Download presentation
Presentation is loading. Please wait.
Published byDoreen Lawrence Modified over 9 years ago
1
Lindenmayer systems Martijn van den Heuvel May 26th, May 26th, 2011
2
Outline Origin Origin Definition Definition Characteristics Characteristics Extensions Extensions Comparision with other models Comparision with other models Conclusion Conclusion
3
Origin - Aristid Lindenmayer 1925 – 1989 (63 years old) 1925 – 1989 (63 years old) Hungarian/US Hungarian/US Biologist at Utrecht University Biologist at Utrecht University L-systems (1968) L-systems (1968) Biological development of multicellular organisms (algae) Biological development of multicellular organisms (algae) Chomsky (formal) grammars (1957) Chomsky (formal) grammars (1957)
4
Definition Rewriting system: Rewriting system: Define complex objects by rewriting simple objects Define complex objects by rewriting simple objects Tuple: G = (A,s,R) Tuple: G = (A,s,R) G = grammar A = alphabet (symbol set/states) s = starting axiom (seed) R= set of production/transition rules ‘Constants’ are identity functions; a a
5
Characteristics Recursive Recursive Parallel applying of productions Parallel applying of productions Rules can be deterministic/non-deterministic Rules can be deterministic/non-deterministic Context-free / context sensitive states Context-free / context sensitive states Namecontext -system Rules 0L-systemfree -L-systema z IL-systemsensitive -L-system z 2L-systemsensitive -L-system z
6
Example D0L-system States: { a,b,c,d,k } Rules: a cbc b dad c k d a k k Tuple: G = (A,s,R) G = grammar A = alphabet (symbol set/states) s = starting axiom (seed) R= set of production/transition rules acbckdadkkacbcakkcbckdadkcbckkkdadkkacbcakkdadkkkkacbcakkcbckdadkcbckkacbcakk… Repeating structure from 4th array: S n = kS n-3 S n-2 S n-3 k
7
Example D0L-system Geometrical interpretation: Geometrical interpretation: a and b are sharp tips a and b are sharp tips c and d are lateral margins of lobes c and d are lateral margins of lobes k represents a notch k represents a notch
8
Geometrical extensions: Turtle graphics Turtle has: A position (coordinates; x,y) A position (coordinates; x,y) An orientation (angle; a) An orientation (angle; a) A state s=(x,y,a) A state s=(x,y,a) Also (possible): Angle increment b Angle increment b Step size d Step size d
9
Geometrical extensions: Turtle graphics Example: Sierpinski triangle Example: Sierpinski triangle variables: A Bboth mean “draw forward” variables: A Bboth mean “draw forward” constants: + means “turn left by angle” constants: + means “turn left by angle” −means “turn right by angle” −means “turn right by angle” start: A start: A rules: (A → B−A−B), (B → A+B+A) rules: (A → B−A−B), (B → A+B+A) angle: 60° angle: 60° Fixed step size (d), no angle increment (b) Fixed step size (d), no angle increment (b)
10
Geometrical extensions: Turtle graphics Result for n=2, n=4, n=6 and n=8
11
Geometrical extensions: Bracketing Extends previous turtle extension to split Extends previous turtle extension to split Uses brackets [,] to delimit branches: Uses brackets [,] to delimit branches: [ means“Pop a state from the stack and make it the current state of the turtle.” [ means“Pop a state from the stack and make it the current state of the turtle.” ] means“Push the current state of the turtle onto a pushdown stack.” ] means“Push the current state of the turtle onto a pushdown stack.”
12
Geometrical extensions: Bracketing Start: F Start: F Rule: F F[-F]F[+F][F] Angle: 25°
13
Comparision to other MOC Very similar to semi-Thue & Markov systems Very similar to semi-Thue & Markov systems L-systems parallel/simultaneous, others sequential. L-systems parallel/simultaneous, others sequential. L-systems can be both deterministic/non-deterministic L-systems can be both deterministic/non-deterministic All are Turing-complete All are Turing-complete Cellular automata Cellular automata Also simultaneous application of rules Also simultaneous application of rules Ozhigov proves: Ozhigov proves: The set of languages, computed in a polynomial time on L- systems, is exactly NP- languages. Even states L-systems might be a more powerful MOC than a non-deterministic Turing machine
14
Conclusions Powerful systems Powerful systems Distinguishing by parallel appliance of rules Distinguishing by parallel appliance of rules As formal language: As formal language: 0L-languages are context free type-2 in Chomsky hierarchy 0L-languages are context free type-2 in Chomsky hierarchy IL-languages are context sensitive type-1 Chomsky hierarchy IL-languages are context sensitive type-1 Chomsky hierarchy Interesting readings: Interesting readings: The algorithmic beauty of plants – Lindenmayer & Prusinkiewicz (1990) The algorithmic beauty of plants – Lindenmayer & Prusinkiewicz (1990) The algorithmic beauty of plants – Lindenmayer & Prusinkiewicz (1990) The algorithmic beauty of plants – Lindenmayer & Prusinkiewicz (1990) Lindenmayer systems as a model of computations – Y. Ozhigov (1998) Parametric L-systems and their application to the modelling and visualization of plants – J.S. Hanan (1992) http://www.kevs3d.co.uk/dev/lsystems/# (online L-system renderer) http://www.kevs3d.co.uk/dev/lsystems/#
Similar presentations
© 2024 SlidePlayer.com. Inc.
All rights reserved.