1. Lindenmayer Systems 

2. Lindenmayer systems There is no distinction between terminal and non-terminal symbols. All the words derived from a given word are in the language. All letters of a word are rewritten simultaneously. Motivation - the description of the development of filamentous organisms - words represent stages of development of disposition of cells - productions correspond to the instructions for generating organisms and are applied simultaneously to letters as developmenof cells is simultaneous in organisms -there are no terminals because they would correspond to dead cells; disappearance of a cell is represented by the empty word. The various parts of a developing organism might communicate among themselves. The basic system model assumes that there is no communication. 

3. OL systems(1) An OL system is OLS = (V, P 0, F), where V is an alphabet, P 0 is a non-empty word on V (axiom or initial word), F is a set of ordered pairs (a,P) with a  V, P  (V +  { } such that for every a  V there is at least a word P in (a, P)  F (we write a  P and call it a production). The relation  OLS is defined as follows: P  Q holds if and only if P = a 1 … a n, Q = Q 1 … Q n and a i  Q i, n  i  1 The generated language is defined as L (OLS) = {P | P 0  *  P} An OL system is deterministic (DOL) if and only if for every a  V there is exactly one word P such that (a,P)  F. An OL system is -free or propagating (POL) if and only if for every production a  P in F, P is not. Two systems are equivalent if they generate the same language. 

4. OL systems (2) Examples. 1)OLS = ({a}, a, {a  a 2 }) L(OLS) = {a 2n | n  0} 2) OLS = ({a, b}, a, {a  b, b  ab}) L(OLS) = {a, b, ab, bab, abbab, bababbab, …} (word lenghts are the Fibonacci’s numbers) 3)OLS = ({a, b, c}, a, {a  abcc, b  bcc, c  c}) L(OLS) = {a, abcc, abccbcccc, abccbccccbcccccc, …} (word lenghts are the squares of natural numbers) Note. All the systems are DOL systems and the languages are context dependent. 

5. OL systems (3) Example (red alga). 1)Consider the propagating DOL system with alphabet and productions as in the table 1 23 4567( )  2  3 2 2  4 504678(1)( )  The words derived from the axiom P 0 = 1 are P 0 = 1 P 1 = 2  3 P 2 = 2  2  4 P 3 = 2  2  504 P 4 = 2  2  60504 P 5 = 2  2  7060504 P 6 = 2  2  8(1)07060504 P n+6 = 2  2  8(P n )08(P n-1 ) 0 … 08 P 0 07060504 

6. OL systems (4) Representation Parenthesized expressions are branches whose position is indicated by the 8’s. The 0’s are represented as oblique walls alternatively bending to the left and to the right. Branches are attached alternatively on one and on the other side of the branch where they are born. The symbols  are represented as vertical walls.  2 28         2 2 4  3 

7. Properties of OL languages (1) We call anti-AFL a family of languages which is not closed under union, -free concatenation, -free omomorphism, inverse omomorphism, and intersection with regular languages. Theorem. The family of OL languages is anti-AFL. Teorema. The family of OL languages is closed under specular image. Proof. If a language is generated by the system (V, P 0, F), its specular image is generated by the system (V, mi(P 0 ), mi(F)) where mi(F) is obtained from F by replacing right parts of productions with their specular images. 

8. Properties of OL languages (2) Theorem. Every OL language is context dependent. Proof. Let L be generated by the OL system H = (V, P 0, F). Let us take the grammar G = ({X 0, X 1, X 2, X 3 }, V, X 0, F 1 ) where F 1 consists of the productions X 0  X 1 P 0 X 1 X 1  X 1 a  X 1 X 2 a for every a  V X 2 X 1  X 3 X 1 a X 3  X 3 a for every a  V X 1 X 3  X 1 X 2 a  P X 2 for every production a  P in F. It can be proved that L(G) = L(H). 

9. Properties of OL languages (3) Example. Let us take the system OLS = ({a,b}, a, {a  b, b  ab}. The context dependent grammar generating the same language is G = ({X 0, X 1, X 2, X 3 }, {a,b}, X 0, F 1 ) where F 1 consists of the productions X 0  X 1 a X 1 X 1  X 1 a  X 1 X 2 a X 1 b  X 1 X 2 b X 2 a  b X 2 X 2 b  ab X 2 X 2 X 1  X 3 X 1 a X 3  X 3 a b X 3  X 3 b X 1 X 3  X 1 

10. TOL systems (1) A TOL system is H = (V, P 0, T) where V and P 0 are as in the OL systems and T is a collection of subsets of V  (V +  { }. Every t  T satisfies the condition: for every a  V there is P  (V +  { }) such that t contains the pair (a, P). P  Q if and only if there esists k  1, letters a 1, …, a k, words Q 1, …, Q k and t contains the pair (a i, Q i ) per k  i  1. The generated language is L (H) = {P | P 0  *  P} Motivation. At every step only productions are used which belong to the same table. The biological motivation is that at different stages of development of an organism sets of different rules may be needed. 

11. TOL systems (2) Example. Let us take the system H = ({a}, a, {{a  a 2 }, {a  a 3 }}). We have L(H) = {a i | i = 2 m 3 n, m,n  0. L(H) is not an OL language. Theorem. OL languages are properly contained in TOL languages. 

12. 1L e 2L systems (1) A 2L system is H = (V, P 0, a 0, F), where: -V and P 0 are as in OL systems -a 0  V is the input from the environment -F is a subset of V  V  V  (V +  { } such that for all the (non necessarily distinct) letters a, b, c  V there is a word P  V +  { } such that F contains the quadruple (a,b,c,P). P  Q if and only if there exists n  1, letters a 1, …, a n, words Q 1, …, Q n such that F contains the quadruples (a 0, a 1, a 2, Q 1 ), …, (a i-1, a i, a i+1, Q i ), …, (a n-1, a n, a 0, Q n ) per n-1  i  2 P = a 1 …a n and Q = Q 1 …Q n. For n= 1 F contains the quadruple (a 0, a 1, a 0, Q 1 ). The language generated by H is L(H) = {P | P 0  *  P}. 

13. 1L and 2L systems (2) A 2Lsystem is a 1L system if and only if one of the following conditions holds: -for every a, b, c, P F contains the quadruple (a,b,c,P) if and only if it contains the quadruple (a,b,d,P) or -for every a, b, c, P F contains the quadruple (a,b,c,P) if and only if it contains the quadruple (d,b,c,P). Theorem. There is a 1L language which is non an Ol language and there is a 22L language which is non a 1L language. 

14. References  Lindenmayer A., Mathematical Models for Cellular Interaction in Development, Journal of Theoretical Biology 18(1968), 280-289. Rozenberg G., Salomaa A., The Mathematical Theory of L Systems, Academic Press, New York, 1980. Prusinkiewicz P., Lindenmayer A., The Algorithmic Beauty of Plants. Springer Verlag, New York, 1990.