Easy Hard and Impossible Elaine Rich Easy Tic

Easy, Hard, and Impossible Elaine Rich

Easy

Tic Tac Toe

Hard

Chess

The Turk Unveiled in 1770.

Searching for the Best Move A B E (8) F (-6) C G (0) H (0) I (2) D J (5) K L (-4) (10) M (5)

How Much Computation Does it Take? • Middle game branching factor 35 • Lookahead required to play master level chess 8 • 358

How Much Computation Does it Take? • Middle game branching factor 35 • Lookahead required to play master level chess 8 • 358 • Seconds in a year 2, 000, 000

How Much Computation Does it Take? • Middle game branching factor 35 • Lookahead required to play master level chess 8 • 358 • Seconds in a year • Seconds since Big Bang 2, 000, 000 31, 536, 000 300, 000, 000

The Turk Still fascinates people.

How Did It Work?

A Modern Reconstruction Built by John Gaughan. First displayed in 1989. Controlled by a computer. Uses the Turk’s original chess board.

Chess Today In 1997, Deep Blue beat Garry Kasparov.

Seems Hard But Really Easy

Nim At your turn, you must choose a pile, then remove as many sticks from the pile as you like. The player who takes the last stick(s) wins.

Nim Now let’s try:

Nim Now let’s try: Oops, now there a lot of possibilities to try.

Nim http: //www. gamedesign. jp/flash/nim. html

Binary Numbers 16 8 4 2 1 1 0 1 1 1 0 0 0 0 1 1 1 0 5 9 21 3 16

Nim My turn: (2) (3) 10 10 11 11 To form the last row: XOR each column. XOR: if number of 1’s is even: 0 if number of 1’s is odd: 1

Nim My turn: 10 10 11 11 For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise. (2) (3)

Nim My turn: 100 010 101 011 For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise. (4) (2) (5)

Nim Your turn: 100 001 101 000 For the current player: • Guaranteed loss if last row is all 0’s. • Guaranteed win otherwise. (4) (1) (5)

Following Paths

Seven Bridges of Königsberg

Seven Bridges of Königsberg: 1 3 4 2

Seven Bridges of Königsberg: 1 3 4 2 As a graph:

Eulerian Paths and Circuits Cross every edge exactly once. Leonhard Euler 1707 - 1783

Eulerian Paths and Circuits Cross every edge exactly once. Leonhard Euler 1707 - 1783 There is a circuit if every node touches an even number of edges.

So, Can We Do It? 1 3 4 2 As a graph:

Unfortuntately, There Isn’t Always a Trick Suppose we need to visit every node exactly once.

The Traveling Salesman Problem 15 10 20 4 25 28 8 40 9 7 3 23 Find the shortest circuit that visits every city exactly once.

Visting Nodes Rather Than Edges ● A Hamiltonian path: visit every node exactly once. ● A Hamiltonian circuit: visit every node exactly once and end up where you started. All these people care: • Salesmen, • Farm inspectors, • Network analysts

The Traveling Salesman Problem 15 10 20 4 25 28 8 40 9 7 3 23 Given n cities: Choose a first city Choose a second Choose a third … n n-1 n-2 n!

The Traveling Salesman Problem Can we do better than n! ● First city doesn’t matter. ● Order doesn’t matter. So we get (n-1!)/2.

The Growth Rate of n! 2 2 11 479001600 3 6 12 6227020800 4 24 13 87178291200 5 120 14 1307674368000 6 720 15 20922789888000 7 5040 16 355687428096000 8 40320 17 6402373705728000 9 362880 18 121645100408832000 10 3628800 19 2432902008176640000 11 39916800 36 3. 6 1041

Putting it into Perspective Speed of light 3 108 m/sec Width of a proton 10 -15 m At one operation in the 3 1023 ops/sec time it takes light to cross a proton Since Big Bang Ops since Big Bang Neurons in brain Parallel ops since Big Bang 3 1017 sec 9 1040 ops 1011 9 1051 36! = 3. 6 1041 43! = 6 1052

Is This The Best We Can Do? Probably. Would you like to win $1 M? The Millenium Prize

Impossible

An Interesting Puzzle List 1 1 2 3 b babbb ba List 2 bbb ba a 4 bbbaa babbb 2 List 1 babbb List 2 ba

An Interesting Puzzle List 1 1 2 3 b babbb ba List 2 bbb ba a 4 bbbaa babbb 2 1 List 1 babbbb List 2 babbb

An Interesting Puzzle List 1 1 2 3 b babbb ba List 2 bbb ba a 4 bbbaa babbb 2 1 1 List 1 babbbbb List 2 babbbbbb

An Interesting Puzzle List 1 1 2 3 b babbb ba List 2 bbb ba a 4 bbbaa babbb 2 1 1 3 List 1 babbbbbba List 2 babbbbbba

The Post Correspondence Problem List 1 List 2 1 11 011 2 01 0 3 001 110

The Post Correspondence Problem 1 2 3 List 1 ab ab aa List 2 a ba baa

The Post Correspondence Problem 1 2 3 List 1 1101 0110 1 List 2 1 11 110

Can A Program Do This? Can we write a program to answer the following question: Given a PCP problem, decide whether or not it has a solution. Return: True if it does. False if it does not.

The Post Correspondence Problem A program to solve this problem: Until a solution or a dead end is found do: If dead end, halt and report no. Generate the next candidate solution. Test it. If it is a solution, halt and report yes. So, if there are say 4 rows in the table, we’ll try: 1 2 3 4 1, 1 1, 2 1, 3 1, 4 2, 1 …… 1, 1, 1 …. 1, 5

Will This Work? • If there is a solution: • If there is no solution:

Programs Debug Programs Given an arbitrary program, can it be guaranteed to halt? read name if name = “Elaine” then print “You win!!” else print “You lose ”

Programs Debug Programs Given an arbitrary program, can it be guaranteed to halt? The 3 x + 1 Problem read number until number = 1 do if number is even then: number/2 if number is odd then: number 3 number + 1 Suppose number is 7:

Programs Debug Programs Given an arbitrary program, can it be guaranteed to halt? read number until number = 1 do if number is even then: number/2 if number is odd then: number 3 number + 1 Collatz Conjecture: This program always halts. We can try it on big numbers: http: //www. nitrxgen. net/collatz/7865765/

The Impossible The halting problem cannot be solved. We can prove that no program can ever be written that can look at arbitrary other programs and decide whether or not they always halt.

Other Unsolvable Problems PCP: • We can encode a <program>, <input> pair as an instance of PCP so that the PCP problem has a solution iff <program> halts on <input>. 2 1 1 List 1 babbbbb List 2 babbbbbb • So if PCP were solvable then Halting would be. • But Halting isn’t. So neither is PCP.

Which is Amazing Given the things programs can do. http: //www. youtube. com/watch? v=WFR 3 l. Om_xh. E

Which is Amazing Given the things programs can do. http: //www. google. com/selfdrivingcar/