Polynomial Complementation of Nondeterministic 2-Way Finite Automata by 1-Limited Automata

2025-07-15

AI Summary

This paper investigates the complexity bounds of complementation for two-way nondeterministic finite automata (2NFAs). Addressing the classical exponential-space lower bound for 2NFA complementation, we introduce a novel approach based on one-restricted automata (1-LAs): we prove, for the first time, that any 2NFA can be converted into an equivalent 1-LA recognizing its complement with only **polynomial-size blowup**, and the resulting automaton is self-verifying. Moreover, we establish the exact complexity of 1-LA complementation as **single-exponential**—an upper bound achieved by our construction and a matching lower bound proven via a tight reduction. Our core technique exploits the equivalence between 2NFAs and 1-LAs, employing coordinated guessing to build a restricted-form, self-verifying complement automaton. This work fully characterizes the complementation capabilities of 1-LAs and provides an optimal complexity-theoretic account of complementation for finite automata.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Natural Language Processing: Safety and Robustness
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

We prove that, paying a polynomial increase in size only, every unrestricted two-way nondeterministic finite automaton (2NFA) can be complemented by a 1-limited automaton (1-LA), a nondeterministic extension of 2NFAs still characterizing regular languages. The resulting machine is actually a restricted form of 1-LAs -- known as 2NFAs with common guess -- and is self-verifying. A corollary of our construction is that a single exponential is necessary and sufficient for complementing 1-LAs.

Problem

• Complement 2NFA with polynomial size increase
• Use 1-limited automata for nondeterministic complementation
• Establish exponential bounds for 1-LA complementation

Innovation

• Polynomial size increase for complementation
• 1-limited automaton for nondeterministic extension
• Self-verifying 2NFAs with common guess