the busy beaver, the placid platypus and the universal unicorn

The Busy Beaver, the Placid Platypus and the Universal

Unicorn

Beaver, Platypus, Unicorn … RMIT 1st August, 2014

James Harland

[email protected]

Overview

Beaver, Platypus, Unicorn …

Busy beaver problem and similar problems

Machine execution techniques

Searching large numbers of machines

Universal Turing machines

Connections with other areas

RMIT 1st August, 2014

Overview


Busy beaver problem and similar problems

Machine execution techniques

Searching large numbers of machines


Connections with other areas

Overview


‘Organic’ theoretical computer science

Exploration of theoretical concepts by software tools

Mix of empirical and theoretical methods

What?


Turing Machines of a particular type:

Deterministic

Symbols are only 0 (blank) and 1 (initially)

Only consider blank input

n states plus a halt state means size is n

a b c h11R01R

00L

10R

01L

11R

What?


What is the largest number of 1’s that can be printed by a terminating n-state machine?

n #1’s (productivity) #steps

1 1 1

2 4 6

3 6 21

4 13 107

5 ≥ 4098 ≥ 47,176,870 (??)6 ≥ 3.51×1018,276 ≥ 7.41×1036,534 (!!)

What?


n m #1’s #steps

2 2 4 6

3 2 6 21

2 3 9 38

4 2 13 107

2 4 2,050 3,932,964

3 3 374,676,383 119,112,334,170,342,540

5 2 ≥ 4098 ≥ 47,176,870

2 5 ≥ 1.7 × 10352 ≥ 1.9 × 10704

6 2 3.51×1018276 ≥ 7.41×1036534

4 3 ≥ 1.383 × 107036 ≥ 1.025 × 1014072

3 4 ≥ 3.743 × 106518 ≥ 5.254 × 1013036

2 6 ≥ 1.903 × 104933 ≥ 2.415 × 109866 HERE BE DRAGONS!

What?


Busy Beaver function is non-computable; it grows faster than any computable function (!!)

Various mathematical bounds known

All surpassed in practice

Seems hopeless for n ≥ 7

Values for n ≤ 5 seem settled (but as yet unproven)

Why?


Some enormous numbers here! `… the poetry of logical ideas’ (Einstein) Evidence for results is somewhat lacking (“The remaining 250+ cases were checked by hand”) Should have test data and evidence available Should be able to automate entire test

`Nothing capable of proof should be accepted as true without it’ (Frege)

`Nothing capable of automation should be accepted as finished without it’

Why?


An n-state (2-symbol) machine of productivity k shows that

bb(n,2) ≥ k

at most n states are needed to print k 1’s

What is the minimum number of states (for an n-state 2-symbol machine) needed to print k 1’s?

We call this the placid platypus problem [ pp(k) ]

Why?


bb(n,2) = k means that pp(k) ≤ n and pp(k+1) ≥ n+1

pp(2) = pp(3) = pp(4) = 2

pp(5) = pp(6) = 3

pp(k) = 4 for k є {7,8,9,10,11,12,13}

pp(k) ≥ 5 for k ≥ 14

…???

There seem to be no 5-state 2-symbol machines with productivity 74, 85, 92-7, 100-110, 113, 114, 115, …

Need complete classification to be sure

Executing Machines


Marxen and Buntrock (1990): macro machines

Treat a sequence of k characters as a single ‘symbol’

Avoids repetition of previous sequences

Significant speed up

Need to choose k in advance

Holkner (2004): loop prediction

{b}(01)1001 -> (11)100 {b}1

State

Input InDir Output

OutDir

NewState

b 01 l 11 r b

Executing Machines


What do the monsters do?

5x2 state champion (terminates in 47,176,870 steps)

Step Configuration

12393 001111{b}(001)66100

12480 001(111)6{b}(001)63100

12657 001(111)11{b}(001)60100

… …

33120 001(111)106{b}(001)3100

… …

47176870 101{h}(001)4095100

20,000+ steps in 1

Executing Machines


5x2 state champion takes 11 otter steps

Otter Steps Predicted

1 267

2 2,235

3 6,840

4 20,463

5 62,895

6 176,040

7 500,271

8 1,387,739

9 3,878,739

10 10,830,672

11 30,144,672

Executing Machines


Macro machines + loop prediction + observant otter can cope with all known monsters (!!)

