Written Assignment 3

The assignment should be submitted via Blackboard.

Some questions in this assignment refer to the same problem as programming assignment 3, please refer to the description of that assignment for additional details.

Problem 1

15 points.

Let S1 be a statement in propositional logic. Write a function CHECK_TRUE(S1) 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.

Remember that a model is a complete assignment of Boolean values for each symbol of S1. In other words, a model corresponds to a row of the truth table. Thus, given a model, for any symbol of S1 we know whether that symbol is true or false in that model.

Problem 2

20 points.

Suppose that an agent using the program described in programming assignment 3 for inference, has gone through squares (1,1), (1,2), (1,3), (1,4) sensing no stench or breeze anywhere, except for a stench at square (1,4). You can assume that an agent dies if it moves to a square containing a monster or a pit, and that this particular agent did not die at any of the four squares. Specify a set of statements that we need to add to the knowledge base to represent the information that the agent collected out of those four squares. The set of statements should be sufficient, meaning that the inference engine will always give results that are correct and consistent with the information that the agent obtained from those four squares. Your set of statements should also be MINIMAL, i.e., it should not be possible to leave out any of those statements without sacrificing the correctness of the inference algorithm. You should assume that the knowledge base already contains the wumpus rules, and in particular the version of the rules specified in programming assignment 3, not the version specified in the textbook.

Problem 3

15 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 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.

Problem 4

15 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 5

10 points.

Suppose that you have a function f(kb, statement) that returns true iff kb (the knowledge base) entails the statement (a logical sentence) and false otherwise. Write pseudocode that, using function f, defines a function g(kb, statement) that returns +1 iff the statement is definitely true given kb, -1 iff the statement is definitely false given kb, and 0 otherwise.

Problem 6

10 points.

Consider a knowledge base with these facts:

Convert the above knowledge-base to a first-order logic knowledge base. For each predicate, function, constant, or variable that you use, explicitly state:

Problem 7

15 points.

Consider the same knowledge base as in the previous task. Assume that the knowledge base has been successfully converted to first-order logic statements. Under the rules of first-order logic, does this knowledge base entail that Shadow is female? Justify your answer.

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