What is the difference between arc consistency and path consistency in constraint satisfaction problems?

What is the difference between arc consistency and path consistency in constraint satisfaction problems?

Arc consistency ensures that for every value of a variable, there is a consistent value in the connected variable. Path consistency, on the other hand, checks for consistency across a path of variables, ensuring that any two variables are consistent with a third variable.

Can you give an example of arc consistency?

Sure! Imagine a simple CSP with variables A and B. If A can take values {1, 2} and B can take values {2, 3}, then A is arc consistent with B because for every value of A, there is a corresponding value in B.

And what about path consistency?

In path consistency, consider three variables A, B, and C. If A can be 1, B can be 2, and C can be 3, path consistency checks if A and C can be consistent through B. If A=1 and B=2, then C must also have a value that is consistent with B.

Why is path consistency more complex than arc consistency?

Path consistency is more complex because it involves checking multiple variables and their relationships, while arc consistency only focuses on pairs of variables. This makes path consistency computationally more intensive.