Assignment 3

Written Assignment - CSP and Propositional Logic

NOTE: THIS ASSIGNMENT IS DUE BEFORE START OF CLASS. NO LATE SUBMISSIONS WILL BE ACCEPTED.

Problem 1

30 points (+10 Points EC).

The following outline map was obtained from a map of the Indian Subcontinent.

Indian Subcontinent Outline
Figure 1. Outline Map of Southern part of Indian Subcontinent.

Your job is to color the various sections such that no two sections sharing a border have the same color. You are allowed to use the colors (Red, Green, Blue).

Part a: Draw the Constraint Graph for this problem. Can you use this information to simplify the problem?

Part b: Assuming you are using Backtracking search to solve this problem and that you are using both MRV and Degree heuristic to select the variable, Which variable will be selected at each level of the search tree [You do not need to draw the tree. Just let me know which variable will be selected and why (MRV and degree values)]. Note: Multiple possible correct answers. You only have to give one.

Part c: Assume you assign the color 'Red' to the first variable selected in part b. Show the steps involved in checking the remaining legal values for all other variables using Arc Consistency.

Part d: EC (10 points): Give one valid solution to this problem. (You just have to give the solution. No need to give all the steps)

Problem 2

10 points.

Two logical statements A and B are logically equivalent if A <=> B. We have two knowledge bases, KB1 and KB2.. Write a function CHECK_EQUIVALENCE(KB1, KB2) that: Your pseudocode can use or modify any code from the textbook or slides, and can call any of the functions given in the textbook or slides, as long as such code and functions are used correctly, with correct names for the functions, and with well-specified values for all variables and arguments.

Problem 3

10 points.

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 False False
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.


Problem 4

10 points.

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 true, C is true, D is true.
- Second case: when A is true, B is false, C is true, D is false.

In all other cases, the knowledge base is true. Write a conjunctive normal form (CNF)  for the knowledge base. [Hint: Generate a statement that is false in the cases given above. Then convert it into CNF].

Problem 5

15 points.

On April 20, 2017, John and Mary sign the following contract:

- If it rains on May 1, 2017, then John must give Mary a check for $10,000 on May 2, 2017
- If John gives Mary a check for $10,000 on May 2, 2017, Mary must mow the lawn on May 3, 2017.

What truly happened those days is the following:
-  it did not rain on May 1, 2017
- John gave Mary a check for $10,000 on May 2, 2017
- Mary mowed the lawn on May 3, 2017.

Part a: Write a propositional-logic statement to express the contract. Make sure that, for each symbol that you use, you clearly define what that symbol stands for.

Part b: Write a logical statement to express what truly happened. When possible, use the same symbols as in question 4a. If you need to define any new symbols, clearly define what those new symbols stand for.

Part c: Was the contract violated or not, Justify your answer (For contract violation, the Contract must be false in a model where the Events are true)

Problem 6

15 points.

Consider the KB

(A <=> B) AND ((B AND C) => D)) AND (C <=> E) AND A AND E

Is this KB in Horn Form? If not, can you convert it into Horn from? If you can, give the Horn form representation.

Using Backward chaining (if possible), check if this KB entails the statement: D

Problem 6

10 points.

Convert the following scentences to CNF

Part a. (A <=> (B AND C))

Part b. (NOT (B AND A)) => C

How to submit