Question 1 Part a Consider the following knowledge base A AND B C OR D A C OR D AND NOT A C How many rows are there in the truth table for this knowledge base How did you determine this number There are four symbols A B C D so there are 2 4 16 rows in the truth table Part b John and Mary sign the following binding contract in front of their parents 1 On Sunday John will mow the lawn or buy groceries 2 On Sunday Mary will mow the lawn or wash the car This is an all inclusive list of what actually happens on Sunday 1 Mary mows the lawn on Sunday 2 Mary washes the car on Sunday 3 Mary buys groceries on Sunday How can the above statements be represented using propositional logic First define literals and specify what English phrase each literal corresponds to Second represent the knowledge base i e what happens on Sunday using those literals Third represent the contract as a single logical statement using those literals Four determine in any way you like whether according to the rules of propositional logic the contract was violated or not We define the following literals JM John mows the lawn on Sunday JB John buys groceries on Sunday MB Mary buys groceries on Sunday MM Mary mows the lawn on Sunday MW Mary washes the car on Sunday This is what happens on Sunday MM and MW and MB and not JM and not JB Note not JM and not JB are included because the question explicitly says that the given list of what happens on Sunday is all inclusive so whatever is not specified there did not happen The contract is represented as JM or JB and MM or MW The contract was violated since JM or JB was false 1 Part c Suppose that a knowledge base contains only symbols A B and C When does such a knowledge base entail the statement D i e the statement consisting of a single symbol that does not appear in the knowledge base Always sometimes or never If sometimes then identify precisely the conditions that determine whether this knowledge base entails the statement D There is only one case where this knowledge base entails D the case where the knowledge base is always false A knowledge base that is always false entails everything 2 Question 2 Determine if the following pairs of sentences are logically equivalent meaning that one is true if and only iff the other is true You do not have to justify your answer Part a Propositional logic A or B or not B or C A or B or C C Logically equivalent they are both always true Part b First order logic x and y are variables f is a predicate for every x for every y f x y for every x for every y f y x Logically equivalent for every x for every y is the same as for every y for every x so for every x for every y f x y is the same as for every y for every x f x y By consistently replacing x with y and y with x in for every y for every x f x y we obtain for every x for every y f y x Part c Propositional logic A and B E and G not A or not B or E and G logically equivalent the second statement is obtained from the first one by applying the rule that X Y is the same as not X or Y 3 Question 3 For of the following pairs of sentences determine if they are logically equivalent i e if each sentence of the pair implies the other sentence in the pair Do not assume anything except the laws of propositional and first order logic Part a for every x exists y color x y not exists x for every y not color x y logically equivalent Part b for every x exists y f x y exists y for every x f x y not logically equivalent Part c for every x exists y color x y Not for every x exists y not color x y not logically equivalent Part d for every x exists y son x y for every x exists y father y x not logically equivalent 4 Question 4 Part a Consider the following set of actions Action PutSockOnFoot a f Precond Sock a Foot f FreeSock a Effect not FreeSock a SockOn f Action PutShoeOnFoot b f Precond Shoe b Foot f SockOn f FreeShoe b Effect not FreeShoe b ShoeOn f Now consider this initial state and this goal InitState Sock sock1 and Sock sock2 and FreeSock sock1 and FreeSock sock2 and Shoe left shoe and Shoe right shoe and FreeShoe left shoe and FreeShoe right shoe and Foot left foot and Foot right foot Goal ShoeOn left foot and ShoeOn right foot Make two different plans to achieve the goal given the initial state First plan PutSockOnFoot sock1 left foot PutSockOnFoot sock2 right foot PutShoeOnFoot left shoe left foot PutShoeOnFoot right shoe right foot Second plan PutSockOnFoot sock1 left foot PutShoeOnFoot left shoe left foot PutSockOnFoot sock2 right foot PutShoeOnFoot right shoe right foot 5 Part b Consider the following set of actions just a little different from Part a Action PutSockOnFoot a f Precond Sock a Foot f Effect SockOn f Action PutShoeOnFoot b f Precond Shoe b Foot f SockOn f Effect ShoeOn f Now consider this initial state and this goal InitState Sock sock1 and Shoe left shoe and Foot left foot and Foot right foot Goal ShoeOn left foot and ShoeOn right foot Make a plan to achieve the goal given the initial state PutSockOnFoot sock1 left foot PutShoeOnFoot left shoe left foot PutSockOnFoot sock1 right foot PutShoeOnFoot left shoe right foot Part c Consider the following set of actions Action PutSockOnFoot a f Precond Sock a Foot f Effect SockOn f or SockOnFloor a Action PutShoeOnFoot b f Precond Shoe b Foot f SockOn f Effect ShoeOn f Now consider this initial state and this goal exactly the same as Part b InitState Sock sock1 and Shoe left shoe and Foot left foot and Foot right foot Goal ShoeOn left foot and ShoeOn right foot Is there a finite conditional plan that always achieves the goal given the initial state If yes describe the plan If not why not 6 There is no finite conditional plan that always achieves the goal There is no guarantee that after any finite number of repeating the PutSockOnFoot sock1 left foot action we will achieve SockOn left foot which is a precondition for putting a shoe on the left foot 7 Part d Consider the following set of actions exactly the same as in Part a Action PutSockOnFoot a f Precond Sock a Foot f FreeSock a Effect not FreeSock a SockOn f Action PutShoeOnFoot b f Precond Shoe b Foot f SockOn f FreeShoe b Effect not FreeShoe b ShoeOn f Now consider this initial state and this goal …