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# Unification

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. 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.
2. An atom can be unified only with the same atom.
3. A term is unified with another term, if the heads and arities of the terms are identic and the parameters are unified (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.

## 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