1 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

2 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

task:

brainstorm large-scale experiment topics doable with simple model

3 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

4 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

19 kb memory

5 of 143

Loose upper bound

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

Stratum

“Fingerprint”

0x3d49

New!

…

idx

0

idx

2

idx

1

6 of 143

retained@t

retained@t+1

What to get rid of at time t?

... set subtraction

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

Stratum

“Fingerprint”

0x3d49

New!

…

idx

0

idx

2

idx

1

7 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

19 kb memory

8 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

19 kb memory

🐁

🐘

Simplify and Optimize for “in practice”

📌‏ ‎

9 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

19 kb memory

🐁

🐘

Simplify and Optimize for “in practice”

📌‏ ‎

10 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

19 kb memory

🐁 💪

🐘

Simplify and Optimize for “in practice”

📌‏ ‎

11 of 143

Goal

Gen 0

Column

…

Gen 1

Gen 2

(Gen 3)

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0xddbd

retain

❌

eliminate

Stratum

“Fingerprint”

0x3d49

New!

…

retain

idx

0

idx

2

idx

1

19 kb memory

🐁 💪

🐘

Simplify and Optimize for “in practice”

📌‏ ‎

?

12 of 143

Design Requirements

Memory efficiency

why? WSE has 19kb per core and messages are sent in 32 bit packets
constant-size annotation
make full use of allotted memory
stateless algorithms (i.e., reversible)

Simple* implementation

minimize data structures: ideally, just an array and a counter
no malloc
avoid floating point computation due to low precision
*is what follows really simple???

Execution efficiency

likely working with simple models: instrumentation overhead therefore relatively large
O(1) operation
avoid slow operations (e.g., division, logarithms, etc.)
ideally, flip at most one bit per generation

13 of 143

0

1

0

1

0

1

gen

987

&

14 of 143

gen

988

⬆️

0

1

0

1

0

1

2

3

4

5

6

7

0

1

0

1

0

1

gen

987

0

&

🎲

15 of 143

gen

988

⬆️

0

1

0

1

0

1

2

3

4

5

6

7

0

1

0

1

0

1

gen

987

0

&

🎲

Column

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0x3d49

Stratum

“Fingerprint”

0xd01a

New!

(eliminated)

…

16 of 143

gen

988

⬆️

0

1

0

1

0

1

2

3

4

5

6

7

0

1

0

1

0

1

gen

987

0

&

🎲

Column

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0x3d49

Stratum

“Fingerprint”

0xd01a

New!

(eliminated)

…

&

17 of 143

gen

988

⬆️

0

1

0

1

0

1

2

3

4

5

6

7

0

1

0

1

0

1

gen

987

0

&

🎲

Column

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0x3d49

Stratum

“Fingerprint”

0xd01a

New!

(eliminated)

…

&

18 of 143

gen

988

⬆️

0

1

0

1

0

1

2

3

4

5

6

7

0

1

0

1

0

1

gen

987

0

&

🎲

Column

Stratum

“Fingerprint”

0x504b

Stratum

“Fingerprint”

0x3d49

Stratum

“Fingerprint”

0xd01a

New!

(eliminated)

…

&

Misc

one generation <= 1 bit diff
If two have same stratum, it’s at the same site
Half of the time do nothing 😎

19 of 143

gen

988

⬆️

0

1

0

1

0

1

2

3

4

5

6

7

0

1

0

1

0

1

gen

987

0

&

🎲

20 of 143

time t

time t+1

gen

988

⬆️

0

1

0

1

0

1

2

3

4

5

6

7

0

1

0

1

0

1

gen

987

0

&

🎲

21 of 143

time t

time t+1

⚙️

gen

988

⬆️

0

1

0

1

0

1

2

3

4

5

6

7

0

1

0

1

0

1

gen

987

0

&

🎲

forward

22 of 143

⚙️

0

1

0

1

0

1

gen

X

🎲

&

🎲

&

🎲

&

🎲

&

🎲

1

0

1

0

1

gen

Y

&

0

1

2

4

5

6

7

3

gen X

…

backward

23 of 143

⚙️⚙️⚙️

0

1

0

1

0

1

time

X

&

🎲

&

🎲

&

🎲

&

🎲

1

0

1

0

1

time

Y

&

0

1

2

4

5

6

7

3

gen X

…

constant

🎲

fixed-width buffer

T

I

M

E

24 of 143

⚙️⚙️⚙️

0

1

0

1

0

1

time

X

&

🎲

&

🎲

&

🎲

&

🎲

1

0

1

0

1

time

Y

&

0

1

2

4

5

6

7

3

gen X

…

constant

🎲

fixed-width buffer

T

I

M

E

25 of 143

0

1

0

1

0

1

time

X

🎲

1

0

1

0

1

time

Y

…

0

1

2

4

5

6

7

idx

3

time X

…

calc

orig

time

.⚙.

