[Paper Review] Ergodicity of Random Walks on Random DFA

Borja Balle|arXiv (Cornell University)|Nov 26, 2013

semigroups and automata theory14 references3 citations

TL;DR

This paper demonstrates that random walks on randomly generated deterministic finite automata (DFA) almost surely induce ergodic Markov chains, meaning the state distribution converges to a unique stationary distribution. The result holds with high probability when the DFA is sampled uniformly at random, and is preserved under minimization, indicating structural regularity in typical DFAs that may facilitate learning under uniform distributions.

ABSTRACT

Given a DFA we consider the random walk that starts at the initial state and at each time step moves to a new state by taking a random transition from the current state. This paper shows that for typical DFA this random walk induces an ergodic Markov chain. The notion of typical DFA is formalized by showing that ergodicity holds with high probability when a DFA is sampled uniformly at random from the set of all automata with a fixed number of states. We also show the same result applies to DFA obtained by minimizing typical DFA.

Motivation & Objective

• To investigate whether typical DFAs—sampled uniformly at random—exhibit structural regularities that make them easier to learn than worst-case bounds suggest.
• To analyze the behavior of random walks on random DFAs and determine whether they induce ergodic Markov chains.
• To examine whether ergodicity is preserved when minimizing randomly generated DFAs, which is crucial for learning since minimization preserves the language.
• To provide foundational insights into the structure of random DFAs, a topic with limited prior understanding compared to random graphs.

Proposed method

• The authors model a random walk on a DFA as a stochastic process starting at the initial state and transitioning via uniformly random transitions at each step.
• They analyze the Markov chain induced by this walk, focusing on communication classes and recurrence properties to establish ergodicity.
• Using probabilistic arguments, they show that with high probability, a uniformly random DFA has a single closed communicating class that is aperiodic and irreducible, ensuring ergodicity.
• They prove that the ergodic property is preserved under state minimization by showing that merge operations preserve irreducibility and aperiodicity of the closed communicating class.
• The analysis relies on structural properties of communicating classes, recurrence, and the invariance of periodicity and irreducibility under state merging.

Experimental results

Research questions

• RQ1 Does a random walk on a uniformly random DFA converge to a unique stationary distribution?
• RQ2 What structural properties of random DFAs ensure the ergodicity of their random walks?
• RQ3 Is the ergodicity of random walks preserved when minimizing the underlying DFA?
• RQ4 How do the communication classes and recurrence properties of random DFAs affect the mixing behavior of random walks?

Key findings

• With high probability, a random walk on a uniformly random DFA converges to a unique stationary distribution, indicating ergodicity.
• The closed communicating class in a typical random DFA is irreducible and aperiodic, which are necessary and sufficient conditions for ergodicity.
• Ergodicity is preserved under minimization of the DFA, meaning the random walk on the minimized automaton also induces an ergodic Markov chain.
• The results imply that the distribution of states reached by uniformly sampled strings in typical DFAs is well-behaved and converges to a stationary distribution.