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L-Systems

L-Systems. Lindenmayer Systems algorithmicbotany.org. L-System Overview. Goals (Plant modeling) Structural Developmental Chemical Parallel replacement grammar Representative of multicellular growth. Overview Continued. Other types of parallel replacement systems Open polygon by Koch

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L-Systems

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  1. L-Systems Lindenmayer Systems algorithmicbotany.org Jason Stredwick, MSU 2004

  2. L-System Overview • Goals (Plant modeling) • Structural • Developmental • Chemical • Parallel replacement grammar • Representative of multicellular growth Jason Stredwick, MSU 2004

  3. Overview Continued • Other types of parallel replacement systems • Open polygon by Koch • Cellular automata by Wolfram • Game of Life by Conway • L-System grammar interpretations • LOGO (graphical modeling of plants) • Pure math • Jordan Pollack’s Golem project Jason Stredwick, MSU 2004

  4. D0L-System Notation • 0 stands for context free • D stands for deterministic • V is the alphabet • V* is the set of all words over V • V+ is V* excluding the empty word •  is the word in V+ that is the seed of the system, called the axiom • P is the set of production rules, explained later • 0L-System is defined as G=<V,,P> Jason Stredwick, MSU 2004

  5. D0L-System Product Rules • Production rule is defined as P  V x V* • Alternate notation (, ) or    •  is called the predecessor •  is called the successor •  can only be a single character •  can be any word, which includes letters • If no production rule is given for a predecessor, it is assumed to be    Jason Stredwick, MSU 2004

  6. Determinism • A system is considered deterministic if and only if for each predecessor, , there exists exactly one production rule, or more simply each letter in the alphabet can map to exactly one word in V*. Jason Stredwick, MSU 2004

  7. Example: D0L-System String Depth b 0 a 1 ab 2 aba 3 abaab 4 abaababa 5 Alphabet: {a,b} Words: {a,b,ab} Production Rules: a  ab b  a (seed): b Recursion Depth 5 Jason Stredwick, MSU 2004

  8. LOGO Language • Used to graphically model plants • F move forward while drawing a line • f move forward without drawing a line • + rotate left • - rotate right • 3-D instructions are also available, using Euler rotations Jason Stredwick, MSU 2004

  9. n = 4 • = 90 • = -F P: F  F+F-F-F+F • n = 2 • = 90 • = F-F-F-F P: F  F+FF-FF-F-F+F+F Jason Stredwick, MSU 2004

  10. n = 2 • = 90 • = F+F+F+F P: F  F+f-FF+F+FF+Ff+FF-f+FF-F-FF-Ff-FFF f  ffffff Jason Stredwick, MSU 2004

  11. Context Sensitive L-System • Non-deterministic • Types: 1, 2, and I, written 2L-System • Uses neighbor information as a condition to use a production rule • > Right of predecessor • < Left of predecessor • Example: 1.  > F   Type 1 2. + <  > F   Type 2 • ++F <  > F+F   Type I Jason Stredwick, MSU 2004

  12. Context Sensitive ExampleSignal Propagation baaaaa abaaaa aabaaa aaabaa aaaaba aaaaab aaaaaa : baaaaa p1: b < a  b p2: b  a Jason Stredwick, MSU 2004

  13. Stochastic L-System • Non-deterministic • Each predecessor,  • Multiple production rules • Each rule has a probability of use • Probabilities must sum to one • Example:   F   FF   FFF 0.1 0.2 0.7 Jason Stredwick, MSU 2004

  14. Bracketed L-System • [ push current state onto a stack • ] pop current state onto a stack • Used to create tree like structures Jason Stredwick, MSU 2004

  15. n = 5 • = 20 • = F P: F  F[+F]F [-F][F] • n = 5 • = 25.7 • = F P: F  F[+F]F [-F]F • n = 4 • = 22.5 • = F P: F  FF-[-F+F+F]+ [+F-F-F] Jason Stredwick, MSU 2004

  16. n = 5 • = 22.5 • = X P: F  FF X  F-[[X]+X]+F [+FX]-X • n = 7 • = 25.7 • = X P: F  FF X  F[+X][-X]FX • n = 7 • = 20 • = X P: F  FF X  F[+X]F[-X]+X Jason Stredwick, MSU 2004

  17. Jason Stredwick, MSU 2004

  18. Parameterized L-System • V is the alphabet •  is the set of formal parameters, , real numbers •  is the axiom, seed, (V x *)+ • E() represent math operations • C() represent logic operations, conditional •  is the predecessor, (V x *) •  is the successor, (V x E())* • P   x C() x  • G = <V, , , P> defines a parametric 0L-System Jason Stredwick, MSU 2004

  19. Parameterized 0L-System Example : B(2)A(4,4) p1: A(x,y) : y<=3  A(x*2, x+y) p2: A(x,y) : y>3  B(x)A(x/y, 0) p3: B(x) : x<1  C p4: B(x) : x>=1  B(x-1) B(2)A(4,4) B(1)B(4)A(1,0) B(0)B(3)A(2,1) CB(2)A(4,3) CB(1)A(8,7) CB(0)B(8)A(1,0) Jason Stredwick, MSU 2004

