Algorithms and fine-grained complexity for nondeterministic and symmetric difference automata

2026-07-01

AI Summary

This work investigates the three fundamental decision problems for symmetric difference nondeterministic finite automata (XNFA)—membership, emptiness, and equivalence—under the lens of fine-grained complexity, establishing tight polynomial-time bounds. By introducing randomized reductions coupled with certificate verification mechanisms and leveraging weighted automata theory over the Boolean semiring and other semirings, the study presents the first randomized reduction from NFA membership to XNFA membership under the polynomial ambiguity hypothesis. For both bounded-ambiguity and general cases, the authors devise more efficient decision and verification algorithms, substantially improving the known upper bounds on time complexity for these problems. Notably, several of these results extend naturally to weighted automata over arbitrary semirings.

Abstract

Symmetric difference automata (XNFA) are a variant of standard finite automata in which an input word is accepted iff the number of accepting runs is odd. Equivalently, these are weighted automata over the two-element field. We study the fine-grained complexity of the basic decision problems for XNFA: acceptance, emptiness, and equivalence, aiming to optimise the degree of the polynomial in their running-time bounds. Under the assumption of polynomial ambiguity, we provide a randomised reduction of NFA acceptance to XNFA acceptance. For automata of bounded ambiguity (e.g., unambiguous automata), we show that acceptance for both NFA and XNFA can be decided faster than in the general case. Without ambiguity assumptions, we give faster algorithms for the verification of suitable certificates for (non)emptiness and (non)equivalence of XNFA. Several of our results extend to weighted automata over other semirings and fields.

Problem

• symmetric difference automata
• fine-grained complexity
• acceptance
• emptiness
• equivalence

Innovation

• symmetric difference automata
• fine-grained complexity
• polynomial ambiguity
• randomised reduction
• weighted automata