5.4. Application of Resolution Rule

So to begin with, let us try to understand what exactly is the resolution rule. It is a very important inference rule and it is used extensively in this programming language called PROLOG. So recall I said that PROLOG is an important programming language, which is used in AI applications. So what exactly is this resolution rule? So it says the following, imagine you are given two clauses. So C1 is the clause and C2 is another clause. And the important property here is that I have a literal L which is present in positive form in C1 and negative form in C2.

The resolution rule is a logical principle that helps derive conclusions from given clauses or propositions. In this context, we utilize two clauses, C1 and C2. The resolution rule comes into play when a literal L appears positively in C1, while simultaneously appearing negatively in C2. This setup allows us to infer a new conclusion using these two premises.

Imagine you’re planning a picnic. You know it's sunny (positive) today and your friend says it won’t rain (negative). Here, the resolution rule helps you conclude that you can go ahead with the picnic safely. The positive and negative elements clarify the situation and help you make an informed decision.

What this resolution rule says is the following. It says that, if it is given that the clause C1 and C2 are true, then based on the truth of these 2 clauses we can conclude, C’ ˅ C’ (C1' disjunction C2'). So in some sense you can imagine that resolution rule is something equivalent to cancellation rule. That means you can cancel out the literal L if it is available in positive form in C1 and negative form in C2.

The resolution process involves combining the remaining parts of C1 and C2 after eliminating the common literal L. In symbolic terms, if both C1 and C2 are true, removing L leads us to a conclusion represented as C' ˅ C''. This new statement effectively simplifies the original clauses and indicates the retained truths from each clause, representing what is left after cancellation.

Consider a cooking scenario where you have two recipes. One includes sugar and the other includes no sugar. If your friends at a dinner party enjoy sweet dishes (positive) and dislike things not sweet (negative), you can conclude that you can mix elements from both recipes minus any bitterness (the sugar element) to create a dessert. Thus, you keep what's essential in the new dish.

In argument form, the resolution rule can be stated as follows. It says that if you are given the clauses C1 and C2 where, C1 is C1’ ˅ L and C2 is C2’ ˅ ¬L. Then based on these two premises, you can conclude the conclusion C’ ˅ C2’. I stress that to apply the resolution rule you need C1 and C2 to be clauses.

The argument form highlights the structure we need for applying the resolution rule: two clauses that include the literal in question and its negation. By expressing C1 and C2 in the specified formats, we can derive a resolvent, reinforcing the validity of our argument. It’s crucial that the inputs are clauses so they can correctly support this process of logical deduction.

Think about a court case. To prove a point, both the prosecution and defense must present solid evidence. Imagine the argument as two ‘evidential clauses’ where one party claims the defendant did something (positive) and the other denies it (negative). Only if the evidence holds up can we reach a solid conclusion or verdict that resolves the case.

And since we are saying that resolution as a valid inference rule we will prove that, assume for the moment resolution is a valid inference rule, it means that we can say that the conjunction of clauses C1 and C2 where C1 and C2 have the common literal L available in positive as well as in negative form in C1 and C2 respectively implies the disjunction of C’ and C2’ is a tautology.

Proving that the resolution is a valid argument requires showing that if the conjunction of C1 and C2 is true, the resulting resolution satisfies the conditions of a tautology. By assessing scenarios where L is true and false, we can consistently derive that C’ ˅ C2’ holds true under those assumptions, establishing its validity irrespective of the circumstances.

Consider a debate where two sides argue facts. If both sides provide true premises regarding the weather and traffic, we pull from both sides to create a widely accepted outcome about whether it’s suitable to leave now or later. If the facts are logically sound, the conclusion drawn from all shared truths becomes an undeniable reality.

So we have seen already how to find or how to resolve a pair of clauses. Now next we want to see how exactly we resolve a set of clauses where we may have more than two clauses. So imagine you are given a set of n clauses and I would be interested to resolve the clauses in this set, which is often called as a resolvent of the set of clauses.

Extending the resolution rule to multiple clauses involves establishing a resolution tree. Starting with a set of n clauses, we systematically resolve pairs of clauses until no further resolutions are possible. This mechanism allows us to find a comprehensive resolvent that captures all significant conclusions drawn from the original set of clauses.

Think of a research project where you gather various studies (clauses) to support your thesis. By repeatedly combining insights from different studies, you can form a well-grounded argument. This process resembles constructing a pyramid: each layer strengthens the overall argument until you reach the convincing conclusion at the top.

So now we want to prove that indeed the resolution is the, indeed resolution in principle that we are stating here is a valid argument form. So what we have to prove is we want to prove this statement that indeed this implication is a tautology.

The implementation of a resolution tree shows how we can visualize the resolution process. By constructing this tree, we assess how resolving pairs leads to a new conclusion. Demonstrating that the empty clause or false statement appears as a resolvent confirms that the original clauses could not satisfy a true statement, hence validating the logical structure of our argument.

Envision a treehouse built from several branches of logic. Each branch represents a potential outcome based on the premises. If you find a broken branch (the empty clause), it means your whole tree (argument) is unstable, confirming that the structure doesn’t stand true, akin to reaching a false endpoint in a logical operation.

Now, based on these two properties, I can next discuss what we call as proof by resolution refutation. So in the proof by resolution refutation the goal is the following you are given an argument form and you have to verify whether this argument form is valid or invalid.

Proof by resolution refutation involves examining whether a collection of premises logically leads to a specific conclusion. This method validates the argument form by converting all components to their clause forms, aiming to derive unsatisfiability when negating the conclusion. If the conclusion cannot stand logically with the premises, it reinforces that the argument presented is valid.

Imagine a jury tasked with determining a case based on presented evidence. By evaluating whether the evidence creates a valid conclusion or not mirrors this process. If the evidence contradicts the conclusion, the jury finds the argument invalid, similar to proving unsatisfiability in logical terms.

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