🎲

fixed-width buffer

T

I

M

E

26 of 143

time

X

🎲

time

Y

…

0

1

2

4

5

1

6

7

idx

3

time X

…

1

…

calc

orig

time

.⚙.

🎲

fixed-width buffer

T

I

M

E

0

1

0

1

0

1

0

1

0

1

27 of 143

time

X

time

Y

…

0

1

2

4

5

1

6

7

idx

3

time X

…

1

…

calc

orig

time

.⚙.

fixed-width buffer

T

I

M

E

0

1

0

1

0

1

0

1

28 of 143

?

29 of 143

Steady vs. Tilted

Steady

Tilted

🚩 🚩 🚩 🚩 🚩 🚩 🚩 🚩 🚩

🚩 🚩 🚩 🚩 🚩 🚩 🚩🚩

🆕…

📍

🕐 🕑 🕒 🕓 🕔 🕕 🕖 🕗 🕘 🕙 🕚 🕛

30 of 143

Steady vs. Tilted

🍋

🍊

🍁

🍇

🍀

🥜

🌼

🌻

🥕

🌵

🌾

🌽

https://github.com/ghuertaramos/PlantPhylomoji

31 of 143

Steady vs. Tilted

🍋

🍊

🍁

🍇

🍀

🥜

🌼

🌻

🥕

🌵

🌾

🌽

32 of 143

Steady vs. Tilted

🍋

🍊

🍁

🍇

🍀

🥜

🌼

🌻

🥕

🌵

🌾

🌽

33 of 143

Steady vs. Tilted

🍋

🍊

🍁

🍇

🍀

🥜

🌼

🌻

🥕

🌵

🌾

🌽

34 of 143

Tilted

35 of 143

36 of 143

By Trixx - I designed this using http://thewalnut.io/, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=43282866

37 of 143

38 of 143

39 of 143

⚙️

(n & -n).bit_length() - 1

40 of 143

⚙️

(n & -n).bit_length() - 1

41 of 143

	Hanoi value 0	Hanoi value 1	Hanoi value 2	Hanoi value 3
Time 0	x
Time 1		x
Time 2	x
Time 3			x
Time 4	x
Time 5		x
Time 6	x

42 of 143

1

-

0

-

0

1

-

0

-

1

-

1

0

-

1

-

0

-

1

Hanoi value 0

Hanoi value 1

Hanoi value 2

Hanoi value 3

Hanoi value 4

Hanoi value 5

Hanoi value 6

Hanoi value 7

🎲

43 of 143

1

-

0

-

0

1

-

0

-

1

-

1

0

-

1

-

0

-

1

Hanoi value 0

Hanoi value 1

Hanoi value 2

Hanoi value 3

Hanoi value 4

Hanoi value 5

Hanoi value 6

Hanoi value 7

🎲

-

…

44 of 143

-

-, -

0

-

0, 1

-, -

0, -

-, -

0, 1

-, -

0, -, -, -

0, 1, 2, -

0, -, -, -

0, 1, 2, -

+0

+1

+0

+2

+0

45 of 143

4

5

6

7

2

3

1

0

2

1

3

0

1

0

46 of 143

-, -

-

-, -

0

-

0, 1

-, -

0, 1

0, -

0, -, -, -

0, 1, 2, -

+0

+1

+0

+2

+0

0, -, -, -

0, 1, 2, -

47 of 143

-, -

-

-, -

0

-

0, 1

-, -

0, 1

0, -

0, -, -, -(1)

0, 1, 2, -(1)

+0

+1

+0

+2

+0

0, -, -, -(1)

0, 1, 2, -(1)

48 of 143

15

8

12

11

0

7

3

4

14

9

10

1

6

13

2

5

descending

49 of 143

15

8

12

11

0

7

3

4

14

9

10

1

6

13

2

5

descending

Index -> gray coding -> bit twiddling -> position

50 of 143

15

8

12

11

0

7

3

4

14

9

10

1

6

13

2

5

descending

Index -> gray coding -> bit twiddling -> position

Hadamard sequence ?????

