Deriving Propositional Resolution
We next consider proof rules for checking Satisfiability of Sets of Formulas.
We are extending the notion of substitution on formulas to sets of formulas by
To make intuition clearer, we will next use quantification over potentially infinitely many variables and conjunctions over infinitely many formulas.
Proof System Based on Projecting Variables
We first derive a more abstract proof system and that show that resolution is a special case of it.
The condition that is satisfiable is equivalent to the truth of
which, by changing the order of quantifiers, is:
By expanding existential quantifier, is eqivalent to
which, by distributivity of through is:
Then we conclude that is equivalent to defined by
Projection Proof Rules
Above we justified the use of as an inference rule. We write such rule:
The soundness of projection rule follows from the fact that for every interpretation , if , then also .
Applying the projection rule we obtain formulas with fewer and fewer variables. We therefore also add the “ground contradiction rule”
where is formula that has no variables and that evaluates to false (ground contradictory formula). This rule is trivially sound: we can never have a model of a ground formula that evaluates to false.
Iterating Rule Application
Given some enumeration of propositional variables in , we define the notion of applying projection along all propositional variables, denoted :
Completeness of Projection Rules
We wish to show that projection rules are a sound and complete approach for checking satisfiability of finite and infinite sets formulas. More precisely:
- if we can derive false from using projection rules, then is unsatisfiable
- if is unsatisfiable, then we can derive false using projection rules
Because we can derive false precisely when we have a ground false formula, these statements become:
- if then is not satisfiable
- if is not satisfiable then
The first statement follows from soundness of projection rules. We next prove the second statement.
Suppose that it . We claim that is satisfiable. We show that every finite set is satisfiable, so the property will follow from the Compactness Theorem.
Consider any finite . We show that it is satisfiable. Let and let . Consider the set
By definition of , we can show that the set contains the conjunctive normal form of the expansion of
Each of these conjuncts is a ground formula (all variables have been instantiated), so the formula evaluates to either true or false. By assumption, and therefore do not contain a ground contradiction. Therefore, each conjunct of is true and is satisfiable.
Improvement: Subsumption Rules
Note also that if , where has been derived before, and , then deriving does not help derive a ground contradiction, because the contradiction would also be derived using . If we derive such formula, we can immediately delete it so that it does not slow us down.
In particular, a ground true formula can be deleted.
Resolution as Projection
Instead of arbitrary formulas, use clauses as sets of literals. The projection rule becomes
If is a clause, then
There are several cases:
Therefore, for clauses, projection (with some elimination of redundant conclusions) is exactly the resolution proof rule.