非标准数制中最终周期集的判定问题

一种使用LSDF约定判定$b$-可识别集合周期性的高效算法

An efficient algorithm to decide periodicity of $b$-recognisable sets using LSDF convention

本文提出一种基于LSDF约定的高效算法，通过刻画接受周期集合的最小自动机结构，在线性时间内验证该结构，从而以$O(b \log(n))$复杂度判定$b$-可识别集合的周期性。

模态μ-演算在几乎周期词上的有限收敛

Finite Convergence of the Modal Mu-Calculus on Almost-Periodic Words

研究模态μ-演算在几乎周期词上的有限收敛性，证明所有几乎周期词都有有限收敛性，刻画了无限词上的有限收敛性，还重新证明了相关可判定性结果。

Palindromic Length of Words with Many Periodic Palindromes

The palindromic length $\text{PL}(v)$ of a finite word $v$ is the minimal number of palindromes whose concatenation is equal to $v$. In 2013, Frid, Puzynina, and Zamboni conjectured that: If $w$ is an infinite word and $k$ is an integer such that $\text{PL}(u)\leq k$ for every factor $u$ of $w$ then $w$ is ultimately periodic. Suppose that $w$ is an infinite word and $k$ is an integer such $\text{PL}(u)\leq k$ for every factor $u$ of $w$. Let $Ω(w,k)$ be the set of all factors $u$ of $w$ that have more than $\sqrt[k]{k^{-1}\vert u\vert}$ palindromic prefixes. We show that $Ω(w,k)$ is an infini…

无限字母表上的（标称）omega-正则语言的可判定类

A decidable class of (nominal) omega-regular languages over an infinite alphabet

我们定义了一类在无限个字母表上有无限个词的语言，以及相应的自动机。用于识别的自动机是确定性Muller自动机的一个推广，以设置名义集。值得注意的是，所获得的语言是由它们最终的周期性片段决定的，就像经典的情况一样。在补、并、交下的封闭性，以及空性和等价性的可判定性都被推广所保持。这是通过使用有限表示（否则无限状态）定义类的自动机。

Polynomial time algorithms for inclusion and equivalence of deterministic omega acceptors

The class of omega languages recognized by deterministic parity acceptors (DPAs) or deterministic Muller acceptors (DMAs) is exactly the regular omega languages. The inclusion problem is the following: given two acceptors A1 and A2, determine whether the language recognized by A1 is a subset of the language recognized by A2, and if not, return an ultimately periodic omega word accepted by A1 but not A2. We describe polynomial time algorithms to solve this problem for two DPAs and for two DMAs. Corollaries include polynomial time algorithms to solve the equivalence problem for DPAs and DMAs, an…

Automatic sequences and generalised polynomials

We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this conjecture, proving that any hypothetical counterexample is periodic away from a very sparse and structured set. In particular, we show that for a polynomial $p(n)$ with at least one irrational coefficient (except for the constant one) and integer $m\geq 2$, the sequence $\lfloor p(n) \rfloor \bmod{m}$ is never automatic. We also prove that the conjecture is equ…

一族用于识别和学习 $\omega$-正则语言的新型有限自动机

A novel family of finite automata for recognizing and learning $ω$-regular languages

本文提出极限FDFA这一新型 $\omega$-正则语言表示，分析其与已有规范FDFA的简洁性和对偶关系，并结合循环FDFA及Myhill-Nerode右同余缩小进展自动机。

用于一元语言的拉脱维亚量子有限状态自动机

Latvian Quantum Finite State Automata for Unary Languages

本文通过组合两个分别处理有限部分和最终周期部分的拉脱维亚量子有限状态自动机，构造了以孤立截点1/2识别一元正则语言的LQFA，其状态数与隔离度对最小DFA大小呈指数依赖。

关于整数集上的方程

On equations over sets of integers

系统的方程组的整数作为未知数。证明了用并加运算$S+T=\makeset{m+n}{m \in S, \: n \in T}$表示的具有最终周期常数的方程的唯一解所表示的集合类恰是超算术集合类.在简单编码下，仅用加法的方程可以表示所有超算术集。所有的超算术集合也可以用自然数集合上的方程来表示，这些自然数集合具有并、加、减运算$S \dotminus T=\makeset{m-n}{m \in S, \: n \in T, \: m \geqslant n}$。测试给定的系统是否有解决方案对于每个模型都是$Σ^1_1$完成的。这些结果，特别是，解决了最一般类型的语言方程的表达能力，以及方程的子集的自由群。