Samples for Quiz 3

Task 1

A B C KB S1
   True       True       True       True       True   
True True False False True
True False True True True
True False False False True
False True True False False
False True False False False
False False True True True
False False False False False

KB and S1 are two propositional logic statements, that are constructed using symbols A, B, C, and using various connectives. The above truth table shows, for each combination of values of A, B, C, whether KB and S1 are true or false.

Part a: Given the above information, does KB entail S1? Justify your answer.

Part b: Given the above information, does statement NOT(KB) entail statement NOT(S1)? Justify your answer.


Task 2

Suppose that some  knowledge base contains various propositional-logic sentences that utilize symbols A, B, C, D (connected with various connectives). There are only two cases when the knowledge base is false:
- First case: when A is true, B is false, C is true, D is false.
- Second case: when A is false, B is false, C is true, D is true.

In all other cases, the knowledge base is true. Write a conjunctive normal form (CNF)  for the knowledge base.

Task 3

Consider the KB

(A => C) AND (B <=> C) AND (D => A) AND E AND [(B AND E) => G] AND (B => F) AND D

Show that this entails G (if possible) by

i. Forward Chaining
ii. Backward Chaining
iii. Resolution

Task 4

John and Mary sign the following contract:

- If it rains on Monday, then John must give Mary a check for $100 on Tuesday
- If John gives Mary a check for $100 on Tuesday, Mary must mow the lawn on Wednesday.

What truly happened those days is the following:
- It did not rain on Monday
- John gave Mary a check for $100 on Tueday
- Mary mowed the lawn on Wednesday.

Part a (10 pts): Write a first order logic statement to express the contract. Make sure that you clearly define what constants and predicates that you use are. (NOTE: DO NOT use functions)

Part b (8 pts): Write a logical statement to express what truly happened. When possible, use the same predicates and constants as in question 6a. If you need to define any new predicates or constants, clearly define what they stand for.

Part c (12 pts): Define the symbols required to convert any KB involved in the above domanin from FOL to Propositional logic. Use this to convert the answers to part a and b to Propositional Logic.

Part d (5 pts) (Extra Credit): Was the contract violated or not, Justify your answer [Note: Contract is definitely not violated if the events entail the contract. Contract is definitly violated if the events entail the opposite of the contract. Unknown otherwise]

Task 5

Does a unifier exist for these pairs of predicates. If they do, give the unifier

i. Taller(x, John); Taller(Bob, y)
ii. Taller(y, Mother(x)); Taller(Bob, Mother(Bob))
iii. Taller(Sam, Mary); Shorter(x, Sam)
iv. Shorter(x, Bob); Shorter(y, z)
v. Shorter(Bob, John); Shorter(x, Mary)

Task 6

Consider the 8-Puzzle problem. There are 8 tiles on a 3 by 3 grid. Your task is to get from some given configuration to a goal configuration. You can move a tile to an adjacent location as long as that location is empty.

Sample Initial configuration:
2
 3
 6
1
7
4
 8
 5

Sample Goal configuration:
1
 2
 3
4
 5
 6
7
 8

Your task is to define this problem in PDDL Describe the initial state and the goal test using PDDL. Define appropriate actions for this planning problem, in the PDDL language. For each action, provide a name, arguments, preconditions, and effects.

Task 7

Suppose that we are using PDDL to describe facts and actions in a certain world called JUNGLE. In the JUNGLE world there are 3 predicates, each predicate takes at most 4 arguments, and there are 5 constants. Give a reasonably tight bound on the number of unique states in the JUNGLE world. Justify your answer.

Task 8

Consider the following PDDL state description for the Blocks world problem.

On(A, B)
On(B, C)
On(C, Table)
On(D, E)
On(D, Table)

Consider the definition of Move(block, from, to) as given in the slides

Can you perform action Move(A, B, D) in this state.

What is the outcome of performing this action in this state.