arXiv subjects

Alexander Thumm

Publications and source records attributed to Alexander Thumm.

9 recordsLinked to original sources

arXiv2026

Quadratic Equations in Graph Products of Groups and the Exponent of Periodicity

In 1977, Makanin established the decidability of equations in free monoids. A key ingredient in his proof is the exponent of periodicity: for a word $w$, it is the largest exponent $e$ such that $w$ contains a nonempty factor of the form $p^e$. Makanin showed the following for a system of equations in free monoids: if the system has a solution with a sufficiently large exponent of periodicity, then it has infinitely many solutions. However, the converse -- whether the existence of infinitely many solutions implies the existence of solutions with arbitrarily large exponent of periodicity -- remains open. In this paper, we investigate the analogous problem for quadratic equations in finitely generated groups. We use normal forms to define the exponent of periodicity. We then identify structural conditions on groups and their normal forms that guarantee that infinite solution sets of quadratic systems have an unbounded exponent of periodicity. We prove that these conditions are preserved under graph products and, in particular, hold for all finitely generated right-angled Artin groups. In addition, we show that they also hold for finitely generated (graph products of) torsion-free nilpotent and hyperbolic groups, and we characterize the Baumslag-Solitar groups satisfying them.

math.GR↗

arXiv2026

Efficient Compression in Semigroups

Straight-line programs are a central tool in several areas of computer science, including data compression, algebraic complexity theory, and the algorithmic solution of algebraic equations. In the algebraic setting, where straight-line programs can be interpreted as circuits over algebraic structures such as semigroups or groups, they have led to deep insights in computational complexity. A key result by Babai and Szemer\'edi (1984) showed that finite groups afford efficient compression via straight-line programs, enabling the design of a black-box computation model for groups. Building on their result, Fleischer (2019) placed the Cayley table membership problem for certain classes (pseudovarieties) of finite semigroups in NPOLYLOGTIME, and in some cases even in FOLL. He also provided a complete classification of pseudovarieties of finite monoids affording efficient compression. In this work, we complete this classification program initiated by Fleischer, characterizing precisely those pseudovarieties of finite semigroups that afford efficient compression via straight-line programs. Along the way, we also improve several known bounds on the length and width of straight-line programs over semigroups, monoids, and groups. These results lead to new upper bounds for the membership problem in the Cayley table model: for all pseudovarieties that afford efficient compression and do not contain any nonsolvable group, we obtain FOLL algorithms. In particular, we resolve a conjecture of Barrington, Kadau, Lange, and McKenzie (2001), showing that the membership problem for all solvable groups is in FOLL.

math.RA↗

arXiv2025

Quadratic word equations with regular constraints and the exponent of periodicity

In this article, we study word equations in free semigroups and the conjecture that the existence of infinitely many solutions entails the existence of solutions with arbitrarily large exponent of periodicity. We examine this question in the broader framework of word equations with regular constraints and establish new positive results: the conjecture holds for all quadratic word equations with constraints in finite semigroups from the variety $\mathbf{DLG}$ and its left-right dual $\mathbf{DRG}$, encompassing, in particular, all finite groups, commutative semigroups, and $\mathcal{J}$-trivial semigroups.

cs.FL↗

arXiv2025

Finite Semigroups Satisfying an Identity $x_1 \dots x_n \approx \rho(x_1, \dots, x_n)$

We determine the maximal pseudovarieties of finite semigroups that satisfy an identity of the form $x_1 \dots x_n \approx \rho(x_1, \dots, x_n)$. Applying this classification, we further show that a pseudovariety of permutative semigroups satisfies a common permutation identity if and only if it satisfies an identity of the above form or, equivalently, if it does not contain $\mathbf{T} = [x^2 \approx xyx \approx 0]$.

