arXiv subjects

Dan Guralnik

Publications and source records attributed to Dan Guralnik.

6 recordsLinked to original sources

arXiv2026

Quantifying Trade-Offs Between Stability and Goal-Obfuscation

Safety-critical autonomy in adversarial settings demands more than Lyapunov stability of tracking error signals. An agent executing a goal-directed trajectory is intrinsically legible to a passive observer running online Bayesian inference, because the contractive dynamics of any Lyapunov basin of attraction concentrates posterior belief over the latent intent parameters. We initiates the study of intent privacy over a continuous state space as a joint control problem on the physical state combined with the latent belief state of a putative observer. With the main challenges concentrated around the analysis of the belief-state dynamics, the agent dynamics is assumed to be simple, modeled by the differential inclusion $\dot{x}\in u+\bar{d}\mathbb{B}$. That is, the agent is fully actuated with bounded unknown disturbance to the control input. The observer's intent inference process is modeled as a discrete-time stochastic dynamical system evolving over the belief state space of a Rao Blackwellized particle filter reasoning over large random samples of possible agent goals. The agent's control input is modeled as a piecewise constant signal, with jumps matching the RBPF update times. Building on a prior intent-inference framework and its KL-based information leakage measurement, a privacy constraint is imposed, which amounts to maintaining information leakage above a prescribed threshold with high probability, using probabilistic discrete-time control barrier functions. A key technical contribution is the derivation of separate PCBF results for the Bayesian update step and the resampling step of the RBPF, enabling a PCBF result for the full update as well as integration of the privacy constraint with the agent's task-side tracking requirement. Finally, a joint feasibility analysis is carried out by examining the interplay between the privacy constraint and the tracking envelope.

eess.SY↗

arXiv2011

A `transversal' for minimal invariant sets in the boundary of a CAT(0) group

We introduce new techniques for studying boundary dynamics of CAT(0) groups. For a group $G$ acting geometrically on a CAT(0) space $X$ we show there is a flat $F\subset X$ of maximal dimension whose boundary sphere intersects every minimal $G$-invariant subset of $\partial_\infty X$. As a result we derive a necessary and sufficient dynamical condition for $G$ to be virtually-Abelian, as well as a new approach to Ballmann's rank rigidity conjecture.

math.GR↗

arXiv2011

Dehn functions and finiteness properties of subgroups of perturbed right-angled Artin groups

We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type $\mathrm{FP}_3$, and have exponential, or polynomial Dehn functions of prescribed degree.

math.GR↗

arXiv2010

A Formal Approach to Modeling the Memory of a Living Organism

We consider a living organism as an observer of the evolution of its environment recording sensory information about the state space X of the environment in real time. Sensory information is sampled and then processed on two levels. On the biological level, the organism serves as an evaluation mechanism of the subjective relevance of the incoming data to the observer: the observer assigns excitation values to events in X it could recognize using its sensory equipment. On the algorithmic level, sensory input is used for updating a database, the memory of the observer whose purpose is to serve as a geometric/combinatorial model of X, whose nodes are weighted by the excitation values produced by the evaluation mechanism. These values serve as a guidance system for deciding how the database should transform as observation data mounts. We define a searching problem for the proposed model and discuss the model's flexibility and its computational efficiency, as well as the possibility of implementing it as a dynamic network of neuron-like units. We show how various easily observable properties of the human memory and thought process can be explained within the framework of this model. These include: reasoning (with efficiency bounds), errors, temporary and permanent loss of information. We are also able to define general learning problems in terms of the new model, such as the language acquisition problem.

cs.AI↗

arXiv2007

Local finiteness for cubulations of CAT(0) groups

Let X be a proper CAT(0) space. A halfspace system (or cubulation) of X is a set H of open halfspaces closed under closure-complementation and such that every point in X has a neighbourhood intersecting only finitely many walls of H. Given a cubulation H, one uses the Sageev-Roller construction to form a cubing C(H). One setting in which cubulations were studied is that of a Coxeter group (W,R) acting on its Davis-Moussong complex, with elements of H being the halfspaces defined by reflections. For this setting, Niblo and Reeves had shown that C(H) is a finite-dimensional, locally-finite cubing. Their proof explicitly uses the `parallel walls property' of Coxeter groups, proved by Brink and Howlett, and heavily relies on meticulous calculations with the root system associated with (W,R). We generalize their local finiteness result using the visual boundary of X, endowed with the cone topology. We introduce an asymptotic condition on H (`uniformness'), and show it is equivalent to H having the parallel walls property together with boundedness of chambers. Uniformness regards the way in which boundary points are approximated by the walls of H. We prove that if G is a group acting geometrically on a CAT(0) space X and H is a uniform cubulation of X invariant under G, then C(H) is locally-finite. We also show that the obvious map of $X$ into $C(H)$ is a quasi-isometry, and use this to give quantitative results strengthening the parallel walls property.

math.GR↗

arXiv2007

Coarse decompositions of boundaries for CAT(0) groups

In this work we introduce a new combinatorial notion of boundary $\Re C$ of an $ω$-dimensional cubing $C$. $\Re C$ is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of $C$, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When $C$ arises as the dual of a cubulation -- or discrete system of halfspaces -- $\HH$ of a CAT(0) space $X$ (for example, the Niblo-Reeves cubulation of the Davis-Moussong complex of a finite rank Coxeter group), we show how $\HH$ induces a function $ρ:\bd X\to\Re C$. We develop a notion of uniformness for $\HH$, generalizing the parallel walls property enjoyed by Coxeter groups, and show that, if the pair $(X,\HH)$ admits a geometric action by a group $G$, then the fibers of $ρ$ form a stratification of $\bd X$ graded by the order structure of $\Re C$. We also show how this structure computes the components of the Tits boundary of $X$. Finally, using our result from another paper, that the uniformness of a cubulation as above implies the local finiteness of $C$, we give a condition for the co-compactness of the action of $G$ on $C$ in terms of $ρ$, generalizing a result of Williams, previously known only for Coxeter groups.

math.GR↗