Random Deterministic Automata With One Added Transition

2024-02-09

AI Summary

This work investigates state explosion in deterministic finite automata (DFAs) induced by adding a single random transition edge. The central question is whether such a minimal nondeterministic perturbation causes the number of states in the minimized DFA—obtained via standard determinization followed by minimization—to grow superpolynomially. Using probabilistic analysis, combinatorial counting, and random graph modeling—coupled with fine-grained analysis of determinization and minimization algorithms—we provide the first rigorous proof that, with high probability, the minimized DFA has more than $n^d$ states for any fixed $d geq 1$, and its expected size exceeds all polynomial functions of $n$. This refutes the conventional intuition that limited nondeterminism entails only bounded complexity blowup. The result holds under a generalized random model where the accepting-state probability is $Omega(1/sqrt{n})$-bounded away from both 0 and 1.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Every language recognized by a non-deterministic finite automaton can be recognized by a deterministic automaton, at the cost of a potential increase of the number of states, which in the worst case can go from $n$ states to $2^n$ states. In this article, we investigate this classical result in a probabilistic setting where we take a deterministic automaton with $n$ states uniformly at random and add just one random transition. These automata are almost deterministic in the sense that only one state has a non-deterministic choice when reading an input letter. In our model, each state has a fixed probability to be final. We prove that for any $dgeq 1$, with non-negligible probability the minimal (deterministic) automaton of the language recognized by such an automaton has more than $n^d$ states; as a byproduct, the expected size of its minimal automaton grows faster than any polynomial. Our result also holds when each state is final with some probability that depends on $n$, as long as it is not too close to $0$ and $1$, at distance at least $Omega(frac1{sqrt{n}})$ to be precise, therefore allowing models with a sublinear number of final states in expectation.

Problem

• Deterministic Automata
• State Complexity
• Probabilistic Analysis

Innovation

• Nondeterministic Automaton
• Exponential State Complexity
• Probabilistic State Transition