Informal Overview
Given two input terms s and t, (syntactic) unification is the process which attempts to find a substitution that structurally identifies s and t. If such a substitution exists, we call this substitution a unifier of s and t.
In theory, some pairs of input terms may have infinitely many unifiers. However, for most applications of unification, it is sufficient to consider a most general unifier (mgu). An mgu is useful because all other unifiers are instances of the mgu.
In particular, first-order unification is the syntactic unification of first-order terms (terms built from variable and function symbols). Higher-order unification, on the other hand, is the unification of higher-order terms (terms containing some higher-order variables).
The set of all syntactically equivalent terms is variously called the free theory (because it is a free object), the empty theory (because the set of equational sentences is empty), or the theory of uninterpreted functions (because unification is done on uninterpreted terms.)
The theoretical properties of a particular unification algorithm depend upon the variety of term used as input. First-order unification, for instance, is decidable, efficiently so, and all unifiable terms have (unique, up to variable renamings) most general unifiers. Higher-order unification, on the other hand, is generally undecidable and lacks most general unifiers, though Huet's higher-order unification algorithm seems to work sufficiently well in practice.
Aside from syntactic unification, another form of unification, called semantic unification (known also as equational-unification, E-unification) is also widely used. The key difference between the two notions of unification is how two unified terms should be considered 'equal'. Syntactic unification attempts to find a substitution making the two input terms structurally identical. However, semantic unification attempts to make the two input terms equal modulo some equational theory. Some particularly important equational theories have been widely studied in the literature. For instance, AC-unification is the unification of terms modulo associativity and commutativity.
Unification is a significant tool in computer science. First-order unification is especially widely used in logic programming, programming language type system design, especially in type inferencing algorithms based on the Hindley-Milner type system, and automated reasoning. Higher-order unification is also widely used in proof assistants, for example Isabelle and Twelf, and restricted forms of higher-order unification (higher-order pattern unification) are used in some programming language implementations, such as lambdaProlog, as higher-order patterns are expressive, yet their associated unification procedure retains theoretical properties closer to first-order unification. Semantic unification is widely used in SMT solvers and term rewriting algorithms.
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