the busy beaver, the placid platypus and the universal unicorn

Beaver, Platypus, Unicorn, Zoo …

Monash 29th June,2011

The Busy Beaver, the Placid Platypus and the Universal Unicorn

James Harland

[email protected]

Using an to catch a

to chase a

otter beaver platypus



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 ≥ 3.51×1018276 ≥ 7.41×1036534 (!!)



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



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) ]

Otter -> Beaver -> Platypus Monash 29th June,2011

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



How?

Easy! :-) To find bb(n,m):

1. Generate all n-state m-symbol machines

2. Classifying 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 …



How?

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

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

`Free’ generation of machines quickly becomes impractical

`Free’ generation creates too many trivial machines

a b c h11R01R

00L, 10R

10R01R



How?

First transition is fixed Run partial machine Add a new transition when a `missing’ case happens Add `halt’ transition last

00La b c h11L

01R 10R

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

Generates machines in tree normal form (tnf)



How Many?

n m tnf free

2 2 32 128

3 2 3,288 82,944

2 3 2,280 82,944

4 2 527,952 100,663,296

2 4 447,664 100,663,296

3 3 24,711,336 ??

5 2 110,537,360 ??

2 5 176,293,040 ??



How?

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



How?

Otherwise .. (ie all the easy cases have been checked): Run the machine for a given number of steps If it halts, done. Examine the execution trace for non-termination

conjectures Attempt to show that the hypothesis pattern repeats

infinitely Execute machine for a large number of steps Give up!



How?

11{C}1 → 11{C}111 → 11{C}11111 …

Conjecture 11{C} 1 (11)N → 11{C} 1(11)N+1

Start engine in 11{C} 1 (11)N Terminate with success if we reach 11{C} 1 (11)N+1

(or 11{C} 1 (11)N+2 or …)

Halt with failure after a certain number of steps Main trick is how to deal with (11)N case where N is a

variable

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 …

(this is why two previous instances are needed)

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

or alternatively

1N{D}0 → (11)N111111{D}0

These are known as killer kangaroos ...





How?

Three main cases:

1) Wild Wombat

2) Stretching Stork

3) Observant Otter

4) (soon also the Ossified Ocelet)

slithering snake

resilient reptile

maniacal monkey



Wild Wombat

X {S} IN Y → X JN {S} Y X IN {S} Y → X {S} JN Y

11 {C} (11)N 011 → 11 (00)N {C} 011 if

{C}11 → 0{D}1 →{B}01 → 0{B}1 → 00 {C}

X {S} IN Y → ?? {S} I → J {S}

Pointer remains within I

First exits I to the right

State on exit is S

``Context-free''



Stretching Stork

X {S} IN Y → X {S} I IN-1 Y

11 {C} (11)N 011 → 11 {C} 11 (11)N-1 011

X {S} IN Y → ??

If the wombat doesn't apply and N > 0

Condition N > 0 required to stop this being infinitely applicable



Observant Otter

11 {C} (11)N 01 → … →

111 {C} 1(11)N-1 01 → … →

1111 {C} 1(11)N-2 01 → … →

Oi! There is a pattern here!

11 (1)N {C} 01 → …



Observant Otter

X {S} IN Y → ??

Looks through execution history to find a similar pattern

Requires two previous instances before a match is made

Currently only allows decrement of 1 Calculates new state from patterns

Ossified Ocelet will store these patterns for potential future use

This technique used by Wood in 2008 to evaluate monsters ...

Observant Otter

1111{C}11

1011{C}111

110{C}1111

111111{C}11

101111{C}111

11011{C}1111

1110{C}11111

11111111{C}11



Sometimes this is not as simple as it may seem …



Implementation

Around 8,000 lines of Ciao Prolog

Needs a thorough clean up; necessary code is around 40% of this size

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-state m-symbol machines, n x m ≤ 6

n m Total Terminated

Going Unclassified

2 2 32 8 24 0

3 2 3,288 831 2,457 0

2 3 2,280 616 1,664 0


Going Unclassified

2 2 128 35 93 0

3 2 82,944 22,034 60,910 0

2 3 82,944 18,613 64,331 0

tnf

free



Ebony Elephant

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


Going Unclassified

4 2 527,952 118,348 408,876 728

2 4 447,664 ?? ?? ??

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 24,711,336

5 2 110,537,360

2 5 176,293,040 (??)

Need to improve the technologybefore taking this on ...

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



Demon Duck of Doom

n m Total

6 2 ??

4 3 ??

3 4 ??

2 6 ??



Universal Turing machines

Quest for the smallest universal TM goes on … Involves searching similar 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



2-D machines

0

0

2

1

1



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 here

Strict generalisation of one-dimensional case

No existing work that I have been able to find …



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 ...



Conclusions & Further Work

Otter still needs some refinements

needs to `dovetail’ with the rest of the engine

some small number of machines resistant to otter (so far)

Some parameters (eg maximum search steps) need to be tweaked

Connections to universal Turing machines and 2-D machines need investigation

New (journal) paper will be out `soon’



Conclusions & Further Work

Ebony elephant is close to capture

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

Demon Duck of Doom should be alert but not alarmed yet

Machine learning techniques may identify criteria which can reduce search space

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

Lock away the beaver and play with the platypus

the busy beaver, the placid platypus and the universal unicorn

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