My Blog.

Explain Unification algorithm with suitable example.

Unification Algorithm

Definition

Unification is a fundamental algorithm in logic programming and automated reasoning. It is the process of finding a substitution that makes two logical expressions identical. A substitution is a set of variable bindings that, when applied to the expressions, makes them syntactically identical.

Key Concepts

  • Substitution: A mapping of variables to terms. For example, the substitution ($$ {x \rightarrow a, y \rightarrow b}$$ ) maps variable ( x ) to term ( a ) and variable ( y ) to term ( b ).
  • Most General Unifier (MGU): The simplest substitution that unifies two expressions. It is "most general" because it introduces the fewest constraints necessary to achieve unification.
  • Occurs Check: A condition that ensures a variable does not get unified with a term that contains that variable, preventing infinite regressions (e.g., ( x ) should not unify with ( f(x) )).

Steps of the Unification Algorithm

  1. Initialization: Start with an empty substitution set.
  2. Decompose: Break down the expressions into their components (functions, predicates, terms).
  3. Compare: Compare the corresponding parts of the expressions.
  4. Substitute: Apply substitutions to make the expressions identical.
  5. Occurs Check: Ensure no variable is unified with a term containing itself.
  6. Iterate: Repeat the process until the expressions are identical or unification fails.

Detailed Explanation with Example

Consider two expressions to be unified:

  1. ( P(f(x), y) )
  2. ( P(z, f(a)) )

Step-by-Step Process:

  1. Initialization:

    • Start with the empty substitution set ( \theta = { } ).

  2. Decompose and Compare:

    • The two expressions have the same predicate ( P ), so compare the arguments: ( f(x) ) with ( z ) and ( y ) with ( f(a) ).

  3. Substitute ( f(x) ) with ( z ):

    • To unify ( f(x) ) with ( z ), set ( z = f(x) ). Now the substitution set is ( \theta = { z \rightarrow f(x) } ).

  4. Apply the Substitution:

    • Apply the substitution to the second expression: ( P(f(x), y) ) and ( P(f(x), f(a)) ).

  5. Compare ( y ) with ( f(a) ):

    • To unify ( y ) with ( f(a) ), set ( y = f(a) ). Update the substitution set: ( \theta = { z \rightarrow f(x), y \rightarrow f(a) } ).

  6. Check for Consistency:

    • Ensure the substitution is consistent with both expressions:
      • First expression after substitution: ( P(f(x), f(a)) )
      • Second expression after substitution: ( P(f(x), f(a)) )
    • Both expressions are now identical.

Final Substitution:

  • The Most General Unifier (MGU) is ( { z \rightarrow f(x), y \rightarrow f(a) } ).

Summary

The unification algorithm successfully finds the substitution set that makes the two expressions ( P(f(x), y) ) and ( P(z, f(a)) ) identical by:

  1. Setting ( z ) to ( f(x) )
  2. Setting ( y ) to ( f(a) )

Another Example

Consider unifying ( Q(g(x), h(y, b)) ) and ( Q(g(a), h(b, z)) ).

Step-by-Step Process:

  1. Initialization:

    • Start with the empty substitution set ( \theta = { } ).

  2. Decompose and Compare:

    • The expressions have the same predicate ( Q ), so compare the arguments: ( g(x) ) with ( g(a) ) and ( h(y, b) ) with ( h(b, z) ).

  3. Substitute ( x ) with ( a ):

    • To unify ( g(x) ) with ( g(a) ), set ( x = a ). Now the substitution set is ( \theta = { x \rightarrow a } ).

  4. Apply the Substitution:

    • Apply the substitution to the first expression: ( Q(g(a), h(y, b)) ) and ( Q(g(a), h(b, z)) ).

  5. Compare ( h(y, b) ) with ( h(b, z) ):

    • Unify ( y ) with ( b ) and ( b ) with ( z ). So, ( y = b ) and ( z = b ).

  6. Final Substitution Set:

    • The substitution set is ( \theta = { x \rightarrow a, y \rightarrow b, z \rightarrow b } ).

Final Substitution:

  • The Most General Unifier (MGU) is ( { x \rightarrow a, y \rightarrow b, z \rightarrow b } ).

Conclusion

The unification algorithm is a powerful tool in logic programming and automated reasoning, allowing for the systematic resolution of variable bindings to unify logical expressions. It is essential in systems such as Prolog, theorem proving, and various AI applications where logical inference is required.