Tutorial for Week 1 material
----------------------------

1. Consider the following claims:

  (A1)  Programs that passed test 1 passed test 2.
  (A2)  Programs that passed test 1 also passed test 2.

(a)

Should we consider A1 and A2 to have different meanings?

What would be an appropriate domain/universe for A1 and A2?

What are the other sets we're interested in for A1 and A2?

Draw the general Venn diagram for P, T1 and T2.
        
What would be a counterexample to (A1)?

Where would the counterexample be in the diagram?

(b)

Suppose claim A1 is true.

Put an X in the region(s) of the Venn diagram from (a) that can't
 contain any programs.

Draw a simpler version of the diagram.

Consider the following programs:

  p1: passed test 1
  p2: passed test 2
  p3: didn't pass test 1
  p4: didn't pass test 2

What else can you say (if anything) about p1, p2, p3, p4?
  Try thinking about it in three ways:
    By thinking about A1.
    From the two Venn diagrams.

2.

(a)

Consider the following database of programs:


  Program  Runtime efficiency  Poly?
           is O(n!)?
  ----------------------------------
  p1       Yes                 Yes
  p2       Yes                 No
  p3       No                  Yes
  p4       No                  No

Which of the programs, if any, would be a counterexample to:

  (S1) The runtime efficiency is O(n!) for Poly programs.

  (S1') When a program's efficiency is O(n!) it's a Poly program.

Write S1 in the form:

  If ... then ...

Is there a claim of the form

  If ... then ...

 for which p2 is a counterexample?

How about p1?

(b)

Consider the following database of programs:

  Program  Poly?  NPoly?
  ----------------------
  p1       Yes     Yes
  p2       Yes     No
  p3       No      Yes
  p4       No      No

Which of the programs, if any, would be a counterexample to:

  (S2) A program is Poly only if it's NPoly.

  (S2') A program is Poly if it's NPoly.

Write S2 in the form:

  Every ... is ...

Is there a claim of the form:

  Every ... is ...

 for which p1 is a counterexample?