arXiv subjects

Xuan Kien Phung

Publications and source records attributed to Xuan Kien Phung.

At least 19 recordsLinked to original sources

arXiv2026

Around Gromov's injectivity lemma and applications to post-injunctive groups

Gottschalk's surjunctivity conjecture states that for all group universes and finite alphabets, every equivariant and continuous selfmap of the full shift, known as cellular automaton, cannot be a strict embedding. Not all surjective cellular automata are injective. However, if the surjectivity condition is replaced by a certain strengthened property called post-surjectivity then all post-surjective cellular automata must be bijective whenever the universe is a sofic group. A group universe is said to be post-injunctive if every post-surjective cellular automaton with finite alphabet over this group universe must be bijective. Gromov's injectivity lemma states each injective cellular automaton over a subshift can be extended to an injective cellular automaton over every subshift which is close enough to the initial subshift. In this paper, we obtain analogous results where injectivity is replaced by other fundamental dynamical properties namely post-surjectivity and pre-injectivity. We also study various stable properties of the class of post-injunctive groups in parallel to properties of surjunctive groups. Among the results, we show that semidirect extensions of post-injunctive groups with residually finite kernels must be post-injunctive.

math.GR↗

arXiv2026

On Gottschalk's surjunctivity conjecture for non-uniform cellular automata

Gottschalk's surjunctivity conjecture for a group $G$ states that it is impossible for cellular automata (CA) over the universe $G$ with finite alphabet to produce strict embeddings of the full shift into itself. A group universe $G$ satisfying Gottschalk's surjunctivity conjecture is called a surjunctive group. The surjunctivity theorem of Gromov and Weiss shows that every sofic group is surjunctive. In this paper, we study the surjunctivity of local perturbations of CA and more generally of non-uniform cellular automata (NUCA) with finite memory and uniformly bounded singularity over surjunctive group universes. In particular, we show that such a NUCA must be invertible whenever it is reversible. We also obtain similar results which extend to the class of NUCA a certain dual-surjunctivity theorem of Capobianco, Kari, and Taati for CA.

math.DS↗

arXiv2026

Efficient space reduction techniques by optimized majority rules for the Kemeny aggregation problem and beyond

The Kemeny aggregation problem consists of computing the consensus rankings of an election with respect to the well-known Kemeny-Young voting method. These consensus rankings satisfy various fundamental properties and are the geometric medians of the votes in the election under the Kendall-tau distance which counts the number of pairwise disagreements. The Kemeny aggregation problem admits important applications in various domains such as computational social choice, machine learning, operations research, and biology but it is unfortunately NP-hard. Recently, Milosz and the second author presented an approach to reduce the search space of the problem by solving the relative order of pairs of elements in those consensus. In this article, we prove an optimized extension of this approach achieving significantly more refined space reduction techniques without adding much to the running time of the algorithms in practice, as illustrated by experimental results and analysis on real and synthetic data. We show how the constraints built by our approach can be used in combination with other methods such as Integer Programming and Finest Condorcet Partitioning to achieve an efficient and scalable solution approach to the Kemeny aggregation problem. Relaxed and approximate versions of our algorithms are also described and evaluated. We also provide practical methods to compute provable guarantees for the quality of the approximate rankings obtained.

cs.DS↗

arXiv2025

Topological stability of semigroup actions and shadowing

We investigate expansiveness, topological stability, and shadowing for continuous actions of semigroups on compact Hausdorff spaces. We characterize semigroups for which all full shifts are expansive. We show that every expansive continuous monoid action on a compact Hausdorff space which has the shadowing property is topologically stable, and that a subshift with finite alphabet over a monoid has the shadowing property if and only if it is of finite type.

math.DS↗

arXiv2025

Strongly sofic monoids, sofic topological entropy, and surjunctivity

We introduce the class of strongly sofic monoids. This class of monoids strictly contains the class of sofic groups and is a proper subclass of the class of sofic monoids. We define and investigate sofic topological entropy for actions of strongly sofic monoids on compact spaces. We show that sofic topological entropy is a topological conjugacy invariant for such actions and use this fact to prove that every strongly sofic monoid is surjunctive. This means that if $M$ is a strongly sofic monoid and $A$ is a finite alphabet set, then every injective cellular automaton $τ\colon A^M \to A^M$ is surjective. As an application, we prove that the monoid algebra of a strongly sofic monoid with coefficients in an arbitrary field is always stably finite. Our results are extensions to strongly sofic monoids of two previously known properties of sofic groups. The first one is the celebrated Gromov-Weiss theorem asserting that every sofic group is surjunctive. The second is the Elek-Szabó theorem which says that group algebras of sofic groups satisfy Kaplansky's stable finiteness conjecture.

math.GR↗

arXiv2024

Stable finiteness of monoid algebras and surjunctivity

A monoid $M$ is said to be surjunctive if every injective cellular automaton with finite alphabet over $M$ is surjective. We show that monoid algebras of surjunctive monoids are stably finite. In other words, given any field $K$ and any surjunctive monoid $M$, every one-sided invertible square matrix with entries in the monoid algebra $K[M]$ is two-sided invertible. Our proof uses first-order model theory.

