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The resolution rule in propositional logic is a single valid inference rule that produces a new clause implied by two clauses containing complementary literals. A literal is a propositional variable or the negation of a propositional variable.
SLD resolution (Selective Linear Definite clause resolution) is the basic inference rule used in logic programming. It is a refinement of resolution , which is both sound and refutation complete for Horn clauses .
Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples: The column-14 operator (OR), shows Addition rule: when p=T (the hypothesis selects the first two lines of the table), we see (at column-14) that p∨q=T.
In fact, the resolution of a goal clause with a definite clause to produce a new goal clause is the basis of the SLD resolution inference rule, used in implementation of the logic programming language Prolog. In logic programming, a definite clause behaves as a goal-reduction procedure.
The resolution rule is a single rule of inference that, together with unification, is sound and complete for first-order logic. As with the tableaux method, a formula is proved by showing that the negation of the formula is unsatisfiable.
The LRES rule resembles the resolution rule for classical propositional logic, where any propositional literals and are eliminated: ′ ′. The LERES rule states that if two propositional names p {\displaystyle p} and p ′ {\displaystyle p'} are equivalent, then p {\displaystyle \Box p} and ¬ p ′ {\displaystyle \neg \Box p'} can be eliminated.
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The resolution step leads to a worst-case exponential blow-up in the size of the formula. The Davis–Putnam–Logemann–Loveland algorithm is a 1962 refinement of the propositional satisfiability step of the Davis–Putnam procedure which requires only a linear amount of memory in the worst case.