly', 'lepidopteran'), ('animal', 'multicellular'), ('insect', 'six-legged')]

Model response:
```python
def successor(state, rules):
    def apply_rules(state, rules):
        successors = set(state)  # Start with current state in successors
        changes = True

        while changes:  # Continue until no new successors are found
            changes = False
            new_successors = set()
            for x, y in rules:
                if x in successors and y not in successors:
                    new_successors.add(y)
                    changes = True
            successors.update(new_successors)

        return successors

    return list(apply_rules(state, rules))
```

System message:
Successor States Soundness Test Passed
Completeness test
Successor Iteration 2

AutoToS prompt:
Successor function when run on the state {'painted lady'} failed to produce all successors.
Missing successors are: [{'painted lady', 'butterfly'}]
First think step by step why the successor function failed to produce all successors of the state.
Then, fix the successor function.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.
Rules: [('arthropod', 'protostome'), ('lepidopteran', 'insect'), ('painted lady', 'butterfly'), ('insect', 'arthropod'), ('invertebrate', 'animal'), ('arthropod', 'not-bony'), ('protostome', 'invertebrate'), ('whale', 'bony'), ('butterfly', 'lepidopteran'), ('animal', 'multicellular'), ('insect', 'six-legged')]

Model response:
```python
def successor(state, rules):
    def apply_rules(state, rules):
        successors = set(state)  # Start with the current state in successors
        changes = True

        while changes:  # Continue until no new successors are found
            changes = False
            new_successors = set()
            for x, y in rules:
                if x in successors and y not in successors:
                    new_successors.add(y)
                    changes = True
            successors.update(new_successors)

        return successors

    return [tuple(successor) for successor in apply_rules(state, rules)]
```

System message:
Successor States Soundness Test Passed
Completeness test
Successor Iteration 3

AutoToS prompt:
Successor function when run on the state {'painted lady'} failed to produce all successors.
Missing successors are: [{'painted lady', 'butterfly'}]
First think step by step why the successor function failed to produce all successors of the state.
Then, fix the successor function.
Remember how you fixed the previous mistakes, if any. Keep the same function signature.
Rules: [('arthropod', 'protostome'), ('lepidopteran', 'insect'), ('painted lady', 'butterfly'), ('insect', 'arthropod'), ('invertebrate', 'animal'), ('arthropod', 'not-bony'), ('protostome', 'invertebrate'), ('whale', 'bony'), ('butterfly', 'lepidopteran'), ('animal', 'multicellular'), ('insect', 'six-legged')]

Model response:
```python
def successor(state, rules):
    def apply_rules(state, rules):
        successors = set(state)  # Start with the current state in successors
        changes = True

        while changes:  # Continue until no new successors are found
            changes = False
            new_successors = set()
            for x, y in rules:
                if x in successors and y not in successors:
                    new_successors.add(y)
                    changes = True
            successors.update(new_successors)

        return successors

    initial_successors = apply_rules(state, rules)
    all_successors = [{*state, s} for s in initial_successors]  # Include all initial states with each successor
    return list(all_successors)
```

System message:
Successor States Soundness Test Passed
Completeness test
Successor Completeness Test Passed
Success: 4000, failure: 0, total expansions 897073, total evaluations 9156989
