What is resolution in AI and give some examples?
Posted: Thu Aug 17, 2023 6:29 am
Resolution is a fundamental inference rule and a key technique used in automated theorem proving and logic programming within the field of artificial intelligence. It is particularly associated with the resolution refutation method, which aims to prove the validity or satisfiability of logical formulas through a process of negation and resolution. The resolution rule allows us to derive new clauses from existing ones by eliminating complementary literals.
Here's how resolution works and a couple of examples to illustrate its usage:
Resolution Rule:
Given two clauses that contain complementary literals (a literal and its negation), resolution allows us to derive a new clause that is a result of resolving the two clauses by eliminating the complementary literals.
Example 1:
Let's consider a simple example with propositional logic. Suppose we want to prove that the following statements are contradictory:
P OR Q
NOT P OR R
NOT Q OR R
To prove contradiction, we can use the resolution rule:
Resolve clauses 1 and 2: (P OR Q) AND (NOT P OR R) => Q OR R
Resolve the result with clause 3: (Q OR R) AND (NOT Q OR R) => R
Since we have derived a clause that contains both R and NOT R (a contradiction), we conclude that the original statements are contradictory.
Example 2:
Let's consider a more complex example using first-order logic. Suppose we want to prove the statement: "All humans are mortal."
For all x, Human(x) => Mortal(x)
We'll assume that our knowledge base includes the following premises:
A. Human(Socrates)
B. NOT Mortal(Socrates)
To prove the statement using resolution:
Convert premise 1 to its negation: Human(S) AND NOT Mortal(S)
Resolve with premise A: (Human(S) AND NOT Mortal(S)) AND Human(S) => NOT Mortal(S)
Resolve the result with premise B: NOT Mortal(S) AND NOT Mortal(S) => Contradiction
Since we've derived a contradiction, we can conclude that our original statement "All humans are mortal" is valid.
Resolution is a crucial technique for automated theorem proving, logical reasoning, and model checking in AI. It allows AI systems to systematically explore the logical relationships between statements and make inferences based on the rules of logic.
Here's how resolution works and a couple of examples to illustrate its usage:
Resolution Rule:
Given two clauses that contain complementary literals (a literal and its negation), resolution allows us to derive a new clause that is a result of resolving the two clauses by eliminating the complementary literals.
Example 1:
Let's consider a simple example with propositional logic. Suppose we want to prove that the following statements are contradictory:
P OR Q
NOT P OR R
NOT Q OR R
To prove contradiction, we can use the resolution rule:
Resolve clauses 1 and 2: (P OR Q) AND (NOT P OR R) => Q OR R
Resolve the result with clause 3: (Q OR R) AND (NOT Q OR R) => R
Since we have derived a clause that contains both R and NOT R (a contradiction), we conclude that the original statements are contradictory.
Example 2:
Let's consider a more complex example using first-order logic. Suppose we want to prove the statement: "All humans are mortal."
For all x, Human(x) => Mortal(x)
We'll assume that our knowledge base includes the following premises:
A. Human(Socrates)
B. NOT Mortal(Socrates)
To prove the statement using resolution:
Convert premise 1 to its negation: Human(S) AND NOT Mortal(S)
Resolve with premise A: (Human(S) AND NOT Mortal(S)) AND Human(S) => NOT Mortal(S)
Resolve the result with premise B: NOT Mortal(S) AND NOT Mortal(S) => Contradiction
Since we've derived a contradiction, we can conclude that our original statement "All humans are mortal" is valid.
Resolution is a crucial technique for automated theorem proving, logical reasoning, and model checking in AI. It allows AI systems to systematically explore the logical relationships between statements and make inferences based on the rules of logic.