Size Ones Hops Otters Otter Steps

2x4 90 7195 2 64%

2x4 2050 3932964 8 99%

3x3 17338 127529109 17 > 99%

3x3 95524079 (16 digits) 17 > 99%

3x3 374676383 (18 digits) 60 > 99%

5x2 4098 47,176,870 11 > 99%

5x2 4098 11,798,796 10 > 99%

5x2 4097 23,554,764 74 > 99%

2x5 (31 digits) (62 digits) 168 > 99%

2x5 (106 digits) (212 digits) 593 > 99%

2x5 (353 digits) (705 digits) 1993 > 99%

Executing Machines



6x2 (18268 digits) (36535 digits) 30345 > 99%

6x2 (10567 digits) (21133 digits) 99697 > 99%

6x2 (1440 digits) (2880 digits) 8167 > 99%

6x2 (882 digits) (1763 digits) 2211 > 99%

4x3 (7037 digits) (14073 digits) 25255 > 99%

4x3 (6035 digits) (12069 digits) 34262 > 99%

4x3 (4932 digits) (9864 digits) 57368 > 99%

3x4 (6519 digits) (13037 digits) 13667 > 99%

3x4 (2373 digits) (4745 digits) 13465 > 99%

3x4 (2356 digits) (4711 digits) 107045 > 99%

2x6 (4934 digits) (9867 digits) 16394 > 99%

2x6 (4932 digits) (9864 digits) 49142 > 99%

2x6 (822 digits) (1644 digits) 2733 > 99%

Executing Machines



6x2 (18268 digits) (36535 digits) 30345 > 99%

6x2 (10567 digits) (21133 digits) 99697 > 99%

6x2 (1440 digits) (2880 digits) 8167 > 99%

6x2 (882 digits) (1763 digits) 2211 > 99%

4x3 (7037 digits) (14073 digits) 25255 > 99%

4x3 (6035 digits) (12069 digits) 34262 > 99%

4x3 (4932 digits) (9864 digits) 57368 > 99%

3x4 (6519 digits) (13037 digits) 13667 > 99%

3x4 (2373 digits) (4745 digits) 13465 > 99%

3x4 (2356 digits) (4711 digits) 107045 > 99%

2x6 (4934 digits) (9867 digits) 16394 > 99%

2x6 (4932 digits) (9864 digits) 49142 > 99%

2x6 (822 digits) (1644 digits) 2733 > 99%

HERE BE DRAGONS!

Executing Machines

Beaver, Platypus, Unicorn … Monash June 2nd, 2014


6x2 (18268 digits) (36535 digits) 30345 > 99%

6x2 (10567 digits) (21133 digits) 99697 > 99%

6x2 (1440 digits) (2880 digits) 8167 > 99%

6x2 (882 digits) (1763 digits) 2211 > 99%

4x3 (7037 digits) (14073 digits) 25255 > 99%

4x3 (6035 digits) (12069 digits) 34262 > 99%

4x3 (4932 digits) (9864 digits) 57368 > 99%

3x4 (6519 digits) (13037 digits) 13667 > 99%

3x4 (2373 digits) (4745 digits) 13465 > 99%

3x4 (2356 digits) (4711 digits) 107045 > 99%

2x6 (4934 digits) (9867 digits) 16394 > 99%

2x6 (4932 digits) (9864 digits) 49142 > 99%

2x6 (822 digits) (1644 digits) 2733 > 99%HERE BE HOBBITS!

Aberrent Albatrosses The observant otter is not a panacea …

Some machines do not produce “neat, compressible” patterns …

3x3 case terminates in 2,315,619 steps with 31 non-zeroes

2x5 case terminates in 26,375,397,569,930 steps with 143 non-zeroes

1004: 012010150130170{b}134334

5000: (011)2101(111)311{a}0433135441100

10001: 01401013010160{a}331374300

???Beaver, Platypus, Unicorn, … RMIT 1st August,

2014

Analysing Machines


Some non-termination cases are easy

perennial pigeon: repeats a particular configuration

phlegmatic phoenix: tape returns to blank

road runner: “straight to the end of the tape”

meandering meerkat: terminal transition cannot be reached

Hence we check for these first

Beaver, Platypus, Unicorn, …

How?

