Papers by Vadim Weinstein
arXiv (Cornell University), Feb 28, 2024
This paper formally defines a robot system, including its sensing and actuation components, as a ... more This paper formally defines a robot system, including its sensing and actuation components, as a general, topological dynamical system. The focus is on determining general conditions under which various environments in which the robot can be placed are indistinguishable. A key result is that, under very general conditions, covering maps witness such indistinguishability. This formalizes the intuition behind the well studied loop closure problem in robotics. An important special case is where the sensor mapping reports an invariant of the local topological (metric) structure of an environment because such structure is preserved by (metric) covering maps. Whereas coverings provide a sufficient condition for the equivalence of environments, we also give a necessary condition using bisimulation. The overall framework is applied to unify previously identified phenomena in robotics and related fields, in which moving agents with sensors must make inferences about their environments based on limited data. Many open problems are identified.
arXiv (Cornell University), Dec 1, 2023
Given a nonempty set of linear orders, we say that the linear order is convex embeddable into ... more Given a nonempty set of linear orders, we say that the linear order is convex embeddable into the linear order ′ if it is possible to partition into convex sets indexed by some element of which are isomorphic to convex subsets of ′ ordered in the same way. This notion generalizes convex embeddability and (finite) piecewise convex embeddability (both studied in [IMMRW23]), which are the special cases = { } and = . We focus mainly on the behavior of these relations on the set of countable linear orders, first characterizing when they are transitive, and hence a quasi-order. We then study these quasi-orders from a combinatorial point of view, and analyze their complexity with respect to Borel reducibility. Finally, we extend our analysis to uncountable linear orders. 1. Introduction 1 2. Preliminaries 4 3. The ccs property 6 4. Some technical lemmas 9 5. Combinatorial properties of ⊴ 12 6. Borel complexity of ⊴ and ⋈ 18 7. Examples of ccs families of countable linear orders 21 8. Uncountable linear orders 27 9. Open problems 32 References 33
arXiv (Cornell University), Sep 17, 2023
We consider countable linear orders and study the quasi-order of convex embeddability and its ind... more We consider countable linear orders and study the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results, and further extend our research to the case of circular orders. These results are then applied to the study of arcs and knots, establishing combinatorial properties and lower bounds (in terms of Borel reducibility) for the complexity of some natural relations between these geometrical objects.
arXiv (Cornell University), Aug 16, 2023
This paper addresses the lower limits of encoding and processing the information acquired through... more This paper addresses the lower limits of encoding and processing the information acquired through interactions between an internal system (robot algorithms or software) and an external system (robot body and its environment) in terms of action and observation histories. Both are modeled as transition systems. We want to know the weakest internal system that is sufficient for achieving passive (filtering) and active (planning) tasks. We introduce the notion of an information transition system (ITS) for the internal system which is a transition system over a space of information states that reflect a robot's or other observer's perspective based on limited sensing, memory, computation, and actuation. An ITS is viewed as a filter and a policy or plan is viewed as a function that labels the states of this ITS. Regardless of whether internal systems are obtained by learning algorithms, planning algorithms, or human insight, we want to know the limits of feasibility for given robot hardware and tasks. We establish, in a general setting, that minimal information transition systems (ITSs) exist up to reasonable equivalence assumptions, and are unique under some general conditions. We then apply the theory to generate new insights into several problems, including optimal sensor fusion/filtering, solving basic planning tasks, and finding minimal representations for modeling a system given input-output relations.
Annual Review of Control, Robotics, and Autonomous Systems
This article makes the case that a powerful new discipline, which we term perception engineering,... more This article makes the case that a powerful new discipline, which we term perception engineering, is steadily emerging. It follows from a progression of ideas that involve creating illusions, from historical paintings and film to modern video games and virtual reality. Rather than creating physical artifacts such as bridges, airplanes, or computers, perception engineers create illusory perceptual experiences. The scope is defined over any agent that interacts with the physical world, including both biological organisms (humans and animals) and engineered systems (robots and autonomous systems). The key idea is that an agent, called a producer, alters the environment with the intent to alter the perceptual experience of another agent, called a receiver. Most importantly, the article introduces a precise mathematical formulation of this process, based on the von Neumann–Morgenstern notion of information, to help scope and define the discipline. This formulation is then applied to the ...
Frontiers in Neurorobotics
In this paper we start from the philosophical position in cognitive science known as enactivism. ... more In this paper we start from the philosophical position in cognitive science known as enactivism. We formulate five basic enactivist tenets that we have carefully identified in the relevant literature as the main underlying principles of that philosophy. We then develop a mathematical framework to talk about cognitive systems (both artificial and natural) which complies with these enactivist tenets. In particular we pay attention that our mathematical modeling does not attribute contentful symbolic representations to the agents, and that the agent's nervous system or brain, body and environment are modeled in a way that makes them an inseparable part of a greater totality. The long-term purpose for which this article sets the stage is to create a mathematical foundation for cognition which is in line with enactivism. We see two main benefits of doing so: (1) It enables enactivist ideas to be more accessible for computer scientists, AI researchers, roboticists, cognitive scientist...