math.RA↗

arXiv2024

Generalized Gottschalk's conjecture for sofic groups and applications

We establish generalizations of the well-known surjunctivity theorem of Gromov and Weiss as well as the dual-surjunctivity theorem of Capobianco, Kari and Taati for cellular automata (CA) to local perturbations of CA over sofic group universes. We also extend the results to a class of non-uniform cellular automata (NUCA) consisting of global perturbations with uniformly bounded singularity of CA. As an application, we obtain the surjunctivity of algebraic NUCA with uniformly bounded singularity over sofic groups. Moreover, we prove the stable finiteness of twisted group rings over sofic groups to generalize known results on Kaplansky's stable finiteness conjecture for group rings.

math.DS↗

arXiv2023

Space reduction techniques for the $3$-wise Kemeny problem

Kemeny's rule is one of the most studied and well-known voting schemes with various important applications in computational social choice and biology. Recently, Kemeny's rule was generalized via a set-wise approach by Gilbert et. al. This paradigm presents interesting advantages in comparison with Kemeny's rule since not only pairwise comparisons but also the discordance between the winners of subsets of three alternatives are also taken into account in the definition of the $3$-wise Kendall-tau distance between two rankings. In spite of the NP-hardness of the 3-wise Kemeny problem which consists of computing the set of $3$-wise consensus rankings, namely rankings whose total $3$-wise Kendall-tau distance to a given voting profile is minimized, we establish in this paper several generalizations of the Major Order Theorems, as obtained by Milosz and Hamel for Kemeny's rule, for the $3$-wise Kemeny voting schemes to achieve a substantial search space reduction by efficiently determining in polynomial time the relative orders of pairs of alternatives. Essentially, our theorems quantify precisely the nontrivial property that if the preference for an alternative over another one in an election is strong enough, not only in the head-to-head competition but even when taking into account one or two more alternatives, then the relative order of these two alternatives in all $3$-wise consensus rankings must be as expected. As an application, we also obtain an improvement of the Major Order Theorems for Kememy's rule. Moreover, we show that the well-known $3/4$-majority rule of Betzler et al. for Kemeny's rule is only valid in general for elections with no more than $5$ alternatives with respect to the $3$-wise Kemeny scheme. Several simulations and tests of our algorithms on real-world and uniform data are provided.

cs.DM↗

arXiv2023

First-order model theory and Kaplansky's stable finiteness conjecture

Using algebraic geometry methods, the third author proved that the group ring of a surjunctive group with coefficients in a field is always stably finite. In other words, every group satisfying Gottschalk's conjecture also satisfies Kaplansky's stable finiteness conjecture. Here we present an alternative proof of this result based on first-order model theory.

math.GR↗

arXiv2023

Generalized integral points on abelian varieties and the Geometric Lang-Vojta conjecture

Let $A$ be an abelian variety over the function field $K$ of a compact Riemann surface $B$. Fix a model $f \colon \mathcal{A} \to B$ of $A/K$ and an effective horizontal divisor $\mathcal{D} \subset \mathcal{A}$. We study $(S, \mathcal{D})$-integral sections $σ$ of $\mathcal{A}$ where $S \subset B$ is arbitrary. These sections $σ$ are algebraic and satisfy the geometric condition $f(σ(B) \cap \mathcal{D})\subset S$. Developing the idea of Parshin, we formulate a hyperbolic-homotopic height of such sections as a substitute for intersection theory to establish new results concerning the finiteness and the polynomial growth of large unions of $(S, \mathcal{D})$-integral points where $S$ is only required to be finite in a thin analytic open subset of $B$. Such results are out of reach of purely algebraic methods and imply new evidence and interesting phenomena to the Geometric Lang-Vojta conjecture.

math.AG↗

arXiv2023

Optimal majority rules and quantitative Condorcet properties of setwise Kemeny voting schemes

The important Kemeny problem, which consists of computing median consensus rankings of an election with respect to the Kemeny voting rule, admits important applications in biology and computational social choice and was generalized recently via an interesting setwise approach by Gilbert et. al. Our first results establish optimal quantitative extensions of the Unanimity property and the well-known $3/4$-majority rule of Betzler et al. for the classical Kemeny median problem. Moreover, by elaborating an exhaustive list of quantified axiomatic properties (such as the Condorcet and Smith criteria, the $5/6$-majority rule, etc.) of the $3$-wise Kemeny rule where not only pairwise comparisons but also the discordance between the winners of subsets of three candidates are also taken into account, we come to the conclusion that the $3$-wise Kemeny voting scheme induced by the $3$-wise Kendall-tau distance presents interesting advantages in comparison with the classical Kemeny rule. For example, it satisfies several improved manipulation-proof properties. Since the $3$-wise Kemeny problem is NP-hard, our results also provide some of the first useful space reduction techniques by determining the relative orders of pairs of alternatives. Our works suggest similar interesting properties of higher setwise Kemeny voting schemes which justify and compensate for the more expensive computational cost than the classical Kemeny scheme.

