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Unification in First-Order Logic: A Key Process in AI Inference

Understand the concept of unification in first-order logic and its importance in artificial intelligence (AI) inference algorithms. Learn how unification determines the substitutability of variables to make atomic sentences identical, and explore the concept of the most general unifier (MGU) with practical examples.

Unification in First-Order Logic

What is Unification?

Unification is a process in first-order logic that determines whether two atomic sentences can be made identical by substituting variables with terms. It's a fundamental operation used in many AI inference algorithms. If unification is successful, it finds a substitution (called a unifier) that makes the two sentences identical. The most general unifier (MGU) is a substitution that unifies two sentences and is more general than any other unifier for those sentences.

Unification Example

Let's unify King(x) and King(John).

The substitution θ = {John/x} (replace x with John) unifies these atomic sentences.

The UNIFY Algorithm

The UNIFY algorithm takes two atomic sentences as input and returns a unifier (if one exists). It returns "failure" if the sentences cannot be unified.

Unification Conditions

For unification to succeed, these conditions must be met:

  • The predicate symbols must be the same.
  • The number of arguments must be the same.
  • No variable can be substituted for a term containing itself.

Unification Algorithm Steps

The unification algorithm can be described as follows:

  1. If one expression is a variable or constant:
    1. If they are identical, return NIL (empty substitution).
    2. If the first expression is a variable and doesn't occur in the second expression, return the substitution that replaces that variable with the second expression.
    3. If the second expression is a variable and doesn't occur in the first expression, return the substitution that replaces that variable with the first expression.
    4. Otherwise, return FAILURE.
  2. If the predicate symbols are different, return FAILURE.
  3. If the number of arguments is different, return FAILURE.
  4. Recursively unify the corresponding arguments. If any recursive call returns FAILURE, return FAILURE.
  5. Combine all substitutions into a single substitution set and return it.

Unification Examples

(Several examples demonstrating the unification process and the application of the unification algorithm are provided in the original text, but are omitted here for brevity. These examples, with detailed step-by-step explanations, would be included in the HTML.)