Resolution (logic)


In mathematical logic and automated theorem proving, resolution is a rule of inference leading to a refutation theorem-proving technique for sentences in propositional logic and first-order logic. In other words, iteratively applying the resolution rule in a suitable way allows for telling whether a propositional formula is satisfiable and for proving that a first-order formula is unsatisfiable. Attempting to prove a satisfiable first-order formula as unsatisfiable may result in a nonterminating computation; this problem doesn't occur in propositional logic.
The resolution rule can be traced back to Davis and Putnam ; however, their algorithm required trying all ground instances of the given formula. This source of combinatorial explosion was eliminated in 1965 by John Alan Robinson's syntactical unification algorithm, which allowed one to instantiate the formula during the proof "on demand" just as far as needed to keep refutation completeness.
The clause produced by a resolution rule is sometimes called a resolvent.

Resolution in propositional logic

Resolution rule

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. Two literals are said to be complements if one is the negation of the other. The resulting clause contains all the literals that do not have complements.
Formally:
where
The above may also be written as:
The clause produced by the resolution rule is called the resolvent of the two input clauses. It is the principle of consensus applied to clauses rather than terms.
When the two clauses contain more than one pair of complementary literals, the resolution rule can be applied for each such pair; however, the result is always a tautology.
Modus ponens can be seen as a special case of resolution.
is equivalent to

A resolution technique

When coupled with a complete search algorithm, the resolution rule yields a sound and complete algorithm for deciding the satisfiability of a propositional formula, and, by extension, the validity of a sentence under a set of axioms.
This resolution technique uses proof by contradiction and is based on the fact that any sentence in propositional logic can be transformed into an equivalent sentence in conjunctive normal form. The steps are as follows.
One instance of this algorithm is the original Davis–Putnam algorithm that was later refined into the DPLL algorithm that removed the need for explicit representation of the resolvents.
This description of the resolution technique uses a set S as the underlying data-structure to represent resolution derivations. Lists, Trees and Directed Acyclic Graphs are other possible and common alternatives. Tree representations are more faithful to the fact that the resolution rule is binary. Together with a sequent notation for clauses, a tree representation also makes it clear to see how the resolution rule is related to a special case of the cut-rule, restricted to atomic cut-formulas. However, tree representations are not as compact as set or list representations, because they explicitly show redundant subderivations of clauses that are used more than once in the derivation of the empty clause. Graph representations can be as compact in the number of clauses as list representations and they also store structural information regarding which clauses were resolved to derive each resolvent.

A simple example

In plain language: Suppose is false. In order for the premise to be true, must be true.
Alternatively, suppose is true. In order for the premise to be true, must be true. Therefore, regardless of falsehood or veracity of, if both premises hold, then the conclusion is true.

Resolution in first order logic

Resolution rule can be generalized to first-order logic to:
where is a most general unifier of and, and and have no common variables.

Example

The clauses and can apply this rule with as unifier.
Here x is a variable and b is a constant.
Here we see that
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:
Or, more generally:
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 are simply omitted as understood, while existentially-quantified variables are replaced by Skolem functions.
So the question is, how does the resolution technique derive the last clause from the first two? The rule is simple:
To apply this rule to the above example, we find the predicate P occurs in negated form
in the first clause, and in non-negated form
in the second clause. X is an unbound variable, while a is a bound value. Unifying the two produces the substitution
Discarding the unified predicates, and applying this substitution to the remaining predicates, produces the conclusion:
For another example, consider the syllogistic form
Or more generally,
In CNF, the antecedents become:
Now, unifying Q in the first clause with ¬Q 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:

Factoring

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 only resolution, enhanced by factoring.
An example for an unsatisfiable clause set for which factoring is needed to derive the empty clause is:
Since each clause consists of two literals, so does each possible resolvent. Therefore, by resolution without factoring, the empty clause can never be obtained.
Using factoring, it can be obtained e.g. as follows:

Non-clausal resolution

Generalizations of the above resolution rule have been devised that do not require the originating formulas to be in clausal normal form.
These techniques are useful mainly in interactive theorem proving where it is important to preserve human readability of intermediate result formulas. Besides, they avoid combinatorial explosion during transformation to clause-form, and sometimes save resolution steps.

Non-clausal resolution in propositional logic

For propositional logic, Murray and Manna and Waldinger use the rule
where denotes an arbitrary formula, denotes a formula containing as a subformula, and is built by replacing in every occurrence of by ; likewise for.
The resolvent is intended to be simplified using rules like, etc.
In order to prevent generating useless trivial resolvents, the rule shall be applied only when has at least one "negative" and "positive" occurrence in and, respectively. Murray has shown that this rule is complete if augmented by appropriate logical transformation rules.
Traugott uses the rule
where the exponents of indicate the polarity of its occurrences. While and are built as before, the formula is obtained by replacing each positive and each negative occurrence of in with and, respectively. Similar to Murray's approach, appropriate simplifying transformations are to be applied to the resolvent. Traugott proved his rule to be complete, provided are the only connectives used in formulas.
Traugott's resolvent is stronger than Murray's. Moreover, it does not introduce new binary junctors, thus avoiding a tendency towards clausal form in repeated resolution. However, formulas may grow longer when a small is replaced multiple times with a larger and/or.

Propositional non-clausal resolution example

As an example, starting from the user-given assumptions
the Murray rule can be used as follows to infer a contradiction:
For the same purpose, the Traugott rule can be used as follows :
From a comparison of both deductions, the following issues can be seen:
For first-order predicate logic, Murray's rule is generalized to allow distinct, but unifiable, subformulas and of and, respectively. If is the most general unifier of and, then the generalized resolvent is. While the rule remains sound if a more special substitution is used, no such rule applications are needed to achieve completeness.
Traugott's rule is generalized to allow several pairwise distinct subformulas of and of, as long as have a common most general unifier, say. The generalized resolvent is obtained after applying to the parent formulas, thus making the propositional version applicable. Traugott's completeness proof relies on the assumption that this fully general rule is used; it is not clear whether his rule would remain complete if restricted to and.

Paramodulation

Paramodulation is a related technique for reasoning on sets of clauses where the predicate symbol is equality. It generates all "equal" versions of clauses, except reflexive identities. The paramodulation operation takes a positive from clause, which must contain an equality literal. It then searches an into clause with a subterm that unifies with one side of the equality. The subterm is then replaced by the other side of the equality. The general aim of paramodulation is to reduce the system to atoms, reducing the size of the terms when substituting.

Implementations