arXiv subjects

Richard Mandel

Publications and source records attributed to Richard Mandel.

4 recordsLinked to original sources

arXiv2026

The complexity of downward closures of indexed languages

Indexed languages are a classical notion in formal language theory, which has attracted attention in recent decades due to its role in higher-order model checking: They are precisely the languages accepted by order-2 pushdown automata. The downward closure of an indexed language -- the set of all (scattered) subwords of its members -- is well-known to be a regular over-approximation. It is known since 2015 that the downward closure of a given indexed language is effectively computable. However, the algorithm comes with no complexity bounds, and it has remained open whether a primitive-recursive construction exists. We settle this question and provide a triply (resp. quadruply) exponential construction of a non-deterministic (resp. deterministic) automaton. We also prove (asymptotically) matching lower bounds. For the upper bounds, we rely on recent advances in semigroup theory, which let us compute bounded-size summaries of words with respect to a finite semigroup. By replacing stacks with their summaries, we are able to transform an indexed grammar into a context-free one with the same downward closure, and then apply existing bounds for context-free grammars.

cs.FL↗

arXiv2023

Quadratic equations in metabelian Baumslag-Solitar groups

For a finitely generated group $G$, the \emph{Diophantine problem} over $G$ is the algorithmic problem of deciding whether a given equation $W(z_1,z_2,\ldots,z_k) = 1$ (perhaps restricted to a fixed subclass of equations) has a solution in $G$. In this paper, we investigate the algorithmic complexity of the Diophantine problem for the class $\mathcal{C}$ of quadratic equations over the metabelian Baumslag-Solitar groups $\mathbf{BS}(1,n)$. We prove that this problem is $\mathbf{NP}$-complete whenever $n\neq \pm 1$, and determine the algorithmic complexity for various subclasses (orientable, nonorientable etc.) of $\mathcal{C}$.

math.GR↗

arXiv2022

The Diophantine problem for systems of algebraic equations with exponents

Consider the equation $q_1\alpha^{x_1}+\dots+q_k\alpha^{x_k} = q$, with constants $\alpha \in \overline{\mathbb{Q}} \setminus \{0,1\}$, $q_1,\ldots,q_k,q\in\overline{\mathbb{Q}}$ and unknowns $x_1,\ldots,x_k$, referred to in this paper as an \emph{algebraic equation with exponents}. We prove that the problem to decide if a given equation has an integer solution is $\textbf{NP}$-complete, and that the same holds for systems of equations (whether $\alpha$ is fixed or given as part of the input). Furthermore, we describe the set of all solutions for a given system of algebraic equations with exponents and prove that it is semilinear.

math.NT↗

arXiv2016

Computing the Fixed Field of $\text{Aut}(\mathbb{F}(x)/\mathbb{F})$

For a finite field $\mathbb{F}$, it is a basic result of Galois theory that the fixed field $E$ of $\text{Aut}(\mathbb{F}(x)/\mathbb{F})$ is a proper extension of $\mathbb{F}$. In this expository paper we construct, for all finite fields, an element $f\in \mathbb{F}(x)$ such that $E = \mathbb{F}(f)$. This element is shown to be easily computable in all cases using a simple formula. The approach is entirely elementary, and aimed at the student level; only a basic familiarity with Galois theory and field extensions is assumed.

math.NT↗