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A positive integer $n$ is defined to be cyclic if and only if every group of size $n$ is cyclic. Equivalently, $n$ is cyclic if and only if $n$ is relatively prime to the number of positive integers less than $n$ that are relatively prime to $n$. Because every prime number is cyclic, it is natural to ask whether a (proved or conjectured) property of primes extends to cyclic numbers. I review proved or conjectured properties of primes (including some new conjectures about primes) and propose analogous conjectures about cyclic numbers. Using the 28,488,167 cyclic numbers less than $10^8$, I test the conjectures about cyclic numbers and disprove the cyclic analog of the second conjecture about primes of Hardy and Littlewood. Proofs or disproofs of the remaining conjectures are invited.
In experimental applications of bounded-reasoning models, behavior is often summarized by distributions of "levels". We argue that such summaries conflate two conceptually distinct dimensions: a player's type, capturing beliefs about what types their opponents might be, and the depth of higher-order reasoning about rationality. Distinguishing these dimensions matters for interpreting experimental evidence and for understanding when cross-environment variation should be read as changes in beliefs versus changes in cognitive depth, but existing frameworks provide no language to do so. We develop a unified framework by "lifting" static complete-information games into incomplete-information versions in which players are explicitly uncertain about opponents' types. Within this framework, bounded reasoning about opponents' types is represented by transparent first-order belief restrictions, while (higher-order) reasoning depth is captured by bounds on belief in rationality. We analyze three benchmark instances: downward rationalizability, a robust baseline, and two refinements, $\mathsf{L}$-rationalizability and $\mathsf{C}$-rationalizability, which provide epistemic foundations -- with
Increasing complexity in software systems places a growing demand on reasoning tools that unlock vulnerabilities manifest in source code. Many current approaches focus on vulnerability analysis as a classifying task, oversimplifying the nuanced and context-dependent real-world scenarios. Even though current code large language models (LLMs) excel in code understanding, they often pay little attention to security-specific reasoning. We propose LLaVul, a multimodal LLM tailored to provide fine-grained reasoning about code through question-answering (QA). Our model is trained to integrate paired code and natural queries into a unified space, enhancing reasoning and context-dependent insights about code vulnerability. To evaluate our model performance, we construct a curated dataset of real-world vulnerabilities paired with security-focused questions and answers. Our model outperforms state-of-the-art general-purpose and code LLMs in the QA and detection tasks. We further explain decision-making by conducting qualitative analysis to highlight capabilities and limitations. By integrating code and QA, LLaVul enables more interpretable and security-focused code understanding.
We study a sequential social learning model in which there is uncertainty about the informativeness of a common signal-generating process. Rational agents arrive in order and make decisions based on the past actions of others and their private signals. We show that, in this setting, asymptotic learning about informativeness is not guaranteed and depends crucially on the relative tail distributions of the private beliefs induced by uninformative and informative signals. We identify the phenomenon of perpetual disagreement as the cause of learning and characterize learning in the canonical Gaussian environment.
Learning about the relationship between distance to landmarks and events and phenomena of interest is a multi-faceted problem, as it may require taking into account multiple dimensions, including: spatial position of landmarks, timing of events taking place over time, and attributes of occurrences and locations. Here I show that tree-based methods are well suited for the study of these questions as they allow exploring the relationship between proximity metrics and outcomes of interest in a non-parametric and data-driven manner. I illustrate the usefulness of tree-based methods vis-à-vis conventional regression methods by examining the association between: (i) distance to border crossings along the US-Mexico border and support for immigration reform, and (ii) distance to mass shootings and support for gun control.
This paper formulates a conjectural description of of the space of weightless functions (see\cite{BK}) and raises a question about a possibility of extending such a description in a more general context.
This paper presents a little reflection about the Sleeping Beauty Problem, maybe contributing to shed light on it and perhaps helping to find a simple and elegant solution that could definitively resolve the controversies about it.
It is often useful, if not necessary, to reason about the syntactic structure of an expression in an interpreted language (i.e., a language with a semantics). This paper introduces a mathematical structure called a syntax framework that is intended to be an abstract model of a system for reasoning about the syntax of an interpreted language. Like many concrete systems for reasoning about syntax, a syntax framework contains a mapping of expressions in the interpreted language to syntactic values that represent the syntactic structures of the expressions; a language for reasoning about the syntactic values; a mechanism called quotation to refer to the syntactic value of an expression; and a mechanism called evaluation to refer to the value of the expression represented by a syntactic value. A syntax framework provides a basis for integrating reasoning about the syntax of the expressions with reasoning about what the expressions mean. The notion of a syntax framework is used to discuss how quotation and evaluation can be built into a language and to define what quasiquotation is. Several examples of syntax frameworks are presented.
Much of the theoretical work on strategic voting makes strong assumptions about what voters know about the voting situation. A strategizing voter is typically assumed to know how other voters will vote and to know the rules of the voting method. A growing body of literature explores strategic voting when there is uncertainty about how others will vote. In this paper, we study strategic voting when there is uncertainty about the voting method. We introduce three notions of manipulability for a set of voting methods: sure, safe, and expected manipulability. With the help of a computer program, we identify voting scenarios in which uncertainty about the voting method may reduce or even eliminate a voter's incentive to misrepresent her preferences. Thus, it may be in the interest of an election designer who wishes to reduce strategic voting to leave voters uncertain about which of several reasonable voting methods will be used to determine the winners of an election.