51 of 143

15

8

12

11

0

7

3

4

14

9

10

1

6

5

13

2

descending

52 of 143

15

8

12

11

0

7

3

4

14

9

13

10

1

6

2

5

8

12

11

7

3

4

14

0

9

10

1

6

5

13

2

descending

53 of 143

15

8

12

11

0

7

3

4

14

9

13

10

1

6

2

5

8

12

11

7

3

4

14

0

9

10

1

6

5

8

12

11

0

7

3

4

9

10

6

5

1

13

2

13

2

descending

54 of 143

15

8

12

11

0

7

3

4

8

12

11

0

7

3

4

14

9

13

10

1

6

2

5

9

10

6

5

2

8

12

11

7

3

4

14

0

9

10

1

6

5

8

12

11

0

7

3

4

9

10

6

5

1

13

2

13

2

descending

55 of 143

15

8

12

11

0

7

3

4

8

12

11

0

7

3

4

8

0

7

3

14

9

13

10

1

6

2

5

9

10

6

5

2

9

1

6

2

11

4

8

12

11

7

3

4

14

0

9

10

1

6

5

8

12

11

0

7

3

4

9

10

6

5

1

13

2

13

2

10

5

descending

56 of 143

15

8

12

11

0

7

3

4

8

12

11

0

7

3

4

8

0

7

3

4

8

0

7

3

14

9

13

10

1

6

2

5

9

10

6

5

9

1

6

2

9

10

1

6

2

5

8

0

7

3

4

1

6

2

5

8

11

0

7

3

4

9

1

6

2

5

8

12

11

7

3

4

14

0

9

10

1

6

5

8

12

11

0

7

3

4

9

10

6

5

1

13

2

13

2

10

5

4

descending

57 of 143

6

15

8

12

11

0

7

3

4

8

12

11

0

7

3

4

8

0

7

3

4

7

3

4

0

8

0

7

3

14

9

13

10

1

6

2

5

9

10

6

5

9

1

6

2

5

1

9

10

1

6

2

5

8

0

7

3

4

1

6

2

5

8

11

0

7

3

4

9

1

6

2

5

8

12

11

7

3

4

14

0

9

10

1

6

5

8

12

11

0

7

3

4

9

10

6

5

1

13

2

13

2

10

5

4

descending

58 of 143

6

15

8

12

11

0

7

3

4

8

12

11

0

7

3

4

8

0

7

3

4

7

3

4

0

8

0

7

3

14

9

13

10

1

6

2

5

9

10

6

5

9

1

6

2

5

1

9

10

1

6

2

5

3

4

0

6

2

5

8

0

7

3

4

1

6

2

5

8

11

0

7

3

4

9

1

6

2

5

1

8

12

11

7

3

4

14

0

9

10

1

6

5

8

12

11

0

7

3

4

9

10

6

5

1

13

2

13

2

10

5

4

descending

59 of 143

6

15

8

12

11

0

7

3

4

8

12

11

0

7

3

4

8

0

7

3

4

7

3

4

0

8

0

7

3

14

9

13

10

1

6

2

5

9

10

6

5

9

1

6

2

5

1

9

10

1

6

2

5

3

4

0

6

2

5

8

0

7

3

4

1

6

2

5

8

11

0

7

3

4

9

1

6

2

5

1

8

12

11

7

3

4

14

0

9

10

1

6

5

8

12

11

0

7

3

4

9

10

6

5

1

13

2

13

2

10

5

4

descending

60 of 143

6

15

8

12

11

0

7

3

4

8

12

11

0

7

3

4

8

0

7

3

4

7

3

4

0

3

4

0

8

0

7

3

14

9

13

10

1

6

2

5

9

10

6

5

9

1

6

2

5

1

5

1

2

1

9

10

1

6

2

5

0

2

1

4

0

3

1

3

4

0

6

2

5

8

0

7

3

4

1

6

2

5

3

0

2

8

11

0

7

3

4

9

1

6

2

5

1

8

12

11

7

3

4

14

0

9

10

1

6

5

8

12

11

0

7

3

4

9

10

6

5

1

13

2

13

2

10

5

4

descending

61 of 143

5

4

6

0

3

1

7

2

62 of 143

5

4

6

0

3

1

7

2

Hanoi 0
Hanoi 1
Hanoi 2
Hanoi 3
…

63 of 143

5

4

6

0

3

1

7

2

Hanoi 0	h0’	h0’’’	h0’’	…
Hanoi 1	h1’	…	h1’’
Hanoi 2	h2’		…
Hanoi 3	…
…

64 of 143

D

E

F

5

4

6

0

3

1

7

B

A

2

C

H

G

65 of 143

D

E

F

5

4

6

0

3

1

7

B

A

2

C

H

G

66 of 143

D

E

F

B

A

C

H

G

5

4

6

0

3

7

2

1

67 of 143

D

E

F

B

A

C

H

G

5

4

6

0

3

1

7

2

68 of 143

D

E

F

B

A

C

H

G

5

4

6

0

3

1

7

2

69 of 143

D

E

F

B

A

C

H

G

5

4

6

0

3

1

7

2

70 of 143

A

B

C

D

E

F

G

H

NEW

5

4

2

6

0

3

1

7

71 of 143

A

B

C

D

E

F

G

H

NEW

A

B

C

D

E

F

G

H

NEW

OLDER

5

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“Fractionally incrementing”

