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First-order ResolutionIn mathematical logic and automated theorem proving, first-order resolution is a theorem-proving technique. It condenses the traditional syllogisms of logical inference down to a single rule. To understand how resolution works, consider the following example syllogism of term logic: All Greeks are Europeans. Homer is a Greek. Therefore, Homer is a European. Or, more generally: ∀X, P(X) implies Q(X). P(A). Therefore, Q(A). To recast the reasoning using the resolution technique, first the clauses must be converted to Conjunctive normal form. In this form, all quantification becomes implicit: universal quantifiers on variables (X, Y...) are simply omitted as understood, while existentially-quantified variables are replaced by Skolem functions. ¬P(X) ∨ Q(X) P(A) Therefore, Q(A) So the question is, how does the resolution technique derive the last clause from the first two? The rule is simple: - Find two clauses containing the same predicate, where it is negated in one clause but not in the other.
- Perform a unification on the two predicates. (If the unification fails, you made a bad choice of predicates. Go back to the previous step and try again.)
- If any unbound variables which were bound in the unified predicates also occur in other predicates in the two clauses, replace them with their bound values (terms) there as well.
- Discard the unified predicates, and combine the remaining ones from the two clauses into a new clause, also joined by the "∨" operator.
To apply this rule to the above example, we find the predicate P occurs in negated form ¬P(X) in the first clause, and in non-negated form P(A) in the second clause. X is an unbound variable, while A is a bound value (atom). Unifying the two produces the substitution X => A Discarding the unified predicates, and applying this substitution to the remaining predicates (just Q(X), in this case), produces the conclusion: Q(A) For another example, consider the syllogistic form All Cretans are islanders. All islanders are liars. Therefore all Cretans are liars. Or more generally, ∀X P(X) implies Q(X) ∀X Q(X) implies R(X) Therefore, ∀X P(X) implies R(X) In CNF, the antecedents become: ¬P(X) ∨ Q(X) ¬Q(Y) ∨ R(Y) (Note I renamed the variable in the second clause to make it clear that variables in different clauses are distinct.) Now, unifying Q(X) in the first clause with ¬Q(Y) in the second clause means that X and Y become the same variable anyway. Substituting this into the remaining clauses and combining them gives the conclusion: ¬P(X) ∨ R(X) The resolution rule (with additional factoring) similarly subsumes all the other syllogistic forms of traditional logic.
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