Core Concepts

Probabilistic automatic complexity of finite strings is characterized by specific forms, revealing structural properties.

Abstract

The content introduces a new complexity measure for finite strings using probabilistic finite-state automata (PFAs). It explores the properties of PFA complexity, including a variant with a real-valued parameter. The article relates PFA complexity to deterministic and nondeterministic finite-state automata complexities. It provides a complete classification of binary strings with complexity 2 and discusses the computability of the PFA complexity. The content also delves into the concept of probabilistic automatic complexity with a lower bound on the gap between acceptance probabilities. It presents theorems, propositions, and proofs related to the classification of binary strings with complexity 2, highlighting the unique structural properties of these strings. The article further discusses the connection between PFAs and iterated function systems, demonstrating how PFAs can be used to generate fractal images. It concludes with a detailed proof of the classification of binary strings with complexity 2.

Key Insights Distilled From

by Kenneth Gill at **arxiv.org** 03-28-2024

Stats

- The PFA complexity AP(x) is defined as the least number of states of a PFA for which x is the most likely string of its length to be accepted.
- The variant AP,δ(x) adds a real-valued parameter δ specifying a lower bound on the gap in acceptance probabilities between x and other strings.
- Theorem 4.1 states that for a binary string w, AP(w) = 2 if and only if w is of specific forms: injm, injmi, in(ji)m, or inj(ij)m for some n, m ≥ 0, where i, j ∈ {0, 1}.
- Corollary 4.13 highlights that the quantity AN(w) - AP(w) may be arbitrarily large among binary w.
- Theorem 5.1 shows that AP,δ(w) is computable for all w and almost all δ ≠ 0.

Quotes

- "AP does appear to capture some intuitive structural properties of strings."
- "The study of AN has been continued by Kjos-Hanssen."

Deeper Inquiries

The concept of probabilistic automatic complexity can extend beyond finite strings by considering more complex structures or systems that can be represented using probabilistic finite-state automata (PFAs). PFAs can be used to model various probabilistic processes or systems, such as probabilistic algorithms, stochastic systems, or even real-world scenarios involving uncertainty and randomness. By analyzing the probabilistic behavior of these systems using PFAs, one can determine the most likely outcomes or sequences of events, similar to how the complexity of finite strings is determined based on acceptance probabilities.

One counterargument against the classification of binary strings with complexity 2 is the potential limitation of the model used to determine complexity. The classification is based on the properties of PFAs and their acceptance probabilities, which may not capture all aspects of the complexity of binary strings. It is possible that certain strings may exhibit more intricate patterns or structures that are not fully captured by the criteria for complexity 2, leading to misclassifications or oversimplifications.

Another counterargument could be the subjective nature of complexity and the interpretation of what constitutes a "complex" string. Different perspectives or definitions of complexity may lead to alternative classifications or disagreements on the categorization of binary strings with complexity 2.

The study of PFAs and iterated function systems can contribute to various areas of computer science and mathematics by providing insights into probabilistic modeling, computational complexity, and dynamical systems.

In computer science, PFAs can be applied in areas such as machine learning, natural language processing, and pattern recognition, where probabilistic models are used to analyze and make predictions based on uncertain data. Understanding the properties and behaviors of PFAs can lead to the development of more efficient algorithms and models in these fields.

In mathematics, the connection between PFAs and iterated function systems can provide a deeper understanding of fractals, chaos theory, and nonlinear dynamics. By studying the relationship between these two concepts, researchers can explore the connections between discrete and continuous dynamical systems, leading to advancements in areas such as topology, geometry, and mathematical modeling.

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Table of Content

Probabilistic Automatic Complexity of Finite Strings

How does the concept of probabilistic automatic complexity extend beyond finite strings?

What counterarguments exist against the classification of binary strings with complexity 2?

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