A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. J. Theoretical Biology 30, 455-484, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....patterns described by the recursive encodings could be scaled up to cope with larger instances. More recent work that uses recursive descriptions of networks [Kitano 1990] describes a context free production grammar that can be used to generate connection matrices. This is a modified L system [Lindenmayer 1971] where each production rule gives rise to a 2X2 matrix of further production rules or terminal symbols. Rules can reference themselves or one another, thus introducing recursion. Such systems give rise to structured (at least visually) connection matrices. A GA was used to search for good sets of ....

A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. J. Theoretical Biology 30, 455-484, 1971.

....two possible ways to shrink the genotype involving different aspects of biological development. As a smaller step towards replicating biological development, we may include some form of growth in the genotype representation. An example of such an abstraction are growth rules based on L systems [10][11]. A first attempt at achieving growth based on L systems for digital design may be found in [8] Here rules fire when their conditional clause is matched to produce additions and or changes to the growing phenotype. A problem here is that it is hard to control the size of the phenotype eventually ....

A. Lindenmayer. Developmental Systems without Cellular Interactions, their Languages and Grammars. Journal of Theoretical Biology, 1971.

....Originally, Lindenmayer described his formalism in terms of cellular automata, in which in contrast to the standard definition the cells could divide. Subsequently he observed that L systems can be formulated in a simpler and more elegant manner in terms of formal language theory [Lin71]. That theory was originally proposed by Chomsky [Cho56, Cho57] to describe the syntax of natural languages. Its fundamental notion is that of a (generative) grammar, which consists of productions or rewriting rules. In general, a production replaces a symbol by zero, one, or several new symbols. ....

A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology, 30:455--484, 1971.

....weak to generate acrotonic and mesotonic branching structures, which are often observed in nature. Their development must therefore be controlled by more involved mechanisms, overriding at least one of the assumptions (a c) above. 1 Introduction Bracketed L systems, introduced by Lindenmayer [8, 9] to model the development of branching structures, have been investigated to a lesser degree from the theoretical point of view than the L systems without brackets (c.f. 10, page 138] In contrast, most practical applications of L systems fall in the areas of modeling, simulation, and ....

....the (first order) lateral branches of #w#. It is known that the standard decomposition of a branch is unique, thus the above definition is unambiguous [6] The terminology corresponds to the standard interpretation of well nested bracketed words as string representations of branching structures [8, 9]. As the empty branch ## appears to have no biological interpretation, we assume in practice that the word w in any branch #w# is not empty. This assumption, however, is not essential to the mathematical reasoning presented in this paper. Example 1. Figure 3 shows a branching structure ....

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A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology, 30:455--484, 1971.

....weak to generate acrotonic and mesotonic branching structures, which are often observed in nature. Their development must therefore be controlled by more involved mechanisms, overriding at least one of the assumptions (a c) above. 1 Introduction Bracketed L systems, introduced by Lindenmayer [8, 9] to model the development of branching structures, have been investigated to a lesser degree from the theoretical point of view than the L systems without brackets (c.f. 10, page 138] In contrast, most practical applications of L systems fall in the areas of modeling, simulation, and ....

.... the (first order) lateral branches of [w] It is known that the standard decomposition of a branch is unique, thus the above definition is unambiguous [6] The terminology corresponds to the standard interpretation of well nested bracketed words as string representations of branching structures [8, 9]. As the empty branch [ appears to have no biological interpretation, we assume in practice that the word w in any branch [w] is not empty. This assumption, however, is not essential to the mathematical reasoning presented in this paper. Example 1. Figure 3 shows a branching structure ....

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A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology, 30:455--484, 1971.

....the branching topology of the modeled structure (b) The same production is applied in both cases, but the rules for incorporating the successor into the structure are different. to maintain the connectivity of the structure (Figure 2. 5b) The bracketed string notation introduced by Lindenmayer [67,68] inherently maintains the branching topology of the modeled structures while their component modules are rewritten. We describe it in detail in the next section. 3. Formal description of branching structures 3.1 Axial trees and bracketed strings In order to consider plantarchitecture at an ....

