Unification

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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. If any common substitution instance of s and t is also an instance of u, u is called minimal unification.

For example, with polynomials, X ² and Y ³ can be unified to Z⁶ by taking X = Z³ and Y = Z².

Unification in logic programming

The concept of unification is one of the main ideas behind logic programming, best known through the language 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 "=".

In traditional Prolog, a variable X which is uninstantiated—i.e. no previous unifications were performed on it—can be unified with an atom, a term, or another uninstantiated variable, thus effectively becoming its alias. In many modern Prolog dialects and in first-order logic, a variable cannot be unified with a term that contains it; this is the so called occurs check.
Two Prolog atoms can only be unified if they are identical.
Similarly, a term can be unified with another term if the top function symbols and arities of the terms are identical and if 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 is (usually) unimportant.

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 because the atoms are different
f(A) = f(B) : A is unified with B
f(A) = g(B) : Fails because the heads of the terms are different
f(A) = f(B, C) : Fails to unify because the 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(...)))). In proper first-order logic and many modern Prolog dialects this is forbidden (and enforced by the occurs check)
A = abc, xyz = X, A = X : Fails to unify; effectively abc = xyz

References

F. Baader and T. Nipkow, Term Rewriting and All That Cambridge University Press, 1998.
F. Baader and W. Snyder, Unification Theory In J.A. Robinson and A. Voronkov, editors, Handbook of Automated Reasoning, volume I, pages 447–533. Elsevier Science Publishers, 2001.
Joseph Goguen, What is Unification?

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