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85 of 143

Modular arithmetic

86 of 143

Sloppy base 2 lowest common multiple…

Modular arithmetic

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Next Steps

Write notes & make graphics (mostly done)
Finish reverse mapping for more sophisticated algorithm
Put it on a shelf for later
How to distill/streamline a zoo of highly-redundant functions into a single procedure/expression?

Sympy? (i.e., computer algebra)
Throw it through an optimizing compiler and parse the assembly/IR?
By hand?

What to do when rank capacity is exceeded?
Add to hstrat library (Python/C++ & CerebrasLang)
Write a paper etc.

105 of 143

Steady

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r5h1

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r0h0

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r10h0

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r31h5

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r0h0

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r3h2

r7h3

r2h0

r5h1

r11h2

r4h0

r9h1

r6h0

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r8h0

r10h0

r12h0

r14h0

r15h4

r19h2

r23h3

r31h5

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s1

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🌊🌊🌊🌊

🌊

🌊🌊🌊

🌊

🌊🌊

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🌊

Hanoi

Value

x

122 of 143

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h.v.

h

h.v.

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h.v.

h

h.v.

h

memory

buffer

epoch

h.v.

h

h.v.

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123 of 143

…

t = h

t = h-1

t = h+1

bunch 2

bunch 3

t = h-3

bunch 1

bunch 0

t = h-2

h.v.

h

h.v.

h

h.v.

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h

h.v.

h

memory

buffer

epoch

h.v.

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t = h

t = h-1

bunch 2

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t = h-2

memory

buffer

epoch t

<——- seg 0 -——>

<———— seg 0 ————>

<– seg 0 –>

seg 0

seg 1

seg 2

seg 3

<– seg 1 –>

…

h.v. h+1

t = h+1

h.v.

h+4

h.v.

h+2

h.v. h

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h.v.

h+3

h.v.

h

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125 of 143

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t = h

t = h-1

bunch 2

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memory

buffer

epoch t

<——- seg 0 -——>

<———— seg 0 ————>

<– seg 0 –>

seg 0

seg 1

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seg 3

<– seg 1 –>

…

h.v. h+1

h.v.

h+4

h.v.

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h.v. h

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h.v.

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126 of 143

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t = h

t = h-1

bunch 2

bunch 3

t = h-3

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bunch 0

t = h-2

memory

buffer

epoch t

<——- seg 0 -——>

<———— seg 0 ————>

<– seg 0 –>

seg 0

seg 1

seg 2

seg 3

<– seg 1 –>

…

h.v. h+1

t = h+1

h.v.

h+4

h.v.

h+2

h.v. h

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h.v.

h

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127 of 143

…

t = h

t = h-1

bunch 2

bunch 3

t = h-3

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bunch 0

t = h-2

memory

buffer

epoch t

<——- seg 0 -——>

<———— seg 0 ————>

<– seg 0 –>

seg 0

seg 1

seg 2

seg 3

<– seg 1 –>

…

h.v. h+1

t = h+1

h.v.

h+4

h.v.

h+2

h.v. h

h.v.