cs.GT↗

arXiv2023

On the Garden of Eden theorem for non-uniform cellular automata

We establish several extensions of the well-known Garden of Eden theorem for non-uniform cellular automata over the full shifts and over amenable group universes. In particular, our results describe quantitatively the relations between the partial pre-injectivity and the size of the image of a non-uniform cellular automata. A strengthened surjunctivity result is also obtained for multi-dimensional cellular automata over strongly irreducible subshifts of finite type.

math.DS↗

arXiv2022

Some pointwise and decidable properties of non-uniform cellular automata

For non-uniform cellular automata (NUCA) with finite memory over an arbitrary universe with multiple local transition rules, we show that pointwise nilpotency, pointwise periodicity, and pointwise eventual periodicity properties are respectively equivalent to nilpotency, periodicity, and eventual periodicity. Moreover, we prove that every linear NUCA which satisfies pointwise a polynomial equation (which may depend on the configuration) must be an eventually periodic linear NUCA. Generalizing results for higher dimensional group and linear CA, we also establish the decidability results of the above dynamical properties as well as the injectivity for arbitrary NUCA with finite memory which are local perturbations of higher dimensional linear and group CA. Some generalizations to the case of sparse global perturbations of higher dimensional linear and group CA are also obtained.

math.DS↗

arXiv2022

Stable finiteness of twisted group rings and noisy linear cellular automata

For linear non-uniform cellular automata (NUCA) which are local perturbations of linear CA over a group universe $G$ and a finite-dimensional vector space alphabet $V$ over an arbitrary field $k$, we investigate their Dedekind finiteness property, also known as the direct finiteness property, i.e., left or right invertibility implies invertibility. We say that the group $G$ is $L^1$-surjunctive, resp. finitely $L^1$-surjunctive, if all such linear NUCA are automatically surjective whenever they are stably injective, resp. when in addition $k$ is finite. In parallel, we introduce the ring $D^1(k[G])$ which is the Cartesian product $k[G] \times (k[G])[G]$ as an additive group but the multiplication is twisted in the second component. The ring $D^1(k[G])$ contains naturally the group ring $k[G]$ and we obtain a dynamical characterization of its stable finiteness for every field $k$ in terms of the finite $L^1$-surjunctivity of the group $G$, which holds for example when $G$ is residually finite or initially subamenable. Our results extend known results in the case of CA.

math.DS↗

arXiv2022

On linear non-uniform cellular automata: duality and dynamics

For linear non-uniform cellular automata (NUCA) over an arbitrary universe, we introduce and investigate their dual linear NUCA. Generalizing results for linear CA, we show that dynamical properties namely pre-injectivity, resp. injectivity, resp. stably injectivity, resp. invertibility of a linear NUCA is equivalent to surjectivity, resp. post-surjectivity, resp. stably post-surjectivity, resp. invertibility of the dual linear NUCA. However, while bijectivity is a dual property for linear CA, it is no longer the case for linear NUCA. We prove that for linear NUCA, stable injectivity and stable post-surjectivity are precisely characterized respectively by left invertibility and right invertibility and that a linear NUCA is invertible if and only if it is pre-injective and stably post-surjective. Moreover, we show that linear NUCA satisfy the important shadowing property. Applications on the dual surjunctivity are also obtained.

math.DS↗

arXiv2022

Invariant sets and nilpotency of endomorphisms of algebraic sofic shifts

Let $G$ be a group and let $V$ be an algebraic variety over an algebraically closed field $K$. Let $A$ denote the set of $K$-points of $V$. We introduce algebraic sofic subshifts $Σ\subset A^G$ and study endomorphisms $τ\colon Σ\to Σ$. We generalize several results for dynamical invariant sets and nilpotency of $τ$ that are well known for finite alphabet cellular automata. Under mild assumptions, we prove that $τ$ is nilpotent if and only if its limit set, i.e., the intersection of the images of its iterates, is a singleton. If moreover $G$ is infinite, finitely generated and $Σ$ is topologically mixing, we show that $τ$ is nilpotent if and only if its limit set consists of periodic configurations and has a finite set of alphabet values.

math.DS↗

arXiv2022

On reversible asynchronous non-uniform cellular automata

We study the class of asynchronous non-uniform cellular automata (ANUCA) over an arbitrary group universe with multiple local transition rules. We introduce the notion of stable injectivity, stable reversibility, stable post-surjectivity and investigate several dynamical properties of such automata. In particular, we establish the equivalence between reversibility, stable reversibility, and stable injectivity for ANUCA. We also prove the invertibility of several classes of injective and stably injective ANUCA. Counter-examples are given to highlight the differences between cellular automata and ANUCA.

math.DS↗

arXiv2021

On images of subshifts under injective morphisms of symbolic varieties

We show that the image of a subshift $X$ under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, resp. a sofic subshift, if and only if so is $X$. Similarly, let $G$ be a countable monoid and let $A$, $B$ be Artinian modules over a ring. We prove that for every closed subshift submodule $Σ\subset A^G$ and every injective $G$-equivariant uniformly continuous module homomorphism $τ\colon Σ\to B^G$, a subshift $Δ\subset Σ$ is of finite type, resp. sofic, if and only if so is the image $τ(Δ)$. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.

math.DS↗