Classical and Quantum Algorithms for the Deterministic L-system Inductive Inference Problem

L-systems can be made to model and create simulations of many biological processes, such as plant development. Finding an L-system for a given process is typically solved by hand, by experts, in a hugely time-consuming process. It would be significant if this could be done automatically from data, such as from sequences of images. In this paper, we are interested in inferring a particular type of L-system, deterministic context-free L-system (D0L-system) from a sequence of strings. We introduce the characteristic graph of a sequence of strings, which we then utilize to translate our problem (inferring D0L-system) in polynomial time into the maximum independent set problem (MIS) and the SAT problem. After that, we offer a classical exact algorithm and an approximate quantum algorithm for the problem.

Probabilistic Analysis of Least Squares, Orthogonal Projection, and QR Factorization Algorithms Subject to Gaussian Noise

In this paper, we extend the work of Liesen et al. (2002), which analyzes how the condition number of an orthonormal matrix Q changes when a column is added ([Q, c]), particularly focusing on the perpendicularity of c to the span of Q. Their result, presented in Theorem 2.3 of Liesen et al. (2002), assumes exact arithmetic and orthonormality of Q, which is a strong assumption when applying these results to numerical methods such as QR factorization algorithms. In our work, we address this gap by deriving bounds on the condition number increase for a matrix B without assuming perfect orthonormality, even when a column is not perfectly orthogonal to the span of B. This framework allows us to analyze QR factorization methods where orthogonalization is imperfect and subject to Gaussian noise. We also provide results on the performance of orthogonal projection and least squares under Gaussian noise, further supporting the development of this theory.

Optimal L-Systems for Stochastic L-system Inference Problems

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 maximizes the likelihood of producing a given sequence of words through a singular derivation. 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 sophisticated optimization techniques, such as interior point methods, ensuring production of a stochastically optimal stochastic L-system suitable for generating the given sequence. This allows for the use of using stochastic L-systems as model for machine learning using only positive data for training.