Cs3102: Theory of Computation Class 20: Busy Beavers Spring 2010 University of Virginia David Evans Office hours: I am not able to hold my Thursday morning.

cs3102: Theory of Computation

Class 20: Busy Beavers

Spring 2010University of VirginiaDavid Evans

Office hours: I am not able to hold my Thursday morning office hours this week. I will have office hours Thursday 11am-1pm instead.

Exam 2• Out Thursday, due next Tuesday (2:01pm)• Should not take more than two hours to

complete (but don’t wait until 12:01pm Tuesday to start it!)

• Prepare (before class Thursday) one page of notes: you may use this as much as you want during the exam

• You may also use: course book, course handouts, your notes, other resources: but only for up to 30 minutes during exam

Exam 2: Main Topics• Countable and uncountable infinite sets• Turing machines: – Formal definition– Computing model– Robustness of TM model

• Church-Turing thesis and its implications• Turing-recognizable, Turing-decidable classes and

their properties• Using reduction to prove a language is undecidable• Everything from Exam 1 also

“Game Theory” Experiment

Number of “Exempt Me” messages received:

14No exemptions on Exam 2. (Although some more opportunities in today’s class.)

Reduction Proofs

B reduces to Ameans

that can decide B

can be used to make

that can decide A

If we already know A is undecidable, this proves B is undecidable.

Y X

Singleton Language

Proving Undecidability

MSINGLETON

TM that decides SINGLETONTM

Acceptor

Reject

Acceptor

Reject

MH that decides HALTSTM

1. Assume SINGLETONTM is decidable.2. Then, there exists a TM that decides it. We’ll call it MSINGLETON.3. Show how to use MSINGLETON to build a machine that decides HALTSTM.4. Contradiction: we know HALTSTM is undecidable, so if we could use

MSINGLETON to construct a machine that decides it, MSINGLETON must not exist.

Reducing HALTSTM to SINGLETONTM

Reducing HALTSTM to SINGLETONTM

Type Safety

"hello" + 3Traceback (most recent call last): File "<pyshell#1>", line 1, in <module> "hello" + 3TypeError: Can't convert 'int' object to str implicitly

Reducing HALTSTM to WELLTYPEDPython

Reducing HALTSTM to WELLTYPEDPython

Well-Typed Java?public class Test { static public void main(String args[]) { System.out.println("Hello" + 3); }}

> javac Test.java> java TestHello3

public class Test { static public void main(String args[]) { System.out.println("Hello" - 3); }}

> javac Test.javaTest.java:3: operator - cannot be applied to java.lang.String,int

Type Safety

This is decidable: your Java compiler should do this (and should always terminate)!

WELLTYPEDJava--(<P>) = { P is a Java

program that does not use type casts or array assignments and P never produces a run-time type error. }

Reducing HALTSTM to WELLTYPEDJava--?

We claimed on the last slide that it is decidable! Something must be broken with this “proof”.

No such Java program exists!

Busy Beavers

The “Busy Beaver” GameDesign a (2-way infinite tape) Turing Machine that:– Uses k symbols (e.g., “0” and “1”)– Starts with a tape of all “0”s (no

blanks)– Eventually halts (can’t run forever)– Has n states (not counting qAccept

and qReject) Tibor Radó, 1895-1965

Goal is to run for as many steps as possible before eventually halting

Busy Beaver: N = 1

Start

01, R

0 ...0 0 0 0 0 0 0 0 0

BB(1, 2) = 1 Most steps a 1-state machine that halts can make

0 0 0 0 0 0 0 0

Accept

AStart

B

0 ...0 0 0 0 0 0 0 0 0

BB(2, 2) = ?

0 0 0 0 0 0...

Accept1 1, L

1 1, L

0 1, L

0 1, R

Step 0

Step 1

1 ...0 0 0 0 0 0 0 0 00 0 0 0 0 0...

AStart

B

Accept1 1, L

1 1, L

0 1, L

0 1, R

Step 2

1 ...1 0 0 0 0 0 0 0 00 0 0 0 0 0...

AStart

B

Accept1 1, L

1 1, L

0 1, L

0 1, R

Step 3

1 ...1 0 0 0 0 0 0 0 00 0 0 0 0 0...

AStart

B

Accept1 1, L

1 1, L

0 1, L

0 1, R

Step 4

1 ...1 0 0 0 0 0 0 0 00 0 0 0 0 1...

AStart

B

Accept1 1, L

1 1, L

0 1, L

0 1, R

Step 5

1 ...1 0 0 0 0 0 0 0 00 0 0 0 1 1...

