XL as an extension of the L-system formalism

Tutorial "Modelling Plants with GroIMP", Cottbus, March 11-12, 2008

Winfried Kurth Ole Kniemeyer

Reinhard Hemmerling Gerhard Buck-Sorlin

Chair for Graphics Systems, BTU Cottbus

Roots of the language XL:

- object-oriented programming (Java)

- imperative programming (Java)

- rule-based programming (L-systems, graph grammars)

- chemical programming

Roots of the language XL:

- object-oriented programming (Java)

- imperative programming (Java)

- rule-based programming (L-systems, graph grammars)

- chemical programming

Rule-based programming(van Wijngaarden, Lindenmayer)

Computer = transformation machine for structures (or for states).

There exists always a current structure, which is transformed as long as possible.

Working process: Search and application.matching: search for a rule which is applicable to the current structure,rewriting: application of the rule in order to rewrite the current structure.

programme = a set of transformation rules.

finding a programme: specification of the rules.

programming languages: Prolog, L-system languages, Intran

Lindenmayer-Systems (L-Systems)• named after Aristid Lindenmayer (Dutch-Hungarian biologist, 1925-1989)• Data structure and language inseparably connected to each other• Data structure is a sequence of (parameterized) symbols, e.g.: F(1) [ + A(1) ] [ - A(1) ]• Every L-system consists of:

- an alphabet (a set of symbols)- a start word (axiom) (in XL: Axiom)- a set of replacement rules (applied in parallel to the developing string), e.g.: A(x) ==> F(x/2) [ + A(x/2) ] [ - A(x/2) ]

part of the alphabet:

Turtle commands (imperative programming style)

F0 "Forward", including construction of a cylinder, uses the current step width for length

(the zero stands for "no explicit specification of length")

M0 forward without construction (Move command)

L(x) change the current step width (length) to x

LAdd(x) increment the current step width by x

LMul(x) multiply the current step width by x

D(x), DAdd(x), DMul(x) analogously for current thickness

RU(45) Rotation of the turtle around "up" axis by 45°

RL(...), RH(...) analogously around "left" and "head" axis

up-, left- and head axis form an orthogonal, spatial coordinate system, which is always associated with the turtle

RV(x) Rotation "downwards" with a strength specified by x

example:L(100) D(3) RU(-90) F(50) RU(90) M0 RU(90) D(10) F0 F0

D(3) RU(90) F0 F0 RU(90) F(150) RU(90) F(140) RU(90)M(30) F(30) M(30) F(30) RU(120) M0 Sphere(15)

generates

what is the result of the interpretation of the string

L(10) F0 RU(45) F0 RU(45) LMul(0.5) F0 M0 F0 ?

Repetition of parts of the command string with the keyword "for"

e.g. for ((1:3)) ( A B C )

yields A B C A B C A B C

what is the result of the interpretation of

L(10) for ((1:6)) ( F0 RU(90) LMul(0.8) ) ?

Branching: realization with stack commands

[ put current turtle state on stack ("push")

] take current state from stack ("pop") and let this state be the current turtle state (this means: end of the branch, continue with main stem)

Example of interpreted L-system:

rules

A ==> F0 [ RU(45) B ] A ;

B ==> F0 B ;

start word (axiom) L(10) A

(A and B are normally not interpreted geometrically.)

what a structure is the result of the following L-system

A ==> [ LMul(0.25) RU(-45) F0 ] F0 B;

B ==> [ LMul(0.25) RU(45) F0 ] F0 A;

with start word L(10) A ?

what a structure is the result of the following L-system

A ==> [ LMul(0.25) RU(-45) F0 ] F0 B;

B ==> [ LMul(0.25) RU(45) F0 ] F0 A;

with start word L(10) A ?

equivalent rule:

A ==> [ LMul(0.25) RU(-45) F0 ] F0 RH(180) A;

a space-filling curve:

module R extends RU(-45); /* inheritance */

module A extends F(10);

Axiom ==> L(100) R X R A R X;

X ==> X F0 X R A R X F0 X;

Branching, alternating phyllotaxy and shortening of branches:

Axiom ==> L(10) F0 A ;

A ==> LMul(0.5) [ RU(90) F0 ] F0 RH(180) A ;

which structure is given by

Axiom ==> F(10) A ;

A ==> [ RU(-60) F(6) RH(180) A Sphere(3) ] [ RU(40) F(10) RH(180) A Sphere(3) ];

Sphere ==> Z; ?