math.RA↗

arXiv2025

Membership and Conjugacy in Inverse Semigroups

The membership problem for an algebraic structure asks whether a given element is contained in some substructure, which is usually given by generators. In this work we study the membership problem, as well as the conjugacy problem, for finite inverse semigroups. The closely related membership problem for finite semigroups has been shown to be PSPACE-complete in the transformation model by Kozen (1977) and NL-complete in the Cayley table model by Jones, Lien, and Laaser (1976). In the partial bijection model, the membership and the conjugacy problem for finite inverse semigroups were shown to be PSPACE-complete by Birget and Margolis (2008) and by Jack (2023). Here we present a more detailed analysis of the complexity of the membership and conjugacy problems parametrized by varieties of finite inverse semigroups. We establish dichotomy theorems for the partial bijection model and for the Cayley table model. In the partial bijection model these problems are in NC (resp. NP for conjugacy) for strict inverse semigroups and PSPACE-complete otherwise. In the Cayley table model we obtain general LOGSPACE-algorithms as well as NPOLYLOGTIME upper bounds for Clifford semigroups and LOGSPACE-completeness otherwise. Furthermore, by applying our findings, we show the following: the intersection non-emptiness problem for inverse automata is PSPACE-complete even for automata with only two states; the subpower membership problem is in NC for every strict inverse semi-group and PSPACE-complete otherwise; the minimum generating set and the equation satisfiability problems are in NP for varieties of finite strict inverse semigroups and PSPACE-complete otherwise.

cs.CC↗

arXiv2024

Finite groups with geodetic Cayley graphs

A connected undirected graph is called \emph{geodetic} if for every pair of vertices there is a unique shortest path connecting them. It has been conjectured that for finite groups, the only geodetic Cayley graphs are odd cycles and complete graphs. In this article we present a series of theoretical results which contribute to a computer search verifying this conjecture for all groups of size up to 1024. The conjecture is also verified for several infinite families of groups including dihedral and some families of nilpotent groups. Two key results which enable the computer search to reach as far as it does are: if the center of a group has even order, then the conjecture holds (this eliminates all $2$-groups from our computer search); if a Cayley graph is geodetic then there are bounds relating the size of the group, generating set and center (which significantly cuts down the number of generating sets which must be searched).

math.GR↗

arXiv2022

Streaming algorithms for groups and semigroups

We investigate deterministic and randomized streaming algorithms for word problems in finitely generated groups and semigroups. For this we introduce the notion of a distinguisher: a randomized streaming algorithm that processes two input words in parallel and, with high probability, reaches identical memory states if the words represent the same element, and distinct states otherwise. We construct such distinguishers with low error probability using logarithmic, and in some cases doubly logarithmic, space. For example, our results apply to linear semigroups and to semigroups obtained (under suitable restrictions) via standard constructions such as graph products, wreath products, and semilattice decompositions. In case of commutative semigroups and cancellative nilpotent semigroups, we achieve space complexity $\mathcal{O}(\log \log n)$. We complement these upper bounds with lower bounds demonstrating that certain well-known semigroups do not admit sublinear-space distinguishers. This includes, for example, free inverse monoids of rank at least two and Thompson's group $F$. Finally, we study randomized streaming algorithms for subgroup membership problems in free groups and their direct products.

math.GR↗

arXiv2021

The Ordered Join of Impartial Games

Inspired by the theory of poset games, we introduce a new compound of impartial combinatorial games and provide a complete analysis in the spirit of the Sprague-Grundy theory. Furthermore, we establish several substitution and reduction principles for this compound and consider its computational aspects.

math.CO↗

arXiv2016

Solutions of Word Equations over Partially Commutative Structures

Let $M(A,I)$ be a free partially commutative monoid with involution and $G(A,I)$ its quotient group (for example, a right-angled Artin or Coxeter group). We show that for any system of word equations over $M(A,I)$ with recognizable constraints, the solution set - in $M(A,I)$ or in $G(A,I)$ - is an EDT0L language. It is given by an NFA $\mathcal{A}$ recognizing endomorphisms over some extended monoid. Furthermore, if the input size is $n$, then the automaton $\mathcal{A}$ can be constructed effectively by an NSPACE$(n\log n)$-transducer. As a consequence, both Satisfiability (whether the system admits a solution) and Finiteness (whether the solution set is infinite) are decidable in NSPACE$(n \log n)$. For a natural subclass of constraints, we conjecture that these problems are NP-complete.

cs.FL↗