CS 461 – Nov. 2

1. CS 461 – Nov. 2 • Sets • finite vs. infinite • Infinite sets • Countable • Uncountable • Prepare for ATM • Proving undecidable is similar to proving a set uncountable. • Please be sure you understand reasoning.

2. Finite/infinite • Finite set – means that its elements can be numbered. Formally: its elements can be put into a 1-1 correspondence with the sequence 1, 2, 3, … n, where n is some positive integer. • An infinite set is one that’s not finite.  • However, there are 2 kinds of infinity!

3. Countable • Countable set: An infinite set whose elements can be put into a 1-1 correspondence with the positive integers. • Examples we can show are countable: Even numbers, all integers, rational numbers, ordered pairs. • Uncountable set = ∞ set that’s not countable.  • Examples we can show are uncountable: Real numbers, functions of integers, infinite-length strings

4. Examples • The set of even numbers is countable. • The set of integers is countable.

5. Ordered pairs The number assigned to (i , j) is (i + j – 1)*(i + j – 2)/2 + i

6. Real numbers • Suppose real numbers were countable. The numbering might go something like this: • The problem is that we can create a value that has no place in the correspondence!

7. Infinite bit strings

8. Universal TM Let’s design “U” – the Universal TM: • Input consists of <M> and w: • <M> is the encoding of some TM • w is any (binary) string. • Assume: U is a decider (i.e. ATM is decidable.) U <M>,w M yes yes w no* no

9. ATM solution • Start with U, the Universal Turing Machine • Suppose U decides ATM. Let’s build new TM D. • D takes in a Turing machine, and returns opposite of U’s answer. D <M> U no yes <M>,<M> no yes If M accepts its own string rep’n, D rejects <M>. If M doesn’t accept <M>, D accepts <M>. What does D do with <D> as input?

10. For example Contradiction  The TM D can’t exist  So U is not a decider.

11. In other words • Let U = universal TM. • Its input is a TM description <M> and a word <w>. • Determines if M accepts w. • Assume U halts for all inputs. (is a decider) • Create 2nd TM called D. • Its input is a TM description <M>. • Gives <M> to U as the TM to run as well as the input. • D returns the opposite of what U returns. • What happens when the input to D is <D>? • According to U, if D accepts <D>, U accepts, so D must reject! • According to U, if D rejects <D>, U rejects, so D must accept! • Both cases give a contradiction. • Thus, U is not a decider. ATM is undecidable.