Otherwise .. (ie all the easy cases have been checked): Run the machine for a given number of steps

If it halts, done. Check for non-terminating pattern (Verify non-terminating pattern)

Execute machine for a larger number of steps Eventually give up!


Observant Otter

Some surprises can be in store ....

16{D}0 → 118{D}0 → 142{D}0 (!!!) → 190{D}0

130{D}0 does not occur …

So always look for three instances of a pattern

Conjecture in this case is 1N{D}0 → 12N+6{D}0

or alternatively

1N{D}0 → (11)N111111{D}0

Clearly there is exponential growth here …

Beaver, Platypus, Unicorn, … RMIT 1st August, 2014

Aberrent AlbatrossesThese occur in non-termination cases as well …

36: {a}0333

54: {a}030323

74: {a}032333

88: {a}033333

110: {a}03030323

136: {a}03232333

150: {a}03332333

168: {a}03303333

204: {a}0332330323


??

Aberrent AlbatrossesThese occur in non-termination cases as well …

36: {a}0333

54: {a}030323

74: {a}032333

88: {a}033333

110: {a}03030323

136: {a}03232333

150: {a}03332333

168: {a}03303333

204: {a}0332330323


Searching for Machines


To find bb(n,m):

1. Generate all n-state m-symbol machines

2. Classify them as terminating or non-terminating

3. Find the terminating one which produces the largest number of 1’s (or non-zero characters)

Only problem is that there are lots of traps …

Executing Machines



6x2 (18268 digits) (36535 digits) 30345 > 99%

6x2 (10567 digits) (21133 digits) 99697 > 99%

6x2 (1440 digits) (2880 digits) 8167 > 99%

6x2 (882 digits) (1763 digits) 2211 > 99%

4x3 (7037 digits) (14073 digits) 25255 > 99%

4x3 (6035 digits) (12069 digits) 34262 > 99%

4x3 (4932 digits) (9864 digits) 57368 > 99%

3x4 (6519 digits) (13037 digits) 13667 > 99%

3x4 (2373 digits) (4745 digits) 13465 > 99%

3x4 (2356 digits) (4711 digits) 107045 > 99%

2x6 (4934 digits) (9867 digits) 16394 > 99%

2x6 (4932 digits) (9864 digits) 49142 > 99%

2x6 (822 digits) (1644 digits) 2733 > 99%

HERE BE DRAGONS!

Searching for Machines


There is a finite but very large set of machines for each size …

#machines of size n is O(nn) [ > O(n!) > O(2n) !!]

‘Free’ generation of machines quickly becomes cumbersome ..

‘Free’ generation creates too many trivial machines

a b c h11R01R

00L, 10R

10R01R

Searching through Machines


First transition is fixed Run partial machine Add a new transition when a ‘missing’ case

happens Add ‘halt’ transition when appropriate

Only one transition left At least n different states for an n-state machine

00L

a b c h11L01R 10R

{a}0 -> 1{b}0 -> {a}1 -> {a}01 -> 1{b}1 -> 10{c}0

Generates machines in tree normal form (tnf)

Searching through Machines


n m tnf free

2 2 36 64

3 2 3,522 55,296

2 3 2,751 41,472

4 2 507,107 ??

2 4 335,795 ??

3 3 25,994,116 ??

5 2 99,174,398 ??

2 5 72,130,469 ??

Implementation


Around 5,000 lines of SWI Prolog

Versions available at my website

Data available as well

Documentation is a little lacking …

Intention is that other researchers can use data and verify results (this is empirical science!)

Blue Bilby


n m Total Terminated Going Unclassified

2 2 36 14 22 0

3 2 3,522 1,188 2,334 0

2 3 2,751 825 1,926 0

The free and arbitrary classes have also been classified for these cases

n-state m-symbol machines, n x m ≤ 6

Beaver, Platypus, Unicorn, Zoo …

Ebony Elephant

n-state m-symbol machines, n x m = 8

n m Total Terminated Going Unclassified

4 2 507,107 150,897 325,621 0

2 4 335,795 111,825 223,565 405 (!!)