Social media (i.e., Reddit) users are overloaded with people's opinions when viewing discourses about divisive topics. Traditional user interfaces in such media present those opinions in a linear structure, which can limit users in viewing diverse social opinions at scale. Prior work has recognized this limitation, that the linear structure can reinforce biases, where a certain point of view becomes widespread simply because many viewers seem to believe it. This limitation can make it difficult for users to have a truly conversational mode of mediated discussion. Thus, when designing a user interface for viewing people's opinions, we should consider ways to mitigate selective exposure to information and polarization of opinions. We conducted a needs-finding study with 11 Reddit users, who follow climate change threads and make posts and comments regularly. In the study, we aimed to understand key limitations in people viewing online controversial discourses and to extract design implications to address these problems. Our findings discuss potential future directions to address these problems.
Since the 1970's the debate about the rising importance of transnational relations has existed in international relations. Apart from states, related research also focuses on other actors, including epistemic communities. The article uses the concept of epistemic communities and finds whether the activity of epistemic communities determines the process of the international management of outer space in the case of the political negotiations relating to space debris in UNCOPUOS and UNOOSA. The activity of epistemic communities exists in the political negotiations relating to space debris in UNCOPUOS and UNOOSA, but it has not been reflected in the related scholarly literature. Epistemic communities from the non-governmental organizations IAF, COSPAR and IISL contributed to setting the space debris problem on the agenda of UNCOPUOS. Also, under the influence of epistemic communities from the governmental organization IADC, UNCOPUOS adopted guidelines preventing the creation of further amounts of space debris.
We examine the long-term behavior of a Bayesian agent who has a misspecified belief about the time lag between actions and feedback, and learns about the payoff consequences of his actions over time. Misspecified beliefs about time lags result in attribution errors, which have no long-term effect when the agent's action converges, but can lead to arbitrarily large long-term inefficiencies when his action cycles. Our proof uses concentration inequalities to bound the frequency of action switches, which are useful to study learning problems with history dependence. We apply our methods to study a policy choice game between a policy-maker who has a correctly specified belief about the time lag and the public who has a misspecified belief.
If an experimental treatment is experienced by both treated and control group units, tests of hypotheses about causal effects may be difficult to conceptualize let alone execute. In this paper, we show how counterfactual causal models may be written and tested when theories suggest spillover or other network-based interference among experimental units. We show that the "no interference" assumption need not constrain scholars who have interesting questions about interference. We offer researchers the ability to model theories about how treatment given to some units may come to influence outcomes for other units. We further show how to test hypotheses about these causal effects, and we provide tools to enable researchers to assess the operating characteristics of their tests given their own models, designs, test statistics, and data. The conceptual and methodological framework we develop here is particularly applicable to social networks, but may be usefully deployed whenever a researcher wonders about interference between units. Interference between units need not be an untestable assumption; instead, interference is an opportunity to ask meaningful questions about theoretically int
A source of much difficulty and confusion in the interpretation of quantum mechanics is a ``naive realism about operators.'' By this we refer to various ways of taking too seriously the notion of operator-as-observable, and in particular to the all too casual talk about ``measuring operators'' that occurs when the subject is quantum mechanics. Without a specification of what should be meant by ``measuring'' a quantum observable, such an expression can have no clear meaning. A definite specification is provided by Bohmian mechanics, a theory that emerges from Schrodinger's equation for a system of particles when we merely insist that ``particles'' means particles. Bohmian mechanics clarifies the status and the role of operators as observables in quantum mechanics by providing the operational details absent from standard quantum mechanics. It thereby allows us to readily dismiss all the radical claims traditionally enveloping the transition from the classical to the quantum realm---for example, that we must abandon classical logic or classical probability. The moral is rather simple: Beware naive realism, especially about operators!
There is a significant body of literature, which includes Itamar Pitowksy's "Betting on Outcomes of Measurements," that sheds light on the structure of quantum mechanics, and the ways in which it differs from classical mechanics, by casting the theory in terms of agents' bets on the outcomes of experiments. Though this approach, by itself, is neutral as to the ontological status of quantum observables and quantum states, some, notably those who adopt the label "QBism" for their views, take this approach as providing incentive to conclude that quantum states represent nothing in physical reality, but, rather, merely encode an agent's beliefs. In this chapter, I will argue that the arguments for realism about quantum states go through when the probabilities involved are taken to be subjective, if the conclusion is about the agent's beliefs: an agent whose credences conform to quantum probabilities should believe that preparation procedures with which she associates distinct pure quantum states produce distinct states of reality. The conclusion can be avoided only by stipulation of limitations on the agent's theorizing about the world, limitations that are not warranted by the empiric
This paper forms part of a wider campaign: to deny pointillisme. That is the doctrine that a physical theory's fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there; so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts. More specifically, this paper argues against pointillisme about the structure of space and-or spacetime itself, especially a paper by Bricker (1993). A companion paper argues against pointillisme in mechanics, especially about velocity; it focusses on Tooley, Robinson and Lewis. To avoid technicalities, I conduct the argument almost entirely in the context of ``Newtonian'' ideas about space and time. But both the debate and my arguments carry over to relativistic, and even quantum, physics.
As phenomena that necessarily emerge from the collective behavior of interacting particles, phase transitions continue to be difficult to predict using statistical thermodynamics. A recent proposal called the topological hypothesis suggests that the existence of a phase transition could perhaps be inferred from changes to the topology of the accessible part of the configuration space. This paper instead suggests that such a topological change is often associated with a dramatic change in the configuration space geometry, and that the geometric change is the actual driver of the phase transition. More precisely, a geometric change that brings about a discontinuity in the mixing time required for an initial probability distribution on the configuration space to reach steady-state is conjectured to be related to the onset of a phase transition in the thermodynamic limit. This conjecture is tested by evaluating the diffusