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Complexity of Unary Exclusive Nondeterministic Finite Automata

Martin Kutrib, Andreas Malcher, Matthias Wendlandt

TL;DR

This paper characterizes the descriptional and computational complexity of unary exclusive nondeterministic finite automata (XNFA). It develops a Chrobak-normal-form framework for unary XNFA to derive tight upper bounds: any unary $n$-state XNFA can be determinized with $e^{\Theta(\sqrt{n\cdot\ln n})}$ states, and similar bounds apply to converting between unary NFAs and XNFAs. The analysis relies on Landau's function $F(n)$ and related constructs such as $G(n)$ to establish both upper and lower bounds, demonstrating tight exponential-in-sqrt growth in the unary setting. On the decision side, the paper shows that membership is NL-complete, while emptiness, universality, inclusion, and equivalence are coNP-complete for unary XNFA, with NP-completeness for non-emptiness. Collectively, these results illuminate how unary constraints shape the cost of descriptional transformations and fundamental decision problems in XNFA models, connecting classical number-theoretic bounds to automata theory and specifying precise complexity landscapes.

Abstract

Exclusive nondeterministic finite automata (XNFA) are nondeterministic finite automata with a special acceptance condition. An input is accepted if there is exactly one accepting path in its computation tree. If there are none or more than one accepting paths, the input is rejected. We study the descriptional complexity of XNFA accepting unary languages. While the state costs for mutual simulations with DFA and NFA over general alphabets differ significantly from the known types of finite automata, it turns out that the state costs for the simulations in the unary case are in the order of magnitude of the general case. In particular, the state costs for the simulation of an XNFA by a DFA or an NFA are $e^{θ(\sqrt{n \cdot ln{n}})}$. Conversely, converting an NFA to an equivalent XNFA may cost $e^{θ(\sqrt{n \cdot ln{n}})}$ states as well. All bounds obtained are also tight in the order of magnitude. Finally, we investigate the computational complexity of different decision problems for unary XNFA and it is shown that the problems of emptiness, universality, inclusion, and equivalence are coNP-complete, whereas the general membership problem is NL-complete.

Complexity of Unary Exclusive Nondeterministic Finite Automata

TL;DR

This paper characterizes the descriptional and computational complexity of unary exclusive nondeterministic finite automata (XNFA). It develops a Chrobak-normal-form framework for unary XNFA to derive tight upper bounds: any unary -state XNFA can be determinized with states, and similar bounds apply to converting between unary NFAs and XNFAs. The analysis relies on Landau's function and related constructs such as to establish both upper and lower bounds, demonstrating tight exponential-in-sqrt growth in the unary setting. On the decision side, the paper shows that membership is NL-complete, while emptiness, universality, inclusion, and equivalence are coNP-complete for unary XNFA, with NP-completeness for non-emptiness. Collectively, these results illuminate how unary constraints shape the cost of descriptional transformations and fundamental decision problems in XNFA models, connecting classical number-theoretic bounds to automata theory and specifying precise complexity landscapes.

Abstract

Exclusive nondeterministic finite automata (XNFA) are nondeterministic finite automata with a special acceptance condition. An input is accepted if there is exactly one accepting path in its computation tree. If there are none or more than one accepting paths, the input is rejected. We study the descriptional complexity of XNFA accepting unary languages. While the state costs for mutual simulations with DFA and NFA over general alphabets differ significantly from the known types of finite automata, it turns out that the state costs for the simulations in the unary case are in the order of magnitude of the general case. In particular, the state costs for the simulation of an XNFA by a DFA or an NFA are . Conversely, converting an NFA to an equivalent XNFA may cost states as well. All bounds obtained are also tight in the order of magnitude. Finally, we investigate the computational complexity of different decision problems for unary XNFA and it is shown that the problems of emptiness, universality, inclusion, and equivalence are coNP-complete, whereas the general membership problem is NL-complete.
Paper Structure (5 sections, 11 theorems, 12 equations, 1 table)

This paper contains 5 sections, 11 theorems, 12 equations, 1 table.

Table of Contents

  1. Introduction
  2. Definitions and Preliminaries
  3. Determinization of unary XNFAs
  4. Converting unary NFAs to XNFAs and Vice Versa
  5. Computational Complexity

Key Result

Lemma 1

Let $0 < c_1 < c_2 < \cdots < c_r\leq n$ be positive integers. Then the set of integers $z > n^2$ that can be written as a non-negative integer linear combination of the $c_i$ is $\{\, t + x\cdot d\mid x\geq 0\,\}$, where $t$ is the least integer greater than $n^2$ that is a multiple of $d = \gcd(c_

Theorems & Definitions (17)

  • Lemma 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • Proposition 5
  • Theorem 6
  • Proposition 7
  • ...and 7 more