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Enumerating Finitary Processes
Benjamin D Johnson1,2, James P Crutchfield1,2,3, Christopher J Ellison1,3
1Complexity Sciences Center, University of California at Davis, One Shields Avenue, Davis, CA 95616, USA.
Entropy (Basel, Switzerland)
|January 8, 2025
Summary
This study presents an efficient method for enumerating finite-memory stochastic processes using epsilon-machines. The approach adapts automata theory algorithms to precisely count these complex systems.
Area of Science:
- Theoretical Computer Science
- Information Theory
- Dynamical Systems
Background:
- Stochastic processes are fundamental to modeling complex systems.
- Finite-memory processes are a key subclass, but efficient enumeration remains challenging.
- Epsilon-machines offer a causal framework for representing these processes.
Purpose of the Study:
- To develop an efficient method for enumerating finite-memory stochastic processes.
- To characterize epsilon-machines within automata theory.
- To apply this framework to enumerate specific classes of epsilon-machines.
Main Methods:
- Utilizing the causal representation of epsilon-machines.
- Adapting algorithms for generating accessible deterministic finite automata.
- Pruning the set of deterministic finite automata to identify epsilon-machines.
- Applying the method to enumerate topological epsilon-machines.
Main Results:
- An efficient algorithm for enumerating epsilon-machines is presented.
- Epsilon-machines are formally characterized using automata theory.
- The method successfully enumerates topological epsilon-machines up to eight states and six-letter alphabets.
Conclusions:
- The developed method provides an efficient means to enumerate finite-memory stochastic processes.
- This work bridges theoretical computer science and the study of complex systems.
- The enumeration results offer a foundation for further analysis of stochastic process complexity.
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