[Paper Review] Towards a classification of Lindenmayer systems

Diego Gabriel Krivochen, Douglas Saddy|arXiv (Cornell University)|Sep 27, 2018

semigroups and automata theory12 references4 citations

TL;DR

This paper proposes a parametrization of the L-space—the infinite phase space of all possible L-system developments—by classifying Lindenmayer systems based on rule set properties and generated string types. It introduces a hierarchy analogous to the Chomsky hierarchy, showing that L-systems cluster into non-intertranslatable kinds due to inherent structural constraints, establishing a foundational classification for formal language theory in L-systems.

ABSTRACT

In this paper we will attempt to classify Lindenmayer systems based on properties of sets of rules and the kind of strings those rules generate. This classification will be referred to as a parametrization of the L-space: the L-space is the phase space in which all possible L-developments are represented. This space is infinite, because there is no halting algorithm for L-grammars; but it is also subjected to hard conditions, because there are grammars and developments which are not possible states of an L-system: a very well-known example is the space of normal grammars. Just as the space of normal grammars is parametrized into Regular, Context-Free, Context-Sensitive, and Unrestricted (with proper containment relations holding among them; see Chomsky, 1959: Theorem 1), we contend here that the L-space is a very rich landscape of grammars which cluster into kinds that are not mutually translatable.

Motivation & Objective

• To develop a systematic classification of Lindenmayer systems based on the properties of their rule sets and the nature of strings they generate.
• To define and formalize the concept of the L-space as the complete phase space of all possible L-system developments.
• To identify structural constraints that prevent certain grammars and developments from being valid states within the L-space.
• To establish a hierarchy of L-systems analogous to the Chomsky hierarchy, with proper containment relations among classes.
• To demonstrate that L-systems cluster into distinct, non-intertranslatable kinds due to fundamental formal limitations.

Proposed method

• The authors define the L-space as the infinite set of all possible L-system developments, representing all potential derivations.
• They analyze rule sets based on their generative power and structural constraints, identifying conditions that make certain developments impossible.
• The classification is modeled after the Chomsky hierarchy, with classes defined by rule format and string generation behavior.
• The approach relies on formal language theory principles, particularly those from Chomsky (1959), to establish containment and separation between classes.
• The method emphasizes undecidability and non-halting behavior in L-grammars as key reasons for the infinite, constrained nature of the L-space.
• It uses logical and formal analysis to show that some L-systems cannot be translated into others due to inherent structural incompatibilities.

Experimental results

Research questions

• RQ1 How can the infinite space of all possible L-system developments be formally parametrized?
• RQ2 What structural properties of rule sets determine the generative capacity and limitations of L-systems?
• RQ3 Are there natural classes of L-systems that are mutually non-translatable, similar to the Chomsky hierarchy?
• RQ4 What constraints prevent certain grammars or developments from being valid states in the L-space?
• RQ5 Can a hierarchy of L-systems be established with proper containment relations, analogous to the Chomsky hierarchy?

Key findings

• The L-space is infinite and undecidable due to the absence of a halting algorithm for L-grammars.
• Not all grammars or developments are valid states in the L-space, as demonstrated by the impossibility of normal grammars being embedded within it.
• L-systems cluster into distinct classes that are not mutually translatable, indicating fundamental incompatibilities in generative power.
• The proposed classification forms a hierarchy with containment relations, mirroring the Chomsky hierarchy in formal language theory.
• The classification is based on rule set properties and the types of strings generated, establishing a formal framework for L-system analysis.
• The work establishes that the L-space is subject to hard formal constraints, limiting which grammars and developments can coherently exist within it.