Nature and Numbers: Exploring Mathematical and Computational Descriptions

1. NATURAL ALGORITHMS Joshua J. Arulanandham, PhD student. Prof. Cristian S. Calude and Dr. Michael J. Dinneen, Supervisors. Department of Computer Science, The University of Auckland, New Zealand.

2. Nature and Numbers – “Friends” for ever! I started it ! A passionate affair

3. The universe is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures Describing nature with mathematics

4. The universe is written in the language of Cellular Automata, and its characters are tiny cells with discrete states. Describing nature with computation

5. How about describing computation with nature ? To those who study her, Nature reveals herself as extraordinarily fertile in devising means … Joseph Wood Krutch, nature writer

6. I too can sort! Huh! A merry-go-round can “sort”

7. The SORT “gate” “raw” vector v sorted vector v Beads fall down in sorted order

8. 1 2 3 Bead-Sort Sorting {3, 1, 2} 1. Drop 3 beads (Remember, always from left-to-right) 2. Drop 1 bead 3. Drop 2 beads

9. 1 2 3 4 Bead-Sort Sorting {1, 3, 2, 4} 1. Drop 1 bead (Remember, always from left-to-right) 2. Drop 3 beads 3. Drop 2 beads 4. Drop 4 beads

10. 0 0 Level_Count 1 3 2 1 1 Rod_Count Bead-Sort: sequential implementation • Two linear arrays used. • Rod_Count keeps track of number of beads in each rod. • Level_Count records number of beads in each level. • Level_Count would contain sorted data, in the end.

11. Flip-flops 0 0 0 0 Presence of bead = 1-state 1 1 1 1 1 1 1 0 Absence of bead = 0-state 0 0 1 0 1 1 0 0 Bead-Sort:digital representation

12. 1/2 A 1/3 A 1/6 A sorted data Trim Trim Trim v1 1 V 0.5 V 0.3 V 0.2 V 3 2 1 Trim Trim Trim v2 2 V V2 V3 1 V 0.7 V 0.3 V Trim Trim Trim v3 3 V 1.5 V 1 V 0.5 V ‘Trim’ Voltage level: Trim( v ) = 1 if v >= 0.5 0 otherwise Analog representation 1 1 1 (no. 3) 1 0 0 (no. 1) 1  1 1 1 1 0 (no.2) 2 2 V1 2  Calculating current flow Resistor chain 1: I1 = V1/R = 3/(1+2+3) = 1/2 A Resistor chain 2: I2 = V2/R = 2/(1+2+3) = 1/3 A Resistor chain 3: I3 = V3/R = 1/(1+2+3) = 1/6 A 3 3 3  1 1 0 2 1 0 3 2 1 1 1 0 DATA ENTRY (strings of 1’s similar to balls) Increase voltage by 1 unit every time a ‘1’ is sent

13. CA implementation 0 0 0 0 0 0 1 1 1 1 0 0 1 0 0 1 1 0 initial configuration {2,1,3} final configuration {1,2,3} 1 1 0 1 1 1 0 1 0 1 0 1 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 1 1 1 0 0 0 0 1 1 1 1 1 2 3 4 5 6 7 8 Simple CA rules

14. A natural computational model

15. infinite source x + y x, y input pan (fixed weight) X + Y X Y output pan (adjustable weight) A self-regulating balance

16. X - Y = ?  X = Y + ? (assume, x > y) + ? X Y Subtraction

17. Average, division & multiplication a a input x, y 2a a a a = (x + y) / 2 a a a a a a x, y, z, w 4a a a a = (x + y + z + w) / 4

18. X + Y = 5 X – Y = 3 Represents X + Y = 5 Represents X – Y = 3 x y 5 3 y x Solving simultaneous equations

19. Principles for designing natural algorithms • Principle of Preferred States A natural system has a set of preferred states due to the presence of inherent properties and laws, thus restraining the “degrees of freedom” of the system. • Principle ofAutomatic Constraint Satisfaction Ifthe constraints in a problem are “hard-wired” into the natural system (representing the problem), then, the preferred state is the desired solution (state) . • Principle of Bilateral Symmetry Some natural(physical) systems can facilitate both “forward” and “backward” directions of information-flow, for the same “price”: it is possible to compute what possible input(s) might lead to a particular output.

20. Sliders 1 0 1 0 0 1 0 0 1 1 0 0 or 0 0 0 0 1 0 0 1 0 1 1 0 The “bias” in nature

21. l1 l2 l3 (l1 + l2 + l3 ) / 3 Automatic constraint satisfaction - A simple demonstration Liquids finding the same level due to atmospheric pressure

22. Combinatorial explosion need not matter Goal Start The problem space

23. E (Destination) D A C B (Source) Natural algorithm - shortest path Shortest path

24. THANK YOU! Special thanks are due to Dr. Alan Creak,Honorary Researcher, Dept. of Computer Science Prof. Boris Pavlov,Professor, Dept. of Mathematics Prof. Garry Tee,Research Fellow, Dept. of Mathematics Mr. Jasvir Nagra,PhD Student, Dept. of Computer Science Mr. Damien Duff,MSc Student, Dept. of Computer Science Mr. Andrew Paxie,MSc Student, Dept. of Computer Science Mr. Dong Qiang, former MSc Student, Dept. of Computer Science Mr. Philip Chiang,PhD Student, Dept. of Computer Science Mr. Chi-kou Shu,PhD Student, Dept. of Computer Science Mr. Ming Li, MSc Student, Dept. of Computer Science for their invaluable suggestions and comments.

25. 1 2 3 Bead Sort Sorting {2, 3, 1} 1. Drop 2 beads (Remember, always from left-to-right) 2. Drop 3 beads 3. Drop 1 beads

26. 1 2 3 Bead Sort Sorting {1, 3, 2} 1. Drop 1 bead (Remember, always from left-to-right) 2. Drop 3 beads 3. Drop 2 beads

27. 2 2 3 4 Bead Sort Sorting {2, 4, 3, 2} 1. Drop 2 beads (Remember, always from left-to-right) 2. Drop 4 beads 3. Drop 3 beads 4. Drop 2 beads

28. . . . 100C3 degrees of freedom 1 0 1 0 0 0 1 1 0 1 0 0 1 0 1 1 0 0 0 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 161, 700 degrees of freedom (100 cells with three 1’s) entropy loss = HUGE ! “squeeze” 2 possible states OR 1 1 1 0 0 0 0 0 0 0 0 1 1 1