Two-Pointer Resolution Algorithm in Python: Solving 'Howling Hounds' Problem

Two-Pointer Resolution Algorithm in Python: Solving 'Howling Hounds' Problem

This article provides a Python implementation of the Two-Pointer resolution algorithm, demonstrating its application in solving the 'Howling Hounds' logic problem. It explains the core concepts, steps involved, and offers a simple example for understanding.

Two-Pointer Resolution Algorithm

The Two-Pointer resolution algorithm is a fundamental technique used in logic programming for automated theorem proving. It involves iteratively resolving pairs of clauses in a knowledge base to derive new clauses until an empty clause is found, indicating a successful proof.

Python Implementation

Here's a Python implementation of the Two-Pointer resolution algorithm applied to the 'Howling Hounds' problem:

def resolve(clause1, clause2):
    for literal1 in clause1:
        for literal2 in clause2:
            if literal1 == -literal2:  # Opposite literals
                resolvent = clause1 + clause2
                resolvent.remove(literal1)
                resolvent.remove(literal2)
                return resolvent
    return None

def is_resolvable(clause1, clause2):
    for literal1 in clause1:
        for literal2 in clause2:
            if literal1 == -literal2:  # Opposite literals
                return True
    return False

def resolution(knowledge_base, query):
    clauses = knowledge_base + [query]
    new_clause = None
    while True:
        n = len(clauses)
        pairs = [(clauses[i], clauses[j]) for i in range(n) for j in range(i+1, n)]
        for (clause1, clause2) in pairs:
            if is_resolvable(clause1, clause2):
                new_clause = resolve(clause1, clause2)
                if not new_clause:  # Empty clause
                    return True
                if new_clause not in clauses:
                    clauses.append(new_clause)
        if len(clauses) == n:
            return False

# Example usage with 'Howling Hounds' problem
knowledge_base = [[1], [2], [3], [4]]
query = [-2]  # ﾬHasMice(John)
result = resolution(knowledge_base, query)
if result:
    print('John does not have any mice')
else:
    print('Cannot prove that John does not have any mice')

Explanation:

1. resolve(clause1, clause2): This function checks if two clauses (clause1 and clause2) are resolvable. If they are, it returns the resolvent (a new clause formed by removing the opposite literals), otherwise, it returns None.
2. is_resolvable(clause1, clause2): This function determines if two clauses contain opposite literals (one positive and one negative), which is necessary for resolution.
3. resolution(knowledge_base, query): This is the main function that implements the Two-Pointer resolution algorithm. It iterates through pairs of clauses in the knowledge_base and the query, checking if they are resolvable. If a resolvent is found, it is added to the clauses list. The algorithm continues until an empty clause is found (indicating success) or no new clauses are generated (indicating failure).

Application to the 'Howling Hounds' Problem

The provided example uses a simplified knowledge base and query for the 'Howling Hounds' problem. You can modify and extend this example to represent a more comprehensive knowledge base and handle more complex logic expressions.

Conclusion

This article provides a basic understanding of the Two-Pointer resolution algorithm and its Python implementation for solving logic problems like 'Howling Hounds'. It serves as a starting point for exploring and applying this technique for automated theorem proving in various domains. Remember that for complex problems, more sophisticated algorithms and implementations might be required.