CyberSec Research

We give two consistent constructions of trees $T$ whose finite power $T^{n+1}$ is sharply different from $T^n$: 1. An $\aleph_1$-tree $T$ whose interval topology $X_T$ is perfectly normal, but $(X_T)^2$ is not even countably metacompact. 2. For an inaccessible $\kappa$ and a positive integer $n$, a $\kappa$-tree such that all of its $n$-derived trees are Souslin and all of its $(n+1)$-derived trees are special.

We develop techniques at the interface between differential algebra and model theory to study the following problems of exponential algebraicity: Does a given algebraic differential equation admits an exponentially algebraic solution, that is, a holomorphic solution which is definable in the structure of restricted elementary functions? Do solutions of a given list of algebraic differential equations share a nontrivial exponentially algebraic relation, that is, a nontrivial relation definable in the structure of restricted elementary functions? These problems can be traced back to the work of Abel and Liouville on the problem of integration in finite terms. This article concerns generalizations of their techniques adapted to the study of exponential transcendence and independence problems for more general systems of differential equations. As concrete applications, we obtain exponential transcendence and independence statements for several classical functions: the error function, the Bessel functions, indefinite integrals of algebraic expressions involving Lambert's W-function, the equation of the pendulum, as well as corresponding decidability results.

In an o-minimal structure, Wilkie proved that every bounded definable open set is a finite union of definable open cells. We generalze it to weakly o-minimal structures.

Neither the classical nor intuitionistic logic traditions are perfectly-aligned with the purpose of reasoning about computation, in that neither logical tradition can normally permit the direct expression of arbitrary general-recursive functions without inconsistency. We introduce grounded arithmetic or GA, a minimalistic but nonetheless powerful foundation for formal reasoning that allows the direct expression of arbitrary recursive definitions. GA adjusts the traditional inference rules such that terms that express nonterminating computations harmlessly denote no semantic value (i.e., "bottom") instead of leading into logical paradox or inconsistency. Recursive functions may be proven terminating in GA essentially by "dynamically typing" terms, or equivalently, symbolically reverse-executing the computations they denote via GA's inference rules. Once recursive functions have been proven terminating, logical reasoning about their results reduce to the familiar classical rules. A mechanically-checked consistency proof in Isabelle/HOL exists for the basic quantifier-free fragment of GA. Quantifiers may be added atop this foundation as ordinary computations, whose inference rules are thus admissible and do not introduce new inconsistency risks. While GA is only a first step towards richly-typed grounded deduction practical for everyday use in manual or automated computational reasoning, it shows the promise that the expressive freedom of arbitrary recursive definition can in principle be incorporated into formal systems.

We give a unified treatment of the countable dense homogeneity of products of Polish spaces, with a focus on uncountable products. Our main result states that a product of fewer than $\mathfrak{p}$ Polish spaces is countable dense homogeneous if the following conditions hold: (1) Each factor is strongly locally homogeneous, (2) Each factor is strongly $n$-homogeneous for every $n\in\omega$, (3) Every countable subset of the product can be brought in general position. For example, using the above theorem, one can show that $2^\kappa$, $\omega^\kappa$, $\mathbb{R}^\kappa$ and $[0,1]^\kappa$ are countable dense homogeneous for every infinite $\kappa <\mathfrak{p}$ (these results are due to Stepr\={a}ns and Zhou, except for the one concerning $\omega^\kappa$). In fact, as a new application, we will show that every product of fewer than $\mathfrak{p}$ connected manifolds with boundary is countable dense homogeneous, provided that none or infinitely many of the boundaries are non-empty. This generalizes a result of Yang. Along the way, we will discuss and employ several results concerning the general position of countable sets. Finally, we will show that our main result and its corollaries are optimal.

Modules and the notion of Morita equivalence are foundational to the classical study of rings. These concepts extend naturally to semirings and then specialize to Kleene algebrae, and my goal is to investigate Kleene modules and Morita equivalence of Kleene algebrae in the hope that some of the power seen in the context of rings may be found in this new context as well.

