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Unification

Unification Church

Unification can also refer to the German reunification of East and West Germany.

In mathematical logic, in particular as applied to computer science, a unification of two terms is a join (in the lattice sense) with respect to a specialisation order. That is, we suppose a preorder on a set of terms, for which t* ≤ t means that t* is obtained from t by substituting some term(s) for one or more free variables in t. The unification u of s and t, if it exists, is a term that is a substitution instance of both s and t; and such that any common substitution instance of s and t is also an instance of u.

For example, with polynomials, X² and Y³ can be unified to Z⁶ by taking X = Z³ and Y = Z².

Unification in Prolog

The concept of unification is one of the main ideas behind Prolog. It represents the mechanism of binding the contents of variables and can be viewed as a kind of one-time assignment. In Prolog, this operation is denoted by symbol "=".

1. In traditional Prolog, an uninstantiated variable X (i.e. no previous unification were performed on it) can be unified with an uninstantiated variable (and effectively becomes its alias), an atom or a term. In many modern Prolog dialects and in first-order logic calculi, a variable cannot be unified with a term that contains it (the so called occurs-check).
2. A Prolog-atom can be unified only with the same atom.
3. Similarly, a term can be unified with another term, if the top function symbol and arities of the terms are identical and the parameters can be unified simultaneously (note that this is a recursive behaviour).

Due to its declarative nature, the order in a sequence of unifications doesn't play (usually) any role.

Note that in the terminology of first-order logic, an atom is a basic proposition (and is unified similarly to a Prolog term).

Examples of unification

A=A 

Succeeds (tautology)

A=B, B=abc 

Both A and B are unified with the atom abc

xyz=C, C=D 

Unification is symmetric

abc=abc 

Unification succeeds

abc=xyz 

Fails to unify, atoms are different

f(A)=f(B) 

A is unified with B

f(A)=g(B) 

Fails, the heads of terms are different

f(A)=f(B,C) 

Fails to unify, because terms have different arity

f(g(A))=f(B) 

Unifies B with the term g(A)

f(g(A), A)=f(B, xyz) 

Unifies A with the atom xyz and B with the term g(xyz)

A=f(A) 

Infinite unification, A is unified with f(f(f(f(...)))).

A=abc, xyz=X, A=X 

Fails to unify; effectively abc=xyz