Resolution in FOL
Resolution is a theorem proving technique that proceeds by building refutation proofs, i.e., proofs by contradictions. It was invented by a Mathematician John Alan Robinson in the year 1965.
Resolution is used, if there are various statements are given, and we need to prove a conclusion of those statements. Unification is a key concept in proofs by resolutions. Resolution is a single inference rule which can efficiently operate on the conjunctive normal form or clausal form.
Clause: Disjunction of literals (an atomic sentence) is called a clause. It is also known as a unit clause.
Conjunctive Normal Form: A sentence represented as a conjunction of clauses is said to be conjunctive normal form or CNF.
Note: To better understand this topic, firstly learns the FOL in AI.
The resolution inference rule:
The resolution rule for first-order logic is simply a lifted version of the propositional rule. Resolution can resolve two clauses if they contain complementary literals, which are assumed to be standardized apart so that they share no variables.
Where li and mj are complementary literals.
This rule is also called the binary resolution rule because it only resolves exactly two literals.
We can resolve two clauses which are given below:
[Animal (g(x) V Loves (f(x), x)] and [￢ Loves(a, b) V ￢Kills(a, b)]
Where two complimentary literals are: Loves (f(x), x) and ￢ Loves (a, b)
These literals can be unified with unifier θ= [a/f(x), and b/x] , and it will generate a resolvent clause:
[Animal (g(x) V ￢ Kills(f(x), x)].
Steps for Resolution:
To better understand all the above steps, we will take an example in which we will apply resolution.
Step-1: Conversion of Facts into FOL
In the first step we will convert all the given statements into its first order logic.
Step-2: Conversion of FOL into CNF
In First order logic resolution, it is required to convert the FOL into CNF as CNF form makes easier for resolution proofs.
Note: Statements "food(Apple) Λ food(vegetables)" and "eats (Anil, Peanuts) Λ alive(Anil)" can be written in two separate statements.
Step-3: Negate the statement to be proved
In this statement, we will apply negation to the conclusion statements, which will be written as ¬likes(John, Peanuts)
Step-4: Draw Resolution graph:
Now in this step, we will solve the problem by resolution tree using substitution. For the above problem, it will be given as follows:
Hence the negation of the conclusion has been proved as a complete contradiction with the given set of statements.
Explanation of Resolution graph: