Is Natural Language a Perigraphic Process? The Theorem about Facts and Words Revisited.

Area of Science:

• Information Theory
• Computational Linguistics
• Statistical Modeling

Background:

• Stochastic processes are classified as nonergodic if a persistent topic exists in infinite samples.
• Strongly nonergodic processes require infinite probabilistic facts for complete topic description.
• Ergodic processes are adapted using algorithmic facts, leading to the concept of perigraphic processes.

Purpose of the Study:

• To define and explore perigraphic processes, characterized by power-law growth of inferable algorithmic facts.
• To establish the "theorem about facts and words," relating inferable facts to distinct word-like strings.
• To investigate whether natural language exhibits perigraphic properties.

Main Methods:

• Formalizing definitions of nonergodic, strongly nonergodic, and perigraphic processes.
• Developing the "theorem about facts and words" using concepts from information theory and compression algorithms.
• Analyzing text samples, specifically plays by Shakespeare, using the Prediction by Partial Matching (PPM) compression algorithm.

Main Results:

• A simple example of a perigraphic process is presented.
• The theorem about facts and words demonstrates that inferable facts are bounded by the number of distinct word-like strings identified by PPM.
• Analysis of Shakespeare's plays reveals a stepwise power-law distribution of word-like strings, contrasting with Markov processes.

Conclusions:

• Natural language, when viewed as a stochastic process, is likely non-Markovian.
• The observed power-law distribution in natural language suggests it is perigraphic.
• These findings have implications for understanding language complexity and developing advanced natural language processing models.