Langton's AntLangton's Ant

Jacek Strzelczyk

6th of November 2007

Algorithms, Logic and Complexity

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Computer-generated mysteriesComputer-generated mysteries

• integers with interesting but unexplained divisibility properties

• observed geometric configurations with no proof of existence

07-11-6 Jacek Strzelczyk

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The story so farThe story so far

• ant – two-dimensional Turing machine• lives on plane partitioned into squares (cells)• each cell can be in one of several states

Jacek Strzelczyk07-11-6

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Ant's algorithmAnt's algorithmChange the color of the square you are standing on- (if the square is black change it to white and if it is white change it to black)Now Walk Forward to the square in front of you Look at the color of the square you are standing on. If it is black turn Left and if it is white turn Right by 90 degrees. Return to step 1.

Jacek Strzelczyk

http://www.tiac.net/~sw/LangtonsAnt/LangtonsAnt.html

07-11-6

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First phenomenon: patternsFirst phenomenon: patterns

• centrally symmetric „track”

The universe of ant at times 184, 368 and 472

Jacek Strzelczyk07-11-6

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Second phenomenon: highwaySecond phenomenon: highway

• highway to infinity

Jacek Strzelczyk07-11-6

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n-states antsn-states ants

• n different states for cells to be in• rule-string programming• ant description e.g. LLRRRLR• if L = 1 and R = 0 then ant's „genome” LLRRRLR = 98

Jacek Strzelczyk07-11-6

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Phenomenon described by Phenomenon described by ProppPropp

• surprising ants 9 and 12 [LRRL and LLRR]• patterns on visiting starting cell

The universe of ant 12 at time 16,464 The universe of ant 9 at time 38,836

Jacek Strzelczyk07-11-6

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More ants of this sort...More ants of this sort...

• rule strings of length 6• mystery: the rule strings that lead to bilaterally symmetricpatterns patterns are 33, 39, 48, 51 and 60• all divided by three !

Jacek Strzelczyk07-11-6

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Proof idea: Truchet tilesProof idea: Truchet tiles

• cells split into * H-cells (entered horizontally, exited vertically) * V-cells (entered vertically, exited horizontally) Truchet tiles – schematic „switches”

The initial state of universe

Jacek Strzelczyk07-11-6

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Truchet contoursTruchet contours

• closed curves – Truchet contours

Jacek Strzelczyk07-11-6

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Principal contourPrincipal contour

• a contour through starting point

Jacek Strzelczyk07-11-6

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The Even Run-Length PropertyThe Even Run-Length Property

• Why do some ant tracks exhibit recurrent bilateral symmetry and others not?

• Rule-strings for 4-state and 6-state ants that exhibit recurrent symmetry:

Ant Rule-string9 LRRL12 LLRR33 LRRRRL39 LRRLLL48 LLRRRR51 LLRRLL57 LLLRRL60 LLLLRR

Jacek Strzelczyk07-11-6

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Diagonals graphDiagonals graph

• Two types of cells:- cold – if its state is odd (won't change operation after the next visit)- hot – if its state is even (will change operation after the next visit)

• For hot cells display not only the Truchet tile but also its diagonal.

Jacek Strzelczyk07-11-6

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Even diagonal-degree propertyEven diagonal-degree property

• All vertices in the diagonals graph have even degree (0, 2 or 4)

• If the state of the universe satisfies the even diagonals-degree propertywith the ant at home, then the ant must travel along the principal contour,but when it completes this path and returns home, it restores the even diagonals-degree property, so that it must once again travel along the (new) principal contour, and so on, ad infinitum.

Jacek Strzelczyk07-11-6

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ReferencesReferences

• „Further Travels with My Ant”, David Gale, Jim Propp, Scott Sutherland and Serge Troubetzkoy, published in „Math. Intelligencer” 17 (1995)

Jacek Strzelczyk07-11-6