The behavior of self-driving cars must be compatible with an enormous set of conflicting and ambiguous objectives, from law, from ethics, from the local culture, and so on. This paper describes a new way to conveniently define the desired behavior for autonomous agents, which we use on the self-driving cars developed at nuTonomy, an Aptiv company.We define a "rulebook" as a pre-ordered set of "rules", each akin to a violation metric on the possible outcomes ("realizations"). The rules are ordered by priority. The semantics of a rulebook imposes a pre-order on the set of realizations.We study the compositional properties of the rulebooks, and we derive which operations we can allow on the rulebooks to preserve previously-introduced constraints.While we demonstrate the application of these techniques in the self-driving domain, the methods are domain-independent.
We give a construction of two-sided invariant metrics on free products (possibly with amalgamation) of groups with two-sided invariant metrics and, under certain conditions, on HNN extensions of such groups. Our approach is similar to the Graev's construction of metrics on free groups over pointed metric spaces.
The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue Orbit Equivalence, by which we understand an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non smooth R d -flows are shown to be Lebesgue Orbit Equivalence if and only if they admit the same number of invariant ergodic probability measures. arXiv:1504.00958v2 [math.DS] 7 Sep 2015 LEBESGUE ORBIT EQUIVALENCE OF MULTIDIMENSIONAL BOREL FLOWS 2A time-change equivalence between free actions R X and R Y is an OE φ : X → Y such that for each x ∈ X the function f ( · , x) : R → R is a homeomorphism 4 . This is a substantial strengthening of the notion of orbit equivalence. Nevertheless, as proved by B. D. Miller and C. Rosendal [MR10], in the Descriptive Set Theoretic set up the world of free R-flows collapses with respect to time-change equivalence.Theorem (Miller-Rosendal). Any two non smooth free R-flows are time-change equivalent.The difference between continuous and discrete worlds lies in the fact that continuous groups have a lot more non trivial structures on them. The obvious one is topology. Whenever we have a free action G X and an orbit O ⊆ X, we may transfer the topology from G onto O using the correspondence G g → gx ∈ O for any chosen x ∈ O. If groups G and H are discrete, any OE between their free actions respects the topology: restricted onto any orbit O ⊆ X, φ : O → φ(O) is a homeomorphism. When the topology on O is not discrete, the map φ : O → φ(O) has no reasons to preserve the topology, and when this is imposed as an additional assumption on φ, one recovers the concept of time-change equivalence.The structure possessed by all locally compact groups which is responsible for the failure of DJK classification is Haar measure. Being invariant, it can also be transferred 5 onto any orbit of a free action G X. Again, if G and H are discrete (and if one takes counting Haar measures), any OE map φ : X → Y restricts to a measure preserving isomorphism between orbits. When G and H are continuous, this may no longer be the case. This turns out to be an obstacle for cardinality of the set of pie measures to be an invariant of the OE between non discrete locally compact group actions.1.2. Lebesgue Orbit Equivalence. The paper is concerned mainly with free actions of Euclidean spaces R d X on standard Borel spaces. Two such actions R d X and R d Y are Lebesgue Orbit Equivalent (LOE) if there exists an OE φ : X → Y which preserves the Lebesgue measure on each orbit 6 . In Ergodic Theoretical set up, i.e., when X and Y are endowed with probability invariant measures and the map φ needs to be defined almost everywhere, this notion seems to have appeared for the first time in the work of U. Krengel [Kre76], where the following theorem was proved.Theorem (Krengel). Free ergodic flows R X and R Y are always LOE.Still withing the framework of Ergodic Theory, this has later been generalized by D. Rudolph [Rud79] to free actions of R d . In fact, Rudolph proved a much stronger result. Namely for d ≥...
ABSTRACT. We show that every two-dimensional class of topological similarity, and hence every diagonal conjugacy class of pairs, is meager in the group of order preserving bijections of the rationals and in the group of automorphisms of the ordered rational Urysohn space.
Abstract. A homomorphism from a completely metrizable topological group into a free product of groups whose image is not contained in a factor of the free product is shown to be continuous with respect to the discrete topology on the range. In particular, any completely metrizable group topology on a free product is discrete.
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