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. A KFC franchise in Kuwait. ... Mathematical logic is a subfield of mathematics that is concerned with formal systems in relation to the way that they encode intuitive concepts of mathematical objects such as sets and numbers, proofs, and computation. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... The name lattice is suggested by the form of the Hasse diagram depicting it. ... In mathematics, especially in order theory, preorders are certain kinds of binary relations that are closely related to partially ordered sets. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...

For example, with polynomials, X² and Y³ can be unified to Z⁶ by taking X = Z³ and Y = Z². In mathematics, a polynomial is an expression that is constructed from one or more variables and constants, using only the operations of addition, subtraction, multiplication, and constant positive whole number exponents. ...

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 "=". Logic programming (which might better be called logical programming by analogy with mathematical programming and linear programming) is, in its broadest sense, the use of mathematical logic for computer programming. ... Prolog is a logic programming language. ...

1. 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.
2. Two Prolog 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 behaviour.

Due to its declarative nature, the order in a sequence of unifications is (usually) unimportant. In computer science and mathematics, a variable (IPA pronunciation: ) (sometimes called a pronumeral) is a symbolic representation denoting a quantity or expression. ... First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ... In computer science, an occurs check is a feature of some implementations of unification, which causes unification of a logic variable V and a structure S to fail if S contains V. Binding a variable to a structure containing that variable results in a cyclic structure which may subsequently cause... The mathematical term arity sprang from words like unary, binary, ternary, etc. ...

Note that in the terminology of first-order logic, an atom is a basic proposition and is unified similarly to a Prolog term. First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...

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

Within the study of logic, a tautology is a statement containing more than one sub-statement, that is true regardless of the truth values of its parts. ... In computer science, an occurs check is a feature of some implementations of unification, which causes unification of a logic variable V and a structure S to fail if S contains V. Binding a variable to a structure containing that variable results in a cyclic structure which may subsequently cause...

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 at MathWorld.