5.6. Resolving More than Two Clauses

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.

This chunk introduces the concept of resolving multiple clauses rather than just pairs. When we have a set of several clauses (denoted as 'n'), the goal is to find resolutions among these clauses. Each resolution step involves selecting two clauses from the set, resolving them, and generating new clauses. This process continues until no further resolutions can be made, leading to a comprehensive understanding of the relationships and outcomes derived from all the given clauses.

Think of resolving clauses like solving a puzzle. Each piece represents a clause, and the resolution process is akin to connecting pieces together. You don't just connect two pieces at a time; sometimes, multiple pieces must connect to find the larger picture. This is like trying to find a solution or conclusion from multiple statements.

Basically we build what we call as a resolution tree and in the resolution tree we can keep the n clauses that are given in my set S at the root level, meaning that by default we can imagine that the clauses C1, C2, ..., Cn each of them belongs to the resolvent of my set of clauses S because I can always conclude C1, C2, ..., Cn from my set of clauses in S.

The resolution tree is a conceptual representation where all the initial clauses are placed at the base. This structure allows us to visualize how we resolve pairs of clauses step-by-step. As we perform resolutions, the new clauses created from resolving two existing ones become branches of this tree, illustrating how we move towards potential conclusions from the original set of clauses.

Imagine building a family tree where the initial members (clauses) form the foundation and as you discover connections (resolutions), each new connection (resolvent) branches out into the tree. Eventually, you will explore all branches until you can no longer create new connections, representing all possible conclusions from your family history.

Now, next what I have to do is the following I have to resolve a pair of existing clauses which are already there in my tree and whatever is the resolvent I obtain by resolving the pair of clauses, which I have resolved that will be treated as a new clause which will be again added to my tree.

In this part, we focus on the mechanical steps of the resolution process. After identifying two clauses that can be resolved, we achieve a new resolvent through simplification (cancellation of literals in both clauses). This new clause is then integrated back into the resolution tree, allowing for further resolutions with other existing clauses, thus expanding the exploration of possible conclusions.

Consider this process similar to a cooking recipe. You begin with a set of ingredients (initial clauses), and each time you mix (resolve) two ingredients, you create a new dish (resolvent) that can either stand alone or be used to create more complex dishes. Each new dish added is like adding a new branch to your recipe book.

I stress here that there is no restriction at each step regarding the choice of the clauses which you can pick for resolving in what order you have to resolve the clauses and so on. As long as you are picking two clauses which can be resolved and adding the resolvent to the tree you are fine.

This chunk emphasizes the flexibility in the resolution process. There are no strict rules about which clauses to resolve first or how to approach the resolution. As long as the clauses chosen can be resolved, the process remains valid. This openness allows for various strategies to explore potential resolutions in the set, reinforcing the idea that multiple outcomes can arise from differing resolution paths.

Think of this like brainstorming for a project. There's no set order to discuss your ideas (clauses), and you can choose which concepts to elaborate on (resolve) whenever they seem relevant. This freedom can unlock new perspectives and solutions that weren't immediately obvious at the beginning.

So let me give you an example to show how exactly we compute the resolution of a set of clauses, so imagine you are given here compound propositions p → q, r → s, p and r.

Here, we are given a practical example where we initially have compound propositions that need to be converted into clause forms. The conversion process is essential as it enables us to utilize the resolution method effectively. Each proposition is transformed to ensure they adhere to the required structure, from which we can progress to find resolvents through resolutions.

Similar to translating a book into a different language, you must first ensure that the text flows correctly in the new language. Each originally presented idea is turned into a corresponding representation (clause form) that accurately conveys the same meaning but is structured for resolution – just as a translated text needs to maintain the original context.

Next, I can resolve r from these two clauses and a resolvent will be S and now you can see that I can no longer find a pair of clauses which can be resolved further and I stop here and hence I will say that the resolvent of the set of clauses consist of the conjunction of clause q and the conjunction of the clause s.

In this concluding chunk, we identify the achieved resolvent and the stopping point of the resolution process. After conducting resolutions as long as possible, we arrive at a set of resolvents that cannot further be resolved. This marks the end of our logical exploration and indicates the conclusions drawn from the initial set of clauses.

Imagine completing a puzzle after trying various pieces. You might be unable to connect any further pieces because you’ve exhausted all options. What remains exemplifies the picture you aimed to complete – in this case, the final resolvents represent the outcome of the initial set of clauses.

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