Resolution (logic) - Resolution in First Order Logic

Resolution in First Order Logic

In first order logic, resolution 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:

Therefore,

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.

Therefore,

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 (term). 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) → Q(X)

∀X Q(X) → R(X)

Therefore, ∀X P(X) → R(X)

In CNF, the antecedents become:

¬P(X) ∨ Q(X)

¬Q(Y) ∨ R(Y)

(Note that the variable in the second clause was renamed 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, as defined by Robinson, also incorporated factoring, which unifies two literals in the same clause, before or during the application of resolution as defined above. The resulting inference rule is refutation complete, in that a set of clauses is unsatisfiable if and only if there exists a derivation of the empty clause using resolution alone.