1's 1 2 3 4 5 6 7 8 9 10

11

12

13

# 3836 23161

37995

31023 15131

5263 1487

357

74

11

5 3 2


White Whale

n m Total

3 3 25,994,116

5 2 99,174,375

2 5 72,130,469

Need to improve the technologybefore taking this on ...

n-state m-symbol machines, n x m = 9 or 10


Demon Duck of Doomn-state m-symbol machines, n x m = 12 (!??!!)

n m Total

6 2 ??

4 3 ??

3 4 ??

2 6 ??



Quest for the smallest universal TM goes on … Involves searching similar (but larger) spaces

Alain Colmerauer (KR’08 talk) U on code(M)code(w) simulates M on w Let M = U U on code(U)code(w) simulates U on w

Let w = blank (and assume code(blank) = blank) U on code(U) simulates U on blank

Hence pseudo-universality test: M is pseudo-universal if M on code(M) simulates M on blank



What exactly is the definition of a universal Turing machine?

How can such definitions be used to identify universal machines “in the wild”?

What constraints are there on the coding function?

Does a UTM have to terminate? Must a UTM terminate on code(M)code(w) exactly

when M terminates on w?

What is an appropriate “architecture” for a UTM? (code(M)code(w) vs code(w)code(M) )


2-D machines

0

0

2

1

1

Beaver, Platypus, Unicorn … RMIT ??, 2014

2-D machines

Can interpret numbers as images …

Have generated some of the smaller cases here

Very basic implementation of emulation

Potentially very interesting loop behaviour

Strict generalisation of one-dimensional case


Learning?

Ebony Elephant case suggests that `most' machines are boring …

Only 2,667 are either high productivity or unclassified

At most 100,000 of the non-terminators are difficult

The remaining 80% are unexceptional … Can we find a criterion to identify (at least a

substantial fraction) of the unexceptional machines?

Need to identify possible features of machines ...Beaver, Platypus, Unicorn … RMIT ??, 2014

Finite Decidability Anomaly

Termination for TMs on blank input is undecidable

Calculating bb(n) involves “only” a finite # machines …

So there is an algorithm for computing bb(n) for each n (!!)

Claim: Any decision problem over any finite set is decidable

Proof Sketch: For a set of size n, there are only 2n possible decisions. Each such decision can be implemented by a Turing machine, ie. a “table” machine that outputs the decision value for each element i є {1 … n}. This is only a “table”, not an “algorithm”, but it is a Turing machine … QED



Termination for TMs on blank input is undecidable

Calculating bb(n) involves “only” a finite # machines …

So there is an algorithm for computing bb(n) for each n (!!)

Claim: Any decision problem over any finite set is decidable

Proof Sketch: For a set of size n, there are only 2n possible decisions. Each such decision can be implemented by a Turing machine, ie. a “table” machine that outputs the decision value for each element i є {1 … n}. This is only a “table”, not an “algorithm”, but it is a Turing machine … QED



Note: A decision procedure for a decidable problem over an infinite set is much shorter than the set itself

So for a finite set, we should require that the Turing machine that decides it should be quantifiably shorter than the “table” which specifies the decision …

An “algorithm” should be predictive, ie should work for more values than those specified in the table …

(more thought needed!)


Conclusions & Further Work Ebony elephant is close to capture

White whale should be alert and alarmed (“Call me Ishmael’’ )

Demon Duck of Doom … [watch this space]

Machine learning techniques may identify criteria which can reduce search space

Duality between n-state 2-symbol and 2-state n-symbol cases

Otter needs to be “remembered”

Compress machines rather than fix macro size in advanceBeaver, Platypus, Unicorn … RMIT 1st August,

2014

Conclusions & Further Work

Weakening of conjectures for aberrent albatrosses

Proofs desired for

3 instances for otter is sufficient

Calculation of #steps (2nd order difference equations)

Champion machines are always exhaustive

Quadruple vs Quintuple machines

Better “pruning” of search space


the busy beaver, the placid platypus and the universal unicorn

Documents

unicorn rmit

universal unicorn beaver

overview beaver

machines beaver

unicorn busy beaver

busy beaver function

areas rmit

placid platypus problem