We present a direct construction of stationary set preserving forcings that make $\omega$-cofinal all the members of some arbitrary set $\mathcal{K}$ of regular cardinals $\kappa > \omega_1$. In addition, it is made possible to ensure that no other uncountable regular cardinals from the ground model acquire countable cofinality in the forcing extension. Our method is elementary, being based on a combinatorial argument by Foreman and Magidor together with generalizations of typical side-condition arguments and needs no assumptions beyond $\mathsf{ZFC}$.

Given a complemented poset P, we can assign to every element x of P the set x^+ of all its complements. We study properties of the operator ^+ on P, in particular, we are interested in the case when x^+ forms an antichain or when ^+ is involutive or antitone. We apply ^+ to the set Min U(x,y) of all minimal elements of the upper cone U(x,y) of x,y and to the set Max L(x,y) of all maximal elements of the lower cone L(x,y) of x,y. By using ^+ we define four binary operators on P and investigate their properties that are close to adjointness. We present an example of a uniquely complemented poset that is not Boolean. In the last section we study the orthogonality relation induced by complementation. We characterize when two elements of the Dedekind-MacNeille completion of P are orthogonal to each other. Finally, we extend the orthogonality relation from elements to subsets and we prove that two non-empty subsets of P are orthogonal to each other if and only if their convex hulls are orthogonal to each other within the poset of all non-empty convex subsets of P.

Adapting a proof of Bouscaren and Delon, we show that every type-definable connected group in a given stable theory of fields embeds into an algebraic group, under a condition on the definable closure. We also present general hypotheses which yield a uniform description of the definable closure in such theories of fields. The setting includes in particular the theories of separably closed fields of arbitrary degree of imperfection and differentially closed fields of arbitrary characteristic.

In this paper, we consider recurrence sequences $x_n=\xi_1 \alpha_1^n+\xi_2 \alpha_2^n$ ($n=0,1,\ldots$) with companion polynomial $P(X)$. For example, the sequence $x_n=\xi_1(4+\sqrt{2})^n+\xi_2(4-\sqrt{2})^n$ satisfies the recurrence $x_{n+2}-8x_{n+1}+14x_n=0$ and has companion polynomial $P(X)=X^2-8X+14=(X-4-\sqrt{2})(X-4+\sqrt{2})$. We call $(\xi_1,\xi_2)$ normal with respect to the recurrence relation determined by $P(X)$ when $(x_n)_{n\ge 0}$ is uniformly distributed modulo one. Determining the Borel complexity of the set of normal vectors for a fixed recurrence sequence is unresolved even for most geometric progressions. Under certain assumptions, we prove that the set of normal vectors is $\boldsymbol{\Pi}_3^0$-complete. A special case is the new result that the sets of numbers normal in base $\alpha$, i.e. $\{\xi\in \mathbb{R}\mid (\xi\alpha^n)_{n\geq 0}\mbox{ is u.d. modulo one.} \}$, are $\boldsymbol{\Pi}_3^0$-complete for every real number $\alpha$ with $|\alpha|$ Pisot. We analyze the fractional parts of recurrence sequences in terms of finite words via certain numeration systems. One of the difficulties in proving the main result is that even when recurrence sequences are uniformly distributed modulo one, it is not known what the average frequencies of the digits in the corresponding digital expansions are or if they even must exist.

Quasi-Boolean algebras were introduced as the generalization of Boolean algebras in the setting of quantum computation logic. In this paper, we investigate the completeness and congruences of quasi-Boolean algebras. First, we discuss the number of finite quasi-Boolean algebras and characterize the finite irreducible quasi-Boolean algebras. Second, we show the standard completeness of quasi-Boolean algebras. Finally, we prove that the variety of quasi-Boolean algebras satisfies the congruence extension property and provide a complete characterization of how congruences on a Boolean subalgebra can be extended to the whole quasi-Boolean algebra.

In 1901, Bouton proved that a winning strategy of the game of Nim is given by the bitwise XOR, called the nim-sum. But, why does such a weird binary operation work? Led by this question, this paper introduces a categorical reinterpretation of combinatorial games and the nim-sum. The main categorical gadget used here is recursive coalgebras, which allow us to redefine games as ``graphs on which we can conduct recursive calculation'' in a concise and precise way. For game-theorists, we provide a systematic framework to decompose an impartial game into simpler games and synthesize the quantities on them, which generalizes the nim-sum rule for the Conway addition. To read the first half of this paper, the categorical preliminaries are limited to the definitions of categories and functors. For category theorists, this paper offers a nicely behaved category of games $\mathbf{Games}$, which is a locally finitely presentable symmetric monoidal closed category comonadic over $\mathbf{Set}$ admitting a subobject classifier! As this paper has several ways to be developed, we list seven open questions in the final section.