....has order n 1. The order of a branchis equal to the order of its lowest order axis. The terminal node of this axis is called the branch top. 8 Prusinkiewicz et al. 3. 2 The bracketed string notation To represent axial trees, Lindenmayer introduced a bracketed string notation [67,68], whichwepresent here according to [98] An axial tree with edge labels from alphabet V is represented byaword (string of symbols or letters) w over alphabet VE = V [f[# ]g: w = x 1 [ff 1 ]x 2 [ff 2 ] x n [ff n ]x n 1 : 3.1) It is assumed that the subwords x 1 #x 2 #: #x n 1 2 V do not ....

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A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology, 30:455--484, 1971.

....(a) development of a flower, b) development of a branch, and (c) cell division. 2. The modular structure of plants L systems were originally introduced to model the development of simple multicellular organisms (for example, algae) in terms of division, growth, and death of individual cells [1,13]. The range of L system applications has subsequently been extended to higher plants and complex branching structures, in particular inflorescences [14,15] described as configurations of modules in space. In the context of L systems, the term module denotes any discrete constructional unit that ....

A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology, 30:455--484, 1971.

....in combination with the N language grammar and stochastic model. Statistical di erentiation between two cell population is facilitated by the string representation of N neuron models. 7 5 Lindenmayer Systems L systems are a type of formal language rst described by Lindenmayer in [1] and [2]. Since L systems are not well known outside the procedural modeling community a quick tutorial introduction to the concepts and syntax of L systems is presented here. For further information on L systems, both as grammar and in application to the geometric modeling of branching structures, the ....

A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology, 30:455{ 484, 1971.

....are amongst the central themes of artificial life (Problem 1. 1, see Section 11) 2 The modular structure of plants L systems were originally introduced to model the development of simple multicellular organisms (for example, algae) in terms of division, growth, and death of individual cells [36, 37]. The range of L system applications has subsequently been extended to higher plants and complex branching structures, in particular inflorescences [18, 19] described as configurations of modules in space (Problem 2.1) In the context of L systems, the term module denotes any discrete ....

....these two cases is illustrated in Figure 5. The information flow taking place during the development of a branching structure can be expressed directly in the geometric domain, using a proper modification of the Koch construction (Problem 3. 4) A different approach was proposed by Lindenmayer [36, 37] and is essential to the resulting theory of L systems. The generated structures and the rewriting rules are expressed symbolically using a string notation. The geometric interpretation of these strings automatically captures proper positioning of the higher branches on the lower ones. The basic ....

A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology, 30:455--484, 1971.

.... Cohen presented a more realistic model operating in continuous space [13] Linden 1 Department of Computer Science, University of Calgary, Calgary, Alberta, Canada T2N 1N4 (mechjpwp cpsc.ucalgary.ca) mayer proposed the formalism of L systems as a general framework for plant modeling [38, 39], and Honda introduced the first computer model of tree structures [32] From these origins, plant modeling emerged as a vibrant area of interdisciplinary research, attracting the efforts of biologists, applied plant scientists, mathematicians, and computer scientists. Computer graphics, in ....

LINDENMAYER, A. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology 30 (1971), 455--484.

....1990a ] A series of attempts has been made to design network topologies based on GA [ Harp et al. 1989, Miller et al. 1989 ] However, the naive models used 1 to directly encode the network topology suffer from a lack of scalability. Kitano combined GA, the Lsystem [ Lindenmayer, 1968, Lindenmayer, 1971 ] and neural networks to enable modeling of the development process [ Kitano, 1990b ] In [ Kitano, 1990b ] GA is used to acquire graph rewriting rules (the graph Lsystem) instead of directly acquiring the network topology. The introduction of the developmental stage was not only ....

