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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 The Busy Beaver, the Placid Platypus and the Universal Unicorn James Harland james.harland@rmit.edu.au Using anto catch a to chase a otter beaverplatypus

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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 abch11R 01R 00L 10R 01L 11R

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 What? What is the largest number of 1’s that can be printed by a terminating n-state machine? n#1’s (productivity)#steps 111 246 3621 413107 5≥ 4098≥ 47,176,870 (??) 6≥ 3.51×10 18276 ≥ 7.41×10 36534 (!!)

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 What? nm#1’s#steps 2246 32621 23938 4213107 242,0503,932,964 33374,676,383119,112,334,170,342,540 52≥ 4098≥ 47,176,870 25≥ 1.7 × 10 352 ≥ 1.9 × 10 704 623.51×10 18276 ≥ 7.41×10 36534 43≥ 1.383 × 10 7036 ≥ 1.025 × 10 14072 34≥ 3.743 × 10 6518 ≥ 5.254 × 10 13036 26≥ 1.903 × 10 4933 ≥ 2.415 × 10 9866

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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)

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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’

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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) ]

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Otter -> Beaver -> PlatypusMonash 29 th 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

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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 …

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 How? There is a finite but very large set of machines for each size … Number of machines of size n is O(n n ) [ > O(n!) > O(2 n ) !!] `Free’ generation of machines quickly becomes impractical `Free’ generation creates too many trivial machines abch11R 01R 00L, 10R 10R 01R

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 How? First transition is fixed Run partial machine Add a new transition when a `missing’ case happens Add `halt’ transition last 00L abch11L 01R 10R {a}0 -> 1{b}0 -> {a}1 -> {a}01 -> 1{b}1 -> 10{c}0 Generates machines in tree normal form (tnf)

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 How Many? nmtnffree 2232128 323,28882,944 232,28082,944 42527,952100,663,296 24447,664100,663,296 33 24,711,336 ?? 52110,537,360?? 25176,293,040??

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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!

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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

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Observant Otter Some surprises can be in store.... 1 6 {D}0 → 1 18 {D}0 → 1 42 {D}0 (!!!) → 1 90 {D}0 1 30 {D}0 does not occur … (this is why two previous instances are needed) Conjecture in this case is 1 N {D}0 → 1 2N+6 {D}0 or alternatively 1 N {D}0 → (11) N 111111{D}0 These are known as killer kangaroos... Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 Wild Wombat X {S} I N Y → X J N {S} YX I N {S} Y → X {S} J N 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} I N Y → ?? {S} I → J {S} Pointer remains within I First exits I to the right State on exit is S ``Context-free''

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 Stretching Stork X {S} I N Y → X {S} I I N-1 Y 11 {C} (11) N 011 → 11 {C} 11 (11) N-1 011 X {S} I N Y → ?? If the wombat doesn't apply and N > 0 Condition N > 0 required to stop this being infinitely applicable

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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 → …

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 Observant Otter X {S} I N 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...

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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 Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 Sometimes this is not as simple as it may seem …

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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!)

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 Blue Bilby n-state m-symbol machines, n x m ≤ 6 nmTotalTerminatedGoingUnclassified 2232 824 0 323,288 8312,4570 232,280 6161,6640 nmTotalTerminatedGoingUnclassified 22128 3593 0 3282,944 22,03460,9100 2382,944 18,61364,3310 tnf free

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 Ebony Elephant n-state m-symbol machines, n x m = 8 nmTotalTerminatedGoingUnclassified 42 527,952118,348 408,876728 24 447,664 ?? 1's12345678910111213 #383623161379953102315131526314873577411532

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 White Whale nmTotal 3324,711,336 52 110,537,360 25 176,293,040 (??) Need to improve the technology before taking this on... n-state m-symbol machines, n x m = 9 or 10

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 Demon Duck of Doom nmTotal 62?? 43 34 2 6

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 2-D machines 0 0 2 1 1

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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 …

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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...

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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’

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Beaver, Platypus, Unicorn, Zoo …Monash 29 th June,2011 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

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