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h+3

h.v.

h

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128 of 143

a

b

c

d

a

c

a

b

a

b

a

bin 3 (2:1)

bin 2 (2:0)

segment 2

bin 0 (0:0)

segment 0

bin 1 (1:0)

segment 1

bin 4 (3:0)

segment 3

bin 5 (3:1)

bin 6 (3:2)

bin 7 (3:3)

s1

s2

s6

s3

s4

s5

s7

s8

s9

s10

s11

s12

s13

s14

s15

😭

❌

129 of 143

Time since deposition (linear)

Time since deposition (log)

130 of 143

Time since deposition (linear)

Time since deposition (log)

131 of 143

Differentia Value (linear)

Differentia Value (log)

132 of 143

Bin Size Sequence

Highest power of 2 dividing the unique normal number whose unsorted prime signature is the k-th composition in standard order (graded reverse-lexicographic).... https://oeis.org/A065120

bin 3 (2:1)

bin 2 (2:0)

segment n-2

bin 0 (0:0)

segment 0

bin 1 (1:0)

segment 1

bin 4 (3:0)

segment n-1

bin 5 (3:1)

bin 6 (3:2)

bin 7 (3:3)

4

3

2 2

1 1 1 1

0, 1, 2, 1, 3, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 1, 5, 4, 3, 3…

https://muppet.fandom.com/wiki/Elmo_Variants

133 of 143

Enactment: Map Bin # to Site Index

bin 3 (2:1)

bin 2 (2:0)

segment n-2

bin 0 (0:0)

segment 0

bin 1 (1:0)

segment 1

bin 4 (3:0)

segment n-1

bin 5 (3:1)

bin 6 (3:2)

bin 7 (3:3)

4

3

2 2

1 1 1 1

https://muppet.fandom.com/wiki/Elmo_Variants

(1 << n) * mbw - mbw - n * (1 << n) + (1 << (n + 1)) - 2

134 of 143

Decoding: Map Site Index to Bin Number

bin 3 (2:1)

bin 2 (2:0)

segment n-2

bin 0 (0:0)

segment 0

bin 1 (1:0)

segment 1

bin 4 (3:0)

segment n-1

bin 5 (3:1)

bin 6 (3:2)

bin 7 (3:3)

…

Σ

https://mathworld.wolfram.com/EulersNumberTriangle.html

digitaljournal.com, CC BY-SA 2.0 via Wikimedia Commons

Σ

10110000000

11010000000

...

11110111000

11111111111

Eulerian Numbers repeat once

1, 4, 11, 26, 57, 120…

Number of eigenvalues equal to 1 of n X n matrix A(i,j)=1 if j=1 or i divides j

# leading 1’s:

https://oeis.org/A083058

https://oeis.org/A000295

135 of 143

fin.

https://muppet.fandom.com/wiki/Elmo_Variants

136 of 143

0

1

0

1

0

1

2

3

4

5

6

1

7

988

0

1

0

1

0

1

2

3

4

5

6

1

7

987

🎲

⬆️

⚙️

137 of 143

0

1

0

1

0

1

2

3

4

5

6

1

7

987

0

1

0

1

0

1

2

3

4

5

6

1

7

987

🎲

⬆️

138 of 143

7

4

6

5

0

3

1

2

alternating

4

6

5

0

3

1

2

4

0

3

1

2

3

1

2

0

1

2

0

1

0

4

5

0

3

1

2

139 of 143

10

15

8

12

11

0

7

3

4

14

9

10

1

6

13

2

5

10

descending

140 of 143

5

8

10

12

0

2

4

14

10

12

0

6

2

4

14

0

6

2

4

2

6

0

3

7

0

14

0

6

9

13

11

15

1

5

3

7

13

15

5

7

13

1

5

3

2

3

1

6

1

3

1

13

15

1

5

2

0

3

1

7

0

3

1

2

6

0

5

3

7

14

0

6

2

4

1

5

3

2

0

2

14

12

0

6

2

4

1

3

7

1

14

10

12

6

2

4

9

0

13

15

1

5

7

14

11

12

0

6

3

4

13

15

5

7

1

11

3

11

3

15

7

4

5

3

7

15

6

7

15

Piecewise ascending

141 of 143

8

12

10

14

0

4

2

6

9

13

15

1

5

11

3

7

https://oeis.org/A030109

Piecewise ascending