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Unification


Consider expressions built up from variables and constants using function symbols. If v_1, ..., v_n are variables and t_1, ..., t_n are expressions, then a set of mappings between variables and expressions {t_1|v_1,...,t_n|v_n} is called a substitution.

If eta={t_1|v_1,...,t_n|v_n} and E is an expression, then Eeta is called an instance of E if it is received from E by simultaneously replacing all occurrences of v_i (for 0<=i<=n) by the respective t_i.

If eta={t_1|v_1,...,t_n|v_n} and theta={u_1|x_1,...,u_n|x_m} are two substitutions, then the composition of eta and theta (denoted eta*theta) is obtained from {t_1theta|v_1,...,t_ntheta|v_n,u_1|x_1,...,u_n|x_m} by removing all elements t_itheta|v_i such that t_itheta=v_i and all elements u_i|x_i such that x_i is one of v_1, ..., v_n.

A substitution eta is called a unifier for the set of expressions {E_1,...,E_n} if E_1eta=E_2eta=...=E_neta. A unifier eta for the set of expressions {E_1,...,E_n} is called the most general unifier if, for any other unifier for the same set of expressions theta, there is yet another unifier iota such that theta=eta*iota.

A unification algorithm takes a set of expressions as its input. If this set is not unifiable, the algorithm terminates and yields a negative result. If there exists a unifier for the input set of expressions, the algorithm yields the most general unifier for this set of expressions. The unification algorithm serves as a tool for the resolution principle. It is also a basis for term rewriting systems.


See also

Resolution Principle, Term Rewriting System, Unifiable

This entry contributed by Alex Sakharov (author's link)

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References

Chang, C.-L. and Lee, R. C.-T. Symbolic Logic and Mechanical Theorem Proving. New York: Academic Press, 1997.

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Unification

Cite this as:

Sakharov, Alex. "Unification." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Unification.html

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