Resolution Theorem Proving

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AI | Proofs, and Inferences in Proving Propositional Theorem
Wumpus World in Artificial Intelligence

In this article, we will discuss the inference algorithms that use inference rules. Iterative deepening search is a full search algorithm in the sense that it will locate any achievable goal. Nevertheless, if the available inference rules are insufficient, the goal is not reachable — no proof exists that employs just those inference rules. The proof in the preceding section, for example, would fail if the biconditional elimination rule was eliminated. The present part introduces resolution, a single inference rule that, when combined with any full search algorithm, gives a complete inference method.

In the wumpus universe, we start with a simplified version of the resolution rule. Take a look at the steps that led up to the figure above – the agent travels from [2,1] to [1,1], then to [1,2], where it smells a stink but notices no breeze. The following information has been added to the knowledge base:

$\begin{array}{ll} R_{11}: & \neg B_{1,2} . \\ R_{12}: & B_{1,2} \Leftrightarrow\left(P_{1,1} \vee P_{2,2} \vee P_{1,3}\right) \end{array}$

We can now infer the lack of pits in [2,2] and [1,3] (remember that [1,1] is already known to be pitless) using the same approach that leads to R10 earlier:

$\begin{array}{ll} R_{13}: & \neg P_{2,2} \\ R_{14}: & \neg P_{1,3} \end{array}$

To acquire the fact that there is a pit in [1,1], [2,2], or [3,1], we may use biconditional elimination on R3, followed by Modus Ponens on R5, as follows:

$R_{15}: \quad P_{1,1} \vee P_{2,2} \vee P_{3,1}$

The resolution rule is now applied for the first time: the literal P2,2 in R13 resolves with the literal P2,2 in R15, yielding the resolvent.

$R_{16}: \quad P_{1,1} \vee P_{3,1}$

If a pit exists in one of [1,1], [2,2], or [3,1], and it is not in [2,2], it is in [1,1] or [3,1]. Similarly, the literal P1,1 in R1 is resolved by P3,1 when compared to the literal P1,1 in R16 to R17.

$R_{17}: \quad P_{3,1}$

In English, if a pit exists in either [1,1] or [3,1], and it is not in [1,1], it is in [3,1]. The unit resolution inference rule, l1 lk, m, is used in these last two inference stages.

$\frac{\ell_{1} \vee \cdots \vee \ell_{k}, \quad m}{\ell_{1} \vee \cdots \vee \ell_{i-1} \vee \ell_{i+1} \vee \cdots \vee \ell_{k}}$

where each l is a literal and l and m are complimentary literals (In other words, negation).

As a result, the unit resolution rule creates a new clause from a clause (a disjunction of literals) and a literal. A single literal, often known as a unit clause, can be understood as a disjunction of one literal.

The full resolution rule can be generalized to

$\frac{\ell_{1} \vee \cdots \vee \ell_{k}, \quad m_{1} \vee \cdots \vee m_{n}}{\ell_{1} \vee \cdots \vee \ell_{i-1} \vee \ell_{i+1} \vee \cdots \vee \ell_{k} \vee m_{1} \vee \cdots \vee m_{j-1} \vee m_{j+1} \vee \cdots \vee m_{n}}$

where $l_{i}$ and $m_{j}$ are complementary literals. This means that when two clauses are resolved, a new clause is created that contains all of the literal from the two original clauses save the two complimentary literals.

$\frac{P_{1,1} \vee P_{3,1}, \quad \neg P_{1,1} \vee \neg P_{2,2}}{P_{3,1} \vee \neg P_{2,2}}$

There is one additional technical feature to the resolution rule: each literal should only appear once in the resultant clause. Factoring is the process of removing numerous copies of literal. For example, resolving $(A \vee B)$ with $(A \vee \neg B)$ yields $(A \vee A)$ , which may be simplified to just A.

Consider the literal $l_{i}$ , which is complementary to the literal $m_{j}$ in the other sentence, to determine how sound the resolution rule is. If $l_{i}$ is true, then $m_{j}$ is false, and so $m_{1} \vee \cdots \vee m_{j-1} \vee m_{j+1} \vee \cdots \vee m_{n}$ must be true, since $m_{1} \vee \cdots \vee m_{n}$ is supplied. If $l_{i}$ is false, then $\ell_{1} \vee \cdots \vee \ell_{k}$ must be true because l1lk is supplied. Now, because it can be true or false, one of these conclusions must be true—exactly as the resolution rule requires.

The resolution rule is even more startling because it is the foundation for a family of full inference methods. A resolution-based theorem proving can determine if $\alpha \models \beta$ in propositional logic for any statement $\alpha$ and $\beta$ . The following two subsections describe how resolution does this.

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Last Updated : 05 Aug, 2022

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