Logic for Computer Scientists/Predicate Logic/Strategies for Resolution/Input and Unit Resolution

Definition 26

Given a set of clauses $S$. The inference rule input resolution is resolution, such that one parent clause is a clause from $S$. The inference rule unit resolution is resolution, such that at least one parent clause is a unit clause or a unit factor of a parent clause.

In an obvious way the notions of unit (input) derivations and refutations are defined.

Theorem 10

For a set $S$ of clauses there exists a unit refutation iff there exists an input refutation. Unit resolution (and hence input resolution) is not complete for full first order logics!

For Horn clauses, however, it is a complete strategy and indeed, it is the basis for the SLD-resolution principle, which is the core of SLD-Resolution