AStart

B

Accept1 1, L

1 1, L

0 1, L

0 1, R

Step 6:Halted

1 ...1 0 0 0 0 0 0 0 00 0 0 0 1 1...

BB(2, 2) 6

(Actually, it is known that BB(2,2) = 6)

AStart

B

Accept1 1, L

1 1, L

0 1, L

0 1, R

What is BB(6, 2)?

Busy Bunny?

AStart

BC

D E

F

Accept

1 1, L

0 1, R

0 0, R

0 1, L1 0, R1 0, L

1 1, R

1 0, L

0 1, L

0 0, L

0 0, R

1 1, R

6-state machine found by Buntrock and Marxen, 2001

300232771652356282895510301834134018514775433724675250037338180173521424076038326588191208297820287669898401786071345848280422383492822716051848585583668153797251438618561730209415487685570078538658757304857487222040030769844045098871367087615079138311034353164641077919209890837164477363289374225531955126023251172259034570155087303683654630874155990822516129938425830691378607273670708190160525534077040039226593073997923170154775358629850421712513378527086223112680677973751790032937578520017666792246839908855920362933767744760870128446883455477806316491601855784426860769027944542798006152693167452821336689917460886106486574189015401194034857577718253065541632656334314242325592486700118506716581303423271748965426160409797173073716688827281435904639445605928175254048321109306002474658968108793381912381812336227992839930833085933478853176574702776062858289156568392295963586263654139383856764728051394965554409688456578122743296319960808368094536421039149584946758006509160985701328997026301708760235500239598119410592142621669614552827244429217416465494363891697113965316892660611709290048580677566178715752354594049016719278069832866522332923541370293059667996001319376698551683848851474625152094567110615451986839894490885687082244978774551453204358588661593979763935102896523295803940023673203101744986550732496850436999753711343067328676158146269292723375662015612826924105454849658410961574031211440611088975349899156714888681952366018086246687712098553077054825367434062671756760070388922117434932633444773138783714023735898712790278288377198260380065105075792925239453450622999208297579584893448886278127629044163292251815410053522246084552761513383934623129083266949377380950466643121689746511996847681275076313206

(1730 digits)

Best found before 2001, only 925 digits!

http://drb9.drb.insel.de/~heiner/BB/index.html

A

Start

B

C

DE

F

Accept

1 1, L

0 1, R

0 0, R

0 1, L1 0, R

1 0, L

1 1, R

1 0, L

0 1, L

0 0, L

0 0, R

1 1, R

Is this BB(6, 2)?

Busy Beaver NumbersBB(1, 2) = 1BB(2, 2) = 6BB(3, 2) = 21 Proved by Lin and Rado, 1965BB(4, 2) = 107BB(5, 2) = Unknown!

Best so far is 47,176,870BB(6, 2) = Unknown!

Best so far is > 102879 (discovered in 2007)Previous best: > 101730 (discovered in 2001, previous slide)Previous best: > 10925 (discovered in August 2000)

flickr: climbnh2003

Why is BB(5, 2) unknown?

How many (5, 2) TMs are there?

How many (5, 2) TMs are there?

B

E

A

D

C

qAccept

0 1, R

264 Trillion (<1012) is not a very big number! Cryptosystems with 56-bit keys (256 = 1016) are considered easily brute-force breakable, 270K times bigger.

Why BB(5, 2) is Unknown

• Best found so far: 47,176,870• There are about 40 (5,2) TMs that run for

more than 47,176,870 steps without obviously repeating (but haven’t halted yet)– Probably, these machines never halt and

BB(5,2) = 47176870But, its possible one of these machines eventually accepts!

What does this have to do with reviewing reduction proofs?

(or: is BB a language?)

Is LBB Decidable?

Do we know this terminates?

Someone pointed out after class today that this is not quite correct! The problemis we’re simulating M starting with w on the tape, but the BB starts with a blank tape.To fix this, we need to create M’ that first writes w on the tape and then simulates M.

Busy Beaver Challenges

• Determine BB(5, 2)• The standard Busy Beaver problem is defined

for a doubly-infinite tape TM. For the one-way infinite tape TM, what is BB(4, 2)?

Worth an exemption on Exam 2 or the Final Exam

Charge

• Exam 2 will be handed out at end of class Thursday, due on Tuesday

• Prepare your Exam 2 note sheet before class Thursday

• Thursday’s office hours: 11am-1pm (not normal time)