Written Assignment - Propositional and First Order Logic

Max points:
The assignment should be submitted via Canvas.

Instructions

Task 1 (CSE 4308: 10 Points; CSE 5360: 10 points)

Two logical statements S1 and S2 are logically equivalent if (S1 <=> S2) is valid. We have two knowledge bases, KB1 and KB2.. Write a function CHECK_EQUIVALENCE(KB1, KB2) that: Your pseudocode can re-use 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.

Task 2 (CSE 4308: 10 Points; CSE 5360: 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 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 3 (CSE 4308: 10 Points; CSE 5360: 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 false, C is true, D is true.
- Second case: when A is false, 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.

Task 4 (CSE 4308: 20 Points; CSE 5360: 20 points)

Consider the KB

A => B
B <=> C
D => A
E => D
C AND E => F
E

Show that this entails F by

i. Forward Chaining
ii. Backward Chaining
iii. Resolution

Task 5 (CSE 4308: 40 Points; CSE 5360: 40 points)

In April, John and Mary sign the following contract:

- If it rains in May, then John must give Mary a check for $10,000
- If John gives Mary a check for $10,000, Mary must mow the lawn.

What truly happened those days is the following:
- It did not rain in May.
- John gave Mary a check for $10,000
- Mary mowed the lawn.

Part a: 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: 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: Define the symbols required to convert any KB involved in the above domanin from FOL to Propositional logic (Your symbols must allow me to convert ANY KB that uses the predicates and constants as decribed previously).

Part d: Use the sybols given in part c, to convert the answers to part a and b to Propositional Logic.

Part e: Was the contract violated or not, Justify your answer (Note: if the sequence of events that occured entails the contract then it was not violated)

Task 6 (CSE 4308: 10 Points; CSE 5360: 10 Points)

Unify the following if possible: