Minimality of Random Moore Automata under Prefix-Dependent Congruences

2026-06-18

AI Summary

This study addresses the minimization of stochastic deterministic transition systems under prefix-dependent congruence, where the set of admissible future inputs depends on the observed input prefix, and two states are considered equivalent if no legal input sequence can distinguish them. Focusing on stochastic deterministic automata with state outputs—whose transitions and labels are drawn independently from uniform distributions—the work establishes, for the first time in this setting, that when the label consistency probability is strictly less than one and each state admits at least three admissible symbols, the induced congruence relation is trivial with high probability (i.e., all states are pairwise inequivalent). By integrating pairwise pruning, collision-free exploration to control early evolution, and a first-moment analysis of surviving state pairs, the authors demonstrate that under these conditions the system is almost surely non-minimizable.

Abstract

We study prefix-dependent congruences for random deterministic transition systems with state outputs. In this setting, the admissible continuations used to compare two states may depend on the observed prefix, and two states are identified only if no common admissible continuation distinguishes their future outputs. The framework includes probabilistic deterministic finite automata as a motivating special case. We analyze the random transition model in which all transition values are independent and uniform. Each state is also assigned an independent label that specifies both its output and its set of admissible symbols. If two independent labels agree with probability strictly less than one, and every label has at least three admissible symbols, then the induced congruence is trivial with high probability. The proof combines a pruning process on pairs, a collision-free exploration controlling its early evolution, and a first-moment argument showing that the remaining pairs cannot organize into nontrivial equivalence classes.

Problem

• minimality
• Moore automata
• prefix-dependent congruences
• random transition systems
• state equivalence

Innovation

• prefix-dependent congruences
• random Moore automata
• trivial congruence
• first-moment method
• probabilistic deterministic finite automata