(F(n) yields line of the given length n,Sphere(n) a sphere with radius n)

Stochastic L-systemsusing pseudo-random numbersexample:deterministic stochasticfloat c = 0.7;

Axiom ==> L(100) D(5) A;

A ==> F0 LMul(c) DMul(c) [ RU(50) A ] [ RU(-10) A ];

float c = 0.7;

Axiom ==> L(100) D(5) A;

A ==> F0 LMul(c) DMul(c) if (probability(0.5)) ( [ RU(50) A ] [ RU(-10) A ] ) else ( [ RU(-50) A ] [ RU(10) A ] );

Generating a random distribution on a plane:Axiom ==> D(0.5) for ((1:300))

( [ Translate(random(0, 100), random(0, 100), 0)

F(random(5, 30)) ] );

view from above oblique view

Which structure is generated by the following L-system?

Axiom ==> [ RU(90) M(1) RU(90) A(1) ] A(1);

A(n) ==> F(n) RU(90) A(n+1);

variant: replace "RU(90)" in the second rule by "RU(92)"

context sensitivity

Query for a context which must be present in order for a rule to be applicable

Specification of the context in (* .... *)

example:module A(int age);module B(super.length, super.color) extends F(length, 3, color);Axiom ==> A(0);A(t), (t < 5) ==> B(10, 2) A(t+1);A(t), (t == 5) ==> B(10, 4);B(s, 2) (* B(r, 4) *) ==> B(s, 4);B(s, 4) ==> B(s, 3) [ RH(random(0, 360)) RU(30) F(30, 1, 14) ];

Transition to graph grammars:

Rewriting of graphs instead of strings

(strings remain a special case)

Relational growth grammars (RGG): parallel graph grammars with a special graph model

(see Ole Kniemeyer's dissertation for mathematical foundations)

"relational": different edge types of the graph can stand for different relations

an RGG rule and its application in graphical form:

rule:

application:

critical point here:

the connection of the inserted right-hand side with the rest of the graph

("embedding")

XL offers 3 standard types of rules with different solutions for embedding:

L ==> R L-system rules: embedding made compatible with L-systemsL ==>> R SPO rules (single pushout rules, from algebraic graph grammar theory): after insertion, "dangling edges" are removedL ::> C execution rules: no change of the graph topology at all, only parameters are changed (C: command sequence, imperative)

Extended L-Systems (XL)Example for execution rulesgiven: a graph with cycles Spreading rule for a

signal in the network:

XL representation:(* Cell(1) *) c:Cell(0) ::> c[state] := 1;

Extended L-Systems (XL)spreading of signal

output:

1 2 3

Extended L-Systems (XL)The Game of Life

XL code:static Cell* context(Cell c1){

yield (* c2:Cell, ((c2 != c1) && (c2.distanceLinf(c1) < 1.1)) *); }

public void transition()[

x:Cell(1), (!(sum(context(x)[state]) in (2 : 3))) ::> x[state] := 0;

x:Cell(0), (sum(context(x)[state]) == 3) ::> x[state] := 1;]

Extended L-Systems (XL)The Game of Life

Graphical output(a period-8 oscillator):

• global sensitivity, graph queries

Example: growth takes place only when there is enough distance to the next object in space

module A(int s);Axiom ==> F(100) [ RU(-30) A(70) ] RU(30) A(100);a:A(s) ==> if ( forall(distance(a, (* F *)) > 60) ) ( RH(180) F(s) [ RU(-30) A(70) ] RU(30) A(100) )

without the condition with the condition

Interpretive rules

Insertion of an extra rule application immediately before graphical interpretation takes place (without effect on the next generation)

application of interpretive rules

Turtle interpretation

public void run(){ [ Axiom ==> A; A ==> Scale(0.3333) for (i:(-1:1)) for (j:(-1:1)) if ((i+1)*(j+1) != 1) ( [ Translate(i, j, 0) A ] ); ] applyInterpretation();}

public void interpret() [ A ==> Box; ]

Example:

public void run(){ [ Axiom ==> A; A ==> Scale(0.3333) for (i:(-1:1)) for (j:(-1:1)) if ((i+1)*(j+1) != 1) ( [ Translate(i, j, 0) A ] ); ] applyInterpretation();}

public void interpret() [ A ==> Box; ]

(a)

(b) (c)A ==> Sphere(0.5); A ==> Box(0.1, 0.5, 0.1) Translate(0.1, 0.25, 0) Sphere(0.2);

what will be generated by this example?

public void run(){ [ Axiom ==> [ A(0, 0.5) D(0.7) F(60) ] A(0, 6) F(100); A(t, speed) ==> A(t+1, speed); ] applyInterpretation();}public void interpret() [ A(t, speed) ==> RU(speed*t); ]

Extended L-Systems (XL)simple example: railway simulation

Modules: Train Track Switch

Excerpt of the data structure: Train standingbefore a switch

Extended L-Systems (XL)simple example: railway simulation

Rule 1

Rule 2

r1:Rail[t:Train] r2:Rail ==>> r1 r2[t];

r:Rail[t:Train] s:Switch[r2:Rail] r1:Rail ==>> r s[r1[t]] r2;