GANs promise indistinguishability, logic explains it. We put the two on a budget: a discriminator that can only ``see'' up to a logical depth $k$, and a generator that must look correct to that bounded observer. \textbf{LOGAN} (LOGical GANs) casts the discriminator as a depth-$k$ Ehrenfeucht--Fra\"iss\'e (EF) \emph{Opponent} that searches for small, legible faults (odd cycles, nonplanar crossings, directed bridges), while the generator plays \emph{Builder}, producing samples that admit a $k$-round matching to a target theory $T$. We ship a minimal toolkit -- an EF-probe simulator and MSO-style graph checkers -- and four experiments including real neural GAN training with PyTorch. Beyond verification, we score samples with a \emph{logical loss} that mixes budgeted EF round-resilience with cheap certificate terms, enabling a practical curriculum on depth. Framework validation demonstrates $92\%$--$98\%$ property satisfaction via simulation (Exp.~3), while real neural GAN training achieves $5\%$--$14\%$ improvements on challenging properties and $98\%$ satisfaction on connectivity (matching simulation) through adversarial learning (Exp.~4). LOGAN is a compact, reproducible path toward logic-bounded generation with interpretable failures, proven effectiveness (both simulated and real training), and dials for control.

Let $T^*$ be an almost Suslin tree, that is, an Aronszajn tree with no stationary antichains. Krueger introduced a forcing axiom, $\mathrm{PFA}(T^*)$, for the class of proper forcings that preserve that $T^*$ is almost Suslin. He showed that $\mathrm{PFA}(T^*)$ implies several well-known consequences of the Proper Forcing Axiom ($\mathrm{PFA}$), including Suslin's Hypothesis and the P-ideal dichotomy. We extend this list by proving that $\mathrm{PFA}(T^*)$ also implies the Mapping Reflection Principle ($\mathrm{MRP}$) and the Open Graph Axiom ($\mathrm{OGA}$). Additionally, we show that $\mathrm{PFA}(T^*)$ implies that all special Aronszajn trees are club-isomorphic, but it does not imply that all almost Suslin trees are club-isomorphic.

We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of coalgebraic models of algebraic theories endowed with an underlying structure of cocommutative Hopf algebra, and show that these categories are semi-abelian. We call them ``categories of Omega-Hopf algebras'', since it is possible to characterize them as coalgebraic models of algebraic theories of Omega-groups.

We generalize the theory of stable canonical rules by adopting definable filtration, a generalization of the method of filtration. We show that for a modal rule system or a modal logic that admits definable filtration, each extension is axiomatizable by stable canonical rules. Moreover, we provide an algebraic presentation of Gabbay's filtration and generalize stable canonical formulas and the axiomatization results via stable canonical formulas for $\mathsf{K4}$ to pre-transitive logics $\mathsf{K4^{m+1}_{1}} = \mathsf{K} + \Diamond^{m+1} p \to \Diamond p$ $(m \geq 1)$. As consequences, we obtain the fmp of $\mathsf{K4^{m+1}_{1}}$-stable logics and a characterization of splitting and union-splitting logics in the lattice $\mathsf{NExt}\mathsf{K4^{m+1}_{1}}$. Finally, we introduce $m$-stable canonical formulas, strengthening the axiomatization results for these logics.

We present a STIT ('see to it that') logic with discrete temporal operators and deontic operators in which we can formalize and reason about legal concepts such as persistent duty and the dynamic concept of power from Hohfeld. As our main technical contribution, we show that this logic is sound and complete with respect to the semantics based on interpreted systems and is decidable.

Universal algebraic geometry is generalised from solutions of equations in a single algebra to the study of $\varphi$- or $K$-spectra, akin to the prime spectrum of a ring. We explore their basic properties and constructions, give a correspondence between certain quantifier-free propositions and closed sets in the Zariski topology of a free algebra, and show the connection with current UAG. Lastly, equationally Noetherian classes and irreducible spectra are explored.