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Optimal L-Systems for Stochastic L-system Inference Problems

Ali Lotfi, Ian McQuillan

TL;DR

This work tackles stochastic L-system inference by formalizing two optimization-driven problems: (i) constructing an S0L derivation that maximizes the probability of generating a given sequence θ in a single derivation, and (ii) constructing an S0L that maximizes the probability of generating θ across all derivations. It resolves these with two theorems: a sharp upper bound on p(d) and an explicit optimal per-predecessor rule, and a posynomial-optimization formulation that yields a highest-probability S0L over all derivations. An algorithm is proposed that uses nonlinear optimization and interior-point methods to compute the optimal production probabilities p*(a→y) from θ, enabling positive-data-only learning of stochastic grammars. The results have practical implications for automated plant morphology modeling, synthetic data generation for vision, and other domains requiring probabilistic, parallel string rewriting with positive-only training data.

Abstract

This paper presents two novel theorems that address two open problems in stochastic Lindenmayer-system (L-system) inference, specifically focusing on the construction of an optimal stochastic L-system capable of generating a given sequence of strings. The first theorem delineates a method for crafting a stochastic L-system that has the maximum probability of a derivation producing a given sequence of words through a single derivation (noting that multiple derivations may generate the same sequence). Furthermore, the second theorem determines the stochastic L-systems with the highest probability of producing a given sequence of words with multiple possible derivations. From these, we introduce an algorithm to infer an optimal stochastic L-system from a given sequence. This algorithm incorporates advanced optimization techniques, such as interior point methods, to ensure the creation of a stochastic L-system that maximizes the probability of generating the given sequence (allowing for multiple derivations). This allows for the use of stochastic L-systems as a model for machine learning using only positive data for training.

Optimal L-Systems for Stochastic L-system Inference Problems

TL;DR

This work tackles stochastic L-system inference by formalizing two optimization-driven problems: (i) constructing an S0L derivation that maximizes the probability of generating a given sequence θ in a single derivation, and (ii) constructing an S0L that maximizes the probability of generating θ across all derivations. It resolves these with two theorems: a sharp upper bound on p(d) and an explicit optimal per-predecessor rule, and a posynomial-optimization formulation that yields a highest-probability S0L over all derivations. An algorithm is proposed that uses nonlinear optimization and interior-point methods to compute the optimal production probabilities p*(a→y) from θ, enabling positive-data-only learning of stochastic grammars. The results have practical implications for automated plant morphology modeling, synthetic data generation for vision, and other domains requiring probabilistic, parallel string rewriting with positive-only training data.

Abstract

This paper presents two novel theorems that address two open problems in stochastic Lindenmayer-system (L-system) inference, specifically focusing on the construction of an optimal stochastic L-system capable of generating a given sequence of strings. The first theorem delineates a method for crafting a stochastic L-system that has the maximum probability of a derivation producing a given sequence of words through a single derivation (noting that multiple derivations may generate the same sequence). Furthermore, the second theorem determines the stochastic L-systems with the highest probability of producing a given sequence of words with multiple possible derivations. From these, we introduce an algorithm to infer an optimal stochastic L-system from a given sequence. This algorithm incorporates advanced optimization techniques, such as interior point methods, to ensure the creation of a stochastic L-system that maximizes the probability of generating the given sequence (allowing for multiple derivations). This allows for the use of stochastic L-systems as a model for machine learning using only positive data for training.
Paper Structure (9 sections, 4 theorems, 26 equations, 1 algorithm)

This paper contains 9 sections, 4 theorems, 26 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. 0L-systems and D0L-systems
  4. Stochastic 0L-systems
  5. Main Questions and Mathematical Tools
  6. Determining the Optimal Derivation
  7. Determining the Optimal S0L-System
  8. Algorithm
  9. Conclusion

Key Result

Lemma 3.1

Let $\{x_i\}_{i=1}^m$ be a set of variables constrained by where $a_i$ are constants. Then the product $\prod_{i=1}^m x_i^{n_i}$ is maximized when where $N = \sum_{j=1}^m n_j$. The maximum value of $\prod_{i=1}^m x_i^{n_i}$ under these constraints is

Theorems & Definitions (10)

  • Remark
  • Definition 2.1
  • Example 1
  • Example 2
  • Lemma 3.1
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • Theorem 5.1
  • proof