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. 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.

Definition of unification for first-order logic^[1]

Let p and q be sentences in first-order logic.

UNIFY(p,q) = U where subst(U,p) = subst(U,q)

Where subst(U,p) means the result of applying substitution U on the sentence p. Then U is called a unifier for p and q. The unification of p and q is the result of applying U to both of them.

Let L be a set of sentences, for example, L = {p,q}. A unifier U is called a most general unifier for L if, for all unifiers U' of L, there exists a substitution s such that applying s to the result of applying U to L gives the same result as applying U' to L:

subst(U',L) = subst(s,subst(U,L)).

Unification in logic programming and type theory

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 the equality symbol =, but is also done when instantiating variables (see below). It is also used in other languages by the use of the equality symbol =, but also in conjunction with many operations including +, -, *, /. Type inference algorithms are typically based on unification.

In Prolog:

1. A variable 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.
2. Two atoms can only be unified if they are identical.
3. 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 behavior.

In type theory, the analogous statements are:

1. Any type variable unifies with any type expression, and is instantiated to that expression. A specific theory might restrict this rule with an occurs check.
2. Two type constants unify only if they are the same type.
3. Two type constructions unify only if they are applications of the same type constructor and all of their component types recursively unify.

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.

French computer scientist Gérard Huet gave an algorithm for unification in simply typed lambda calculus in 1973.^[2] There have been many developments in unification theory since then.^[3]

Higher-order unification

One of the most influential theories of ellipsis is that ellipses are represented by free variables whose values are then determined using Higher-Order Unification (HOU). For instance, the semantic representation of Jon likes Mary and Peter does too is like(j; m)R(p) and the value of R (the semantic representation of the ellipsis) is determined by the equation like(j; m) = R(j). The process of solving such equations is called Higher-Order Unification.^[4]

Examples of unification

In the convention of Prolog, atoms begin with lowercase letters.

• A, A : Succeeds (tautology)
• A, B, abc : Both A and B are unified with the atom abc
• abc, B, A : As above (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: Fails to unify; effectively abc = xyz

See also

• Admissible rule
• Semantic Unification

Notes

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?.
• Alex Sakharov, "Unification" from MathWorld.