Extended L-Systems (XL)Artificial Chemistry: Prime number generator

A multiset of random integers reacts according to the following rule, thereby distilling prime numbers.

int

Rule:

a b

int int

a b/a

inta divides b

textual output:20 65 68 46 74 53* 22 35 42 21Concentration: 1 / 1020 65 68 46 74 53* 22 35 21 2*Concentration: 2 / 1065 53* 35 21 2* 10 34 23* 37* 11*Concentration: 5 / 1065 53* 35 21 2* 23* 37* 11* 5* 17*Concentration: 7 / 1053* 21 2* 23* 37* 11* 5* 17* 13* 7*Concentration: 9 / 1053* 2* 23* 37* 11* 5* 17* 13* 7* 3*Concentration: 10 / 10

b:int, (* a:int *), (a != b && (b % a) == 0) ==> `b / a`;

Extended L-Systems (XL)A herbivore modelModel of a tree population being grazed upon by phytophagous animals.

• Plants with two parameters: age t and radius r, represented by circles of radius r (proportional to energy budget of plant).• Plants reproduce through seed

• at a certain age and • above a certain energy (radius) threshold (number of offspring = f(r)).

• Plant dies whena) it has reached maximum age or b) when its energy budget is exhausted (r<0).

• If plant does not die it grows by fixed amount at each time step• Animals with two parameters: age t and energy budget e, represented by small circles.• While not in contact with a plant (i. e. within its radius): animal moves and consumes fixed amount of its stored energy. • When in contact with a plant: animal’s radius of movement decreased, starts "grazing"

behaviour of plants

behaviour of animals

/* phytophag.rgg: specification of a grazing and competition model with circular-shaped plants and animals */ module Plant(int t, super.radius) extends Cylinder(1, radius) {{setColor(0x00aa00);}}module Animal(int t, super.radius) extends Cylinder(2, radius) {{setColor(0xff0000); setBaseOpen(true); setTopOpen(true);}}; double pgrow = 0.9; /* regular growth increment per timestep */double seed_rad = 0.1;/* initial radius of a plant */int pmaxage = 30; /* maximal age of a plant */int pgenage1 = 10; /* first reproductive age level */int pgenage2 = 18; /* second reproductive age level */double distmin = 15; /* minimal seed distance */double distmax = 40; /* maximal seed distance */double pminrad = 9; /* necessary plant radius for reproduction */double pgenfac = 0.5; /* ratio #seeds/radius */ int lag = 15; /* sleeping time for animal at start */double shortstep = 0.4;/* movement of animals inside plant canopy */double longstep = 15; /* movement of animals outside */double f_e = 0.2; /* ratio radius / energy of animals */double init_e = 4; /* initial energy amount of animals */double respi = 0.25; /* energy cosumed by animals' respiration */double thr = 7.6; /* energy threshold for reproduction of animals */double eat = 1.1; /* energy transferred during grazing */ 

protected void init(){extent().setDerivationMode(PARALLEL_MODE | EXCLUDE_DELETED_FLAG); [ Axiom ==> Plant(0, seed_rad) [ RH(random(0, 360)) RU(90) M(10) RU(-90) Animal(-lag, f_e*init_e) ]; ]}public void make() { growAnimals(); derive(); growPlants(); } public void growAnimals() [ Animal(t, e), (t < 0) ==> Animal(t+1, e); /* start lag */ Animal(t, e), (e <= 0) ==> ; Animal(t, e), (e > f_e*thr) ==> [ RH(random(0, 360)) RU(90) M(shortstep) RU(-90) Animal(0, e/2 - f_e*respi) ] RH(random(0, 360)) RU(90) M(shortstep) RU(-90) Animal(0, e/2 - f_e*respi); a:Animal(t, e), (* p:Plant(u, r) *), (distance(a, p) < p[radius]) ==> RH(random(0, 360)) RU(90) M(shortstep) RU(-90) Animal(t+1, e + f_e*eat - f_e*respi) { p[radius] :-= eat; }; Animal(t, e) ==> RH(random(0, 360)) RU(90) M(longstep) RU(-90) Animal(t+1, e - f_e*respi); ]

public void growPlants() [ Plant(t, r), (t > pmaxage) ==> ; Plant(t, r), (r < 0) ==> ; p:Plant, (* q:Plant *), (distance(p, q) < q[radius] && p[radius] <= q[radius]) ==> ; Plant(t, r), ((t == pgenage1 || t == pgenage2) && r >= pminrad) ==> for ((1 : (int) (pgenfac*r))) ( [ RH(random(0, 360)) RU(90) M(random(distmin, distmax)) RU(-90) Plant(0, seed_rad) ] ) Plant(t+1, r); Plant(t, r) ==> Plant(t+1, r+pgrow); ]

Extended L-Systems (XL)Herbivore model: output

1219

55 83