The International Journal of Robotics Research, Sep 18, 2023
The Journal of Symbolic Logic, 2015
We prove results that falsify Silver’s dichotomy for Borel equivalence relations on the generaliz... more We prove results that falsify Silver’s dichotomy for Borel equivalence relations on the generalized Baire space under the assumption V = L.
Mathematical Logic Quarterly, 2015
ABSTRACT Working with uncountable structures of fixed cardinality, we investigate the complexity ... more ABSTRACT Working with uncountable structures of fixed cardinality, we investigate the complexity of certain equivalence relations and show that if V=L, then many of them are Sigma(1)(1)-complete, in particular the isomorphism relation of dense linear orders. Then we show that it is undecidable in ZFC whether or not the isomorphism relation of a certain well behaved theory (stable, NDOP, NOTOP) is Sigma(1)(1)-complete (it is, if V=L, but can be forced not to be). (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Fundamenta Mathematicae, 2015
It is shown that the power set of $\k$ ordered by the subset relation modulo various versions of ... more It is shown that the power set of $\k$ ordered by the subset relation modulo various versions of the non-stationary deal can be embedded into the partial order of Borel equivalence relations on $2^\k$ under Borel reducibility. Here $\k$ is uncountable regular cardinal with $\k^{<\k}=\k$.
We start by giving a survey to the theory of Borel*(\kappa) sets in the generalized Baire space B... more We start by giving a survey to the theory of Borel*(\kappa) sets in the generalized Baire space Baire({\kappa}) = {\kappa}^{\kappa}. In particular we look at the relation of this complexity class to other complexity classes which we denote by Borel({\kappa}), \Delta^1_1({\kappa}) and {\Sigma}^1_1({\kappa}) and the connections between Borel*(\kappa)-sets and the infinitely deep language M_{{\kappa}^+{\kappa}}. In the end of the paper we prove the consistency of Borel*(\kappa) \ne {\Sigma}^1_1({\kappa}).
Descriptive set theory is mainly concerned with studying subsets of the space of all countable bi... more Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized Baire and Cantor spaces, equivalence relations and their Borel reducibility. The study shows that the descriptive set theory looks very different in this generalized setting compared to the classical, countable case. We also draw the connection between the stability theoretic complexity of first-order theories and the descriptive set theoretic complexity of their isomorphism relations. Our results suggest that Borel reducibility on uncountable structures is a model theoretically natural way to compare the complexity of isomorphism relations.
We investigate regularity properties derived from tree-like forcing notions in the setting of &qu... more We investigate regularity properties derived from tree-like forcing notions in the setting of "generalized descriptive set theory", i.e., descriptive set theory on $\kappa^\kappa$ and $2^\kappa$, for regular uncountable cardinals $\kappa$.
We prove results that falsify Silver's dichotomy for Borel equivalence relations on the gener... more We prove results that falsify Silver's dichotomy for Borel equivalence relations on the generalised Baire space under the assumption V=L.
In this paper we study the Borel reducibility of Borel equivalence relations, including some orbi... more In this paper we study the Borel reducibility of Borel equivalence relations, including some orbit equivalence relations, on the generalised Baire space $\kappa^\kappa$ for an uncountable $\kappa$ with the property $\kappa^{<\kappa}=\kappa$. The theory looks quite different from its classical counterpart where $\kappa=\omega$, although some basic theorems do generalise.
Transactions of the American Mathematical Society, 2011
In this paper we define a game which is played between two players I and II and two mathematical ... more In this paper we define a game which is played between two players I and II and two mathematical structures A and B. The players choose elements from both structures in α moves, and at the end of the game player II wins if the chosen structures are isomorphic. Thus the difference between this and the ordinary Ehrenfeucht-Fraïssé game is that the isomorphism can be arbitrary, whereas in the ordinary EF-game it is determined by the moves of the players. We investigate determinacy of the weak EF-game for different α (the length of the game) and its relation to the ordinary EF-game.
The Journal of Symbolic Logic, 2013
It is shown that the power set of κ ordered by the subset relation modulo various versions of the... more It is shown that the power set of κ ordered by the subset relation modulo various versions of the non-stationary ideal can be embedded into the partial order of Borel equivalence relations on 2κ under Borel reducibility. Here κ is an uncountable regular cardinal with κ<κ = κ.
We start by giving a survey to the theory of Borel * (κ) sets in the generalised Baire space Bair... more We start by giving a survey to the theory of Borel * (κ) sets in the generalised Baire space Baire(κ) = κ κ . In particular we look at the relation of this complexity class to other complexity classes which we denote by Borel(κ), ∆ 1 1 (κ) and Σ 1 1 (κ) and the connections between Borel * (κ) sets and the infinitely deep language M κ + κ . In the end of the paper we will prove the consistency of Borel * (κ) = Σ 1 1 (κ).
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Papers by Vadim Weinstein