Deriving Propositional Resolution
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 and the ground rule, then is unsatisfiable
- if is unsatisfiable, then we can derive false using projection rules and one application of the ground rule
Because we can derive false precisely when we have a ground false formula, these statements become:
- if contains a ground contradiction, then is not satisfiable
- if is not satisfiable then contains a ground contradiction for some natural number .
The first statement follows from soundness of projection rules. We next prove the second statement.
Suppose that is not satisfiable. By the Compactness Theorem, a finite subset is not satisfiable. Let .
Consider the set
By definition of , we can show that contains a subset of ground propositional formulas equivalent to
Because the above formula is contradictory, is a contradictory set of ground propositional formulas, so it contains a ground contradiction. Thus, we can set in the above theorem.
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.