7.7. Question 13

In question number 13, we have to use resolution to show that the following compound proposition is not satisfiable and as per the properties of resolution basically to show that the conjunction of these clauses is unsatisfiable.

In this section, we are tasked with proving that a compound proposition is not satisfiable using resolution. Unsatisfiability means that there is no assignment of truth values to the variables in the propositions that can make the entire statement true. To show this using resolution, we will utilize the properties of resolution which allow us to derive new conclusions from existing clauses. Essentially, if we can derive a contradiction (the 'FALSE' constant), it confirms that the original clauses cannot all be true simultaneously, thus demonstrating unsatisfiability.

Imagine you and your friends are planning a picnic, but everyone has a different idea of when they are free, and the times can't overlap. If you continue to argue about times and eventually conclude that no one can agree on a time, it's like arriving at the constant 'FALSE.' Just like this scenario reveals that a picnic is impossible with everyone busy, reaching 'FALSE' in resolutions indicates that the compound proposition is unsatisfiable.

So let us build a resolvent or resolution tree for this set of clauses here. So I can pick the first two clauses and resolve p and then I can pick the last two clauses and I can again resolve p.

To apply resolution, we start by building a resolution tree, which is a graphical representation of how we resolve the clauses systematically. In this case, we take the first two clauses of our compound proposition and look for a common literal (in this case, 'p'). When we resolve these two clauses, we eliminate 'p' and combine the remaining parts of the clauses. We perform the same operation with the last two clauses that also involve 'p'. This ongoing process will generate new clauses that might lead us to a contradiction or to the empty clause, indicating that our original proposition cannot hold true.

Think of building a resolution tree like solving a puzzle. Each time you resolve two clauses by removing a common piece (like 'p'), you're finding how the remaining pieces fit together. If you keep putting pieces together and eventually find that you can't complete the picture (no more pieces left to fit), it reveals that there is no way to fit them all together into a coherent whole, illustrating that the original proposition is unsatisfiable.

So the resolvents are now added to the tree and now I can pick these two resolvents for resolving and I obtain the conclusion, the constant, F which shows that the conjunction of these four clauses is not satisfied.

After building our resolution tree and continually resolving clauses, we eventually arrive at a stage where we derive the constant 'F' (false). This constant indicates that it is impossible for all the clauses in the compound proposition to be true at the same time. Hence, we conclude that the original proposition is not satisfiable because reaching 'F' means we can't satisfy all parts of the equation simultaneously.

Imagine you are trying to fit different shaped blocks into a single box that only fits one shape perfectly. As you try each piece (resolving clauses), you might find that no matter how you place them, none of the shapes fit without overflowing. If you finally conclude that bringing all those shapes into that box is impossible, you've reached a state analogous to 'constant F.' It shows that your attempt to make everything fit (satisfy) is futile, aptly demonstrating unsatisfiability.

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