Question

Unify (if possible) the following pairs of predicates and give the resulting substitutions, assume b is a constant.

a.	P(x, f(x), z)
	¬ P(g(y),f(g(b)),y)
b.	P(x, f(x))
	¬ P(f(y), y)
c.	P(x, f(z))
	¬ P(f(y), y)

Answer 1

a) Here,1 = P(x,f(x),z) and  2 = P(g(y),f(g(b)),y)

Step 1: Check if the predicates match. Atoms with different predicate symbols cannot be unified.[here both the predicates match].

Step 2:To unify each pair of terms.

• The first pair is (x,g(y)).Since one of the elements is a variable, they can be unified by substituting the other term for the variable. The substitution for the above pair is g(y)/x.

This substitution must be applied to both the clauses which give,

P(g(y),f(g(y)),z)

P(g(y),f(g(b)),y)

• The second pair is (f(g(y)),f(g(b)).Here, substitute b/y

The substitution must be applied for both the clauses which give,

P(g(b),f(g(b)),z)

P(g(b),f(g(b)),y)

• The third pair is(z,y). Here, substitute z/y

The substitution must be applied bor both the clauses which give,

P(g(b),f(g(b)),y)

P(g(b),f(g(b)),y)

Unified successfully

The substitutions are g(y)/x, b/y and  z/y .

b) Here,1 = P(x,f(x)) and  2 = P(f(y),y)

• The first pair is (x,f(y)).Since one of the elements is a variable, they can be unified by substituting the other term for the variable. The substitution for the above pair is f(y)/x.

The substitution must be applied bor both the clause which give,

P(f(y),f(f(y)))

P(f(y),y)

• The second pair is (f(f(y)),y).

It is not possible to substitute f(f(y))/y.

Hence Unification failed.

c)Here,1 = P(x,f(z)) and  2 = P(f(y),y)

• The first pair is (x,f(y)).Since one of the elements is a variable, they can be unified by substituting the other term for the variable. The substitution for the above pair is f(y)/x.

The substitution must be applied bor both the clause which give,

P(f(y),f(z))

P(f(y),y)

• The second pair is (f(z),y). Since one of the elements is a variable, they can be unified by substituting the other term for the variable. The substitution for the above pair is f(z)/y.

The substitution must be applied bor both the clause which give,

P(f(y),f(z))

P(f(y),f(z))

Hence Unified successfully.

The substitutions are f(y)/x and f(z)/y.