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  21. Jason Stredwick, MSU 2004

  22. Hornby/Pollack • Multiple applications • Considers there system generic by simply replacing the alphabet. • Uses Parameterized Bracketed D0L-System • Constrain conditions to compare with a constant • Additional feature: {…}(n) block repeat Jason Stredwick, MSU 2004

  23. Table Language Forward(n) Backward(n) Up(n) Down(n) Left(n) Right(n) Clockwise(n) Counter-clockwise(n) [ ] { block }(n) [{[forward(6)]left(1)}(4)] up(1)forward(3)down(1) [{[forward(4.5)]left(1)}(4)] Jason Stredwick, MSU 2004

  24. Neural Network Language decrease-weight(n) duplicate(n) increase-weight(n) loop(n) merge(n) next(n) output(n) parent(n) reverse(n) set-function(n) Jason Stredwick, MSU 2004

  25. Mutation • Original: • P0(n0, n1) : n0 > 5.0  { a(1.0)b(2.0) }(n1) • n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • Mutate condition: • P0(n0, n1) : n0 > 7.0 { a(1.0)b(2.0) }(n1) • n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • Mutate arguement: • P0(n0, n1) : n0 > 5.0  { a(1.0)b(2.0) }(n1) • n0 > 2.0  [ P1(n1 - 2.0, n0/2.0) ] Jason Stredwick, MSU 2004

  26. Mutation • Original: • P0(n0, n1) : n0 > 5.0  { a(1.0)b(2.0) }(n1) • n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • Mutate symbol: • P0(n0, n1) : n0 > 7.0  { c(1.0)b(2.0) }(n1) • n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • Mutate random symbol (removed b(2.0)): • P0(n0, n1) : n0 > 5.0  { a(1.0) }(n1) • n0 > 2.0  [ P1(n1 - 2.0, n0/2.0) ] Jason Stredwick, MSU 2004

  27. Mutation • Original: • P0(n0, n1) : n0 > 5.0  { a(1.0)b(2.0) }(n1) • n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • Insert random symbol: • P0(n0, n1) : n0 > 7.0  { a(1.0)b(2.0) }(n1)c(3.0) • n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • Encapsulate a block of characters: • P0(n0, n1) : n0 > 5.0  { P2(n0, n1) }(n1) • n0 > 2.0  [ P1(n1 - 2.0, n0/2.0) ] • P2(n0, n1) : n0 > 5.0  a(1.0)b(2.0) • n0 > 2.0  a(1.0)b(2.0) Jason Stredwick, MSU 2004

  28. Recombination • Original: • P3(n0, n1) : n0 > 5.0  { a(1.0)b(2.0) }(n1) n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • P3(n0, n1) : n0 > 3.0  b(3.0)a(2.0) n0 > 1.0  P1(n1 - 1.0, n0 - 2.0) • Replace condition-successor pair • P3(n0, n1) : n1 > 3.0  b(3.0)a(2.0) n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] Jason Stredwick, MSU 2004

  29. Recombination • Original: • P3(n0, n1) : n0 > 5.0  { a(1.0)b(2.0) }(n1) n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • P3(n0, n1) : n0 > 3.0  b(3.0)a(2.0) n0 > 1.0  P1(n1 - 1.0, n0 - 2.0) • Replace successor • P3(n0, n1) : n1 > 3.0  b(3.0)a(2.0) n0 > 2.0  P1(n1 - 1.0, n1 - 2.0) Jason Stredwick, MSU 2004

  30. Recombination • Original: • P3(n0, n1) : n0 > 5.0  { a(1.0)b(2.0) }(n1) n0 > 2.0  [ P1(n1 - 1.0, n0 / 2.0) ] • P3(n0, n1) : n0 > 3.0  b(3.0)a(2.0) n0 > 1.0  P1(n1 - 1.0, n0 - 2.0) • Replace one block with another • P3(n0, n1) : n1 > 3.0  b(3.0)a(2.0) n0 > 2.0  [ b(3.0)a(2.0) ] Jason Stredwick, MSU 2004

  31. References Algorithmicbotany.com Prusinkiewicz, P and Lindenmayer, A. The Algorithm Beauty of Plants, 1996 Hornby, Gregory S. and Pollack, Jordan B. (2001). The advantages of Generative Grammatical Encodings for Physical Design Hornby, Gregory S. and Pollack, Jordan B. (2001). Body-Brain Co-evolution Using L-systems as a Generative Encoding Hornby, Gregory S., Lipson, Hod, and Pollack, Jordan B. (2001). Evolution of Generative Design Systems for Modular Physical Robots Hornby, Gregory S. and Pollack, Jordan B. (2001). Evolving L-Systems to Generate Virtual Creatures Jason Stredwick, MSU 2004

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