Optimal L-Systems for Stochastic L-system Inference Problems

2024-09-03

arXiv.org

AI Summary

This paper addresses two fundamental challenges in stochastic L-system inference: (i) constructing the optimal stochastic L-system that maximizes the probability of generating a given sequence of graphical structures, and (ii) guaranteeing global optimality of the solution under both single- and multiple-derivation-path semantics. We present the first theoretically grounded solution with provable optimality guarantees. Specifically, we establish two key theorems characterizing the structure of optimal production rules for single-path and multi-path derivations, respectively. Building on formal language theory, probabilistic modeling, and interior-point optimization, we design a parameter inference algorithm that learns from positive-only examples and is provably optimal. The inferred stochastic L-system is guaranteed to maximize the generation probability of the target sequence globally. This framework provides the first stochastic grammar learning method for biological morphogenesis modeling and structured sequence generation that simultaneously ensures theoretical rigor and computational tractability.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Learning to Search
• Machine Learning: Probabilistic Circuits and Graphical Models

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Random L-Systems
• Optimal Rule Discovery
• Probability Maximization

Innovation

• Random L-Systems
• Optimization Algorithms
• Machine Learning Training