CS 343

Homework 3

 

You may do this assignment either by yourself or in a team.  But each person must write up his or her own solution.

 

Part A: Inference methods and algorithms for reasoning with knowledge that is represented in logical formalisms. 

 

Propositional Logic:

 

1)      (R & N 7.8) Decide whether each of the following sentences is valid, unsatisfiable, or neither.  You may use truth tables or any of the standard sound rules for propositional inference.  Show your argument.

a)      Smoke ® Smoke

b)      Smoke ® Fire

c)      (Smoke ® Fire) ® (ØSmoke ® ØFire)

d)      Smoke Ú Fire Ú ØFire

e)      ((Smoke Ù Heat) ® Fire) Û ((Smoke ® Fire) Ú (Hear ® Fire))

f)        (Smoke ® Fire) ® ((Smoke Ù Heat) ® Fire)

g)      Big Ú Dumb Ú (Big ® Dumb)

h)      (Big Ù Dumb) Ú ØDumb

 

2)      (R & N 7.9) Given the following, can you prove that the unicorn is mythical?  Magical?  Horned?  Show your work.

 

If the unicorn is mythical, then it is immortal, but if it is not mythical, then it is a mortal mammal.  If the unicorn is either immortal or a mammal, then it is horned.  The unicorn is magical if it is horned.

 

3)      (R & N 7.12b) Show that every propositional logic clause with at least one positive literal and one negative literal can be written in the form (P1 Ù …Ù Pm) ® (Q1 Ú … Ú Qn), where the Ps and Qs are proposition symbols.  A knowledge base consisting of such sentences is in implicative normal form or Kowalski form.

 

4)      Consider the following KB:

winter Ú hot

winter ® Øsummer Ù Øspring Ù Øfall

rainy ® spring Ú winter

pollen ® winter

rivers ® rainy

spring ® bluebonnets

Øbluebonnets

rivers

a)      Convert each of the assertions to clause form.

b)      Use resolution to prove winter.

 

FOPC

 

5)      (R & N 8.4) Write down a logical sentence such that every world in which it is true contains exactly one object.

 

6)      (R & N 9.4, R & K 5.3) Show the result of applying the unification algorithm given in class to each of the following pairs of clauses: (We use the convention that lower case characters are variables; upper case characters are constants, predicates, and functions.  We also use the convention that if x is substituted for y we will write x/y.  Note that R & N write it the other way, as y/x.)

a)      P(A, B, B), P(x, y, z)

b)      Q(y, G(A, B)), Q(G(x, x), y)

c)      Older(Father(y), y), Older(Father(x), John)

d)      Knows(Father(y), y), Knows(x, x)

e)      F(Marcus), F(Caesar)

f)        F(x), F(G(y))

g)      F(Marcus, G(x, y), F(x, G(Caesar, Marcus))

 

7)      Consider the following KB:

·        "x Married(x) ® $y Spouse(x, y)

·        "x $y Spouse(x, y) ® Married(x)

·        "x "y Spouse(x, y) ® Spouse(y, x)

·        "x "y JointTaxFilers(x, y) ® Spouse(x, y)

·        JointTaxFilers(John, Mary)

·        Ø$y Spouse(Sue, y)

a)      Convert each of these formulas to clause form.

b)      Use resolution and this KB to prove Married(Mary)

c)      Use resolution and this KB to prove ØMarried(Sue)

 

8)      Consider again the facts we have about painters, including the ones presented in class and those you added in question 1 of Homework 2.  Use resolution to answer the question, “What language did Leonardo speak?”

 

9)      (R & K 5.9) Suppose you are given the following facts:

a)      "x, y, z gt(x, y) Ù gt(y, z) ® gt(x, z)

b)      "a, b succ(a, b) ® gt(a, b)

c)      "x Øgt(x, x)

 

Using these facts, we want to prove gt(5, 2), which we should be able to do with resolution.  Consider the following attempt at a resolution proof:


d)      What went wrong?  (Hint: watch very carefully what is going on with the use of variable names.)

e)      What needs to be added to the resolution procedure we described in class to make sure that this problem does not occur?

 

B. Representing facts and reasoning with them

 

1)      (R & K 5.4) Consider the following sentences:

·        John likes all kinds of food.

·        Apples are food.

·        Chicken is food.

·        Anything anyone eats and isn't killed by is food.

·        Bill eats peanuts and is still alive.

·        Sue eats everything Bill eats.

a)      Translate these sentences into formulas in FOPC.

b)      Use backward chaining to prove that John likes peanuts.

c)      Convert the formulas of part a into clause form.

d)      Use resolution to prove that John likes peanuts.

e)      Use resolution to answer the question, "What food does Sue eat?"

 

2)      (R & K 5.5) Consider the following facts:

·        The members of the Elm St. Bridge Club are Joe, Sally, Bill and Ellen.

·        Joe is married to Saly.

·        Bill is Ellen's brother.

·        The spouse of every married person in the club is also in the club.

·        The last meeting of the club was at Joe's house.

a)      Represent these facts in FOPC.

b)      From the facts given above, most people would be able to decide on the truth of the following additional statements:

·        The last meeting of the club was at Sally's house.

·        Ellen is not married.

 

Can you construct resolution proofs to demonstrate the truth of each of these statements given the facts above?  Do so if possible.  Otherwise, add the facts you need and then construct the proofs.

 

3)      (R & K 5.11) What is wrong with the following argument:

·        Men are widely distributed over the earth.

·        Socrates is a man.

·        Therefore, Socrates is widely distributed over the earth.