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L-System Aristid Lindenmayer (1968), biologist

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L-System A method of constructing a FRACTAL that is also a MODEL for plant growth. The Computational Beauty of Nature – Gary William Flake

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Fractal graphics

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Plant model

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L-System L-systems use an AXIOM as a starting STRING and ITERATIVELY apply a set of PARALLEL string substitution rules to yield one long string that can be used as instructions for drawing the fractal.

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Production systems String : concatenation of alphabets (symbols). Axiom : the first initial symbol(s). Production rules : the description of how one string of symbols generate another.

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An example Alphabets: a, b Axiom: b Rules: a -> ab b -> a

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Test the generation StepResult 0b 1a 2ab 3aba 4abaab 5abaababa 6abaababaabaab

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Another example Alphabets: a, b Axiom: b Rules: a -> ba b -> ab

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Test the generation StepResult 0b 1ab 2baab 3abbabaab 4baababbaabbabaab

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Terminal and non-terminal symbols We can use lower and UPPER case alphabets to represent the terminal and non-terminal symbols. Terminal symbols are those need no further expansion. Non-terminal symbols are those can further expand using the production rules.

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Terminal and non-terminal symbols Rules: A -> b A -> bA In this case, A is a non-terminal symbol; b is a terminal symbol. The rules state that A can generate b and stop. A can also expand to bA where the A on the right hand side can expand again using the two rules.

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Terminal and non-terminal symbols Rules: A -> b (terminal) A -> bA (non-terminal) The results will be, b, bb, bbb, bbbb, … Remember the regular expression last time, bb*

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Exercise time Alphabets: A, a, b Axiom: A Rules: A -> ab (terminal) A -> aAb (non-terminal) Let’s see what can be generated.

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Answer StepResult 0AA 1abaAb 2aabbaaAbb 3aaabbbaaaAbbb 4aaaabbbbaaaaAbbbb 5aaaaabbbbbaaaaaAbbbbb

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Language Remember that the Finite State Automata can generate a language we called Regular Expression. The L-system can also generate another language we named Context-Free language.

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Language According to Noam Chomsky, there are 4 types/hierarchies of Formal Language. We have learnt the, Regular expression, Context-free language. There are two more, namely the, Context-sensitive language, Unrestricted language.

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Production rules Regular expression uses the rules, A -> b or A -> bC; While Context-Free language uses the rules like, A -> α; where α is any string of both terminal and non- terminal symbols.

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Simple exercise Alphabets: F, -, + Axiom: F Rules: F -> F + F - - F + F We have only one production rule with the symbol F. Let’s see what happens.

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Any meaning? We have not touched on any ideas about meaning – semantics yet. The language we generated is pure syntactic. We can, however, assign external meaning to the symbols and see what happens.

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Any meaning?

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The meaning Now we assign meaning to the symbols. F: draw a line segment forward +: turn anti-clockwise 60° -: turn clockwise 60 ° The rule F -> F + F - - F + F will become:

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The meaning F F F F + -- +

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Turtle graphics That is the Turtle Graphics (Logo) we learnt in primary school. The commands are, CommandTurtle action FDraw forward by a fixed length. fMove forward by a fixed length. +Turn anti-clockwise by a fixed angle. -Turn clockwise by a fixed angle. [Save the current location and orientation. ]Restore the last saved location and orientation.

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L-system applet Try to create the graphics using the applet at http://www.cs.utah.edu/~jshepher/java/Lsystems/Lsystems.html

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Exercise time Can you figure out the rule for this plant?

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9.7: Chomsky Hierarchy.

9.7: Chomsky Hierarchy.

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