Lindenmayer, A., "Developmental Systems without Cellular interactions, their Languages and Grammars," J. theor. Biol., 30, 455-484, 1971.

....outside an m n rectangle [51] We can now summarize our results by saying LLL DFA NFA h(LLL) with each inclusion proper in d 2. There are many interesting examples of h(LLL) s; here are some of our favorites. 1. Two dimensional L systems, produced by some expansion rule. L systems [33] are languages generated by applying some dilation rule simultaneously everywhere in the string. For instance, the rules 0 01; 1 10 generate the Morse sequence 0110 1001 1001 0110 : and the rules a ab; b a generate the Fibonacci sequence abaab aba abaab : that appears in the ....

A. Lindenmayer, \Developmental systems without cellular interaction, their languages and grammars." Journal of Theoretical Biology 30 (1971) 455-484. 38

....are amongst the central themes of artificial life (Problem 1. 1, see Section 11) 2 The modular structure of plants L systems were originally introduced to model the development of simple multicellular organisms (for example, algae) in terms of division, growth, and death of individual cells [36, 37]. The range of L system applications has subsequently been extended to higher plants and complex branching structures, in particular inflorescences [18, 19] described as configurations of modules in space (Problem 2.1) In the context of L systems, the term module denotes any discrete ....

....these two cases is illustrated in Figure 5. The information flow taking place during the development of a branching structure can be expressed directly in the geometric domain, using a proper modification of the Koch construction (Problem 3. 4) A different approach was proposed by Lindenmayer [36, 37] and is essential to the resulting theory of L systems. The generated structures and the rewriting rules are expressed symbolically using a string notation. The geometric interpretation of these strings automatically captures proper positioning of the higher branches on the lower ones. ....

A. Lindenmayer. Developmental systems without cellular interaction, their languages and grammars. Journal of Theoretical Biology, 30:455--484, 1971.

....is different from the previous methods where the structure of the network is not directly encoded in the chromosome. Rather, this method uses a set of re write rules encoded in the chromosome to generate networks. It is based on Graph L system which is an extension of Lindenmayer s L system [28] [29]. Figure 2 shows the generation of a typical XOR network using Kitano s graph generation system. The followings are the rules used in developing the connectivity matrix for the XOR network. Starting from the initial state S the graph is developed by rule matching in cpaa acae aaaa aaab 0 1 2 3 ....

A. Lindenmayer. Developmental systems without cellular interactions, their language and grammars. J. of Theoretical Biology, 30:455--484, 1971.

....Key Words: Formal languages, parallel rewriting systems, k limited systems. Category: F.4.2, F.4. 3 1 Introduction and definitions Recently, interest emerged concerning limited parallel rewriting, which is also well justified considering the original biological motivation of Lindenmayer systems [17, 18], since the idea of unlimited growth is only realistic at certain scales. Especially, realistic growth rates cannot be obtained by unrestricted Lindenmayer systems [21] We assume the reader to be familiar with some basics of formal language theory. Our notational conventions are: the empty word ....

A. Lindenmayer. Developmental systems without cellular interactions, their languages and grammars. Journal of Theoretical Biology, 30:455--484, 1971.

....that is not NFA is given in [23] We can now summarize our results by saying LLL ae DFA ae NFA ae h(LLL) with each inclusion proper in d 2. There are many interesting examples of h(LLL) s; here are some of our favorites. 1. Two dimensional L systems, produced by some expansion rule. L systems [28] are languages generated by applying some dilation rule simultaneously everywhere in the string. For instance, the rules 0 01; 1 10 generate the Morse sequence 0110 1001 1001 0110 : and the rules a ab; b a generate the Fibonacci sequence abaab aba abaab : that appears in the ....

A. Lindenmayer, "Developmental systems without cellular interaction, their languages and grammars." Journal of Theoretical Biology 30 (1971) 455-484.

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