 Unification

Unification finds a substitution making given first-order terms syntactically identical. In first-order unification, a substitution replaces variables by first-order terms while leaving function symbols fixed. A most general unifier captures the necessary identifications: every other unifier follows by a further substitution.^[ref]

For example, $f(x,g(y))$ and $f(g(z),g(a))$ have most general unifier $\{x\mapsto g(z),y\mapsto a\}$. The variable $z$ remains free; assigning it a particular first-order term gives a more specific unifier. Requiring $f(x,x)=f(a,b)$ forces both $x=a$ and $x=b$, so distinct constant symbols $a,b$ make it impossible.

Syntactic identity is essential. With addition treated as a binary function symbol, the first-order terms $x+0$ and $x$ are different even though their values agree under ordinary arithmetic. Equational unification incorporates specified identities and is a different problem. For finite first-order terms, the occurs check rejects $x=f(x)$: a finite first-order term cannot equal one of its proper subterms. Unification makes resolution possible by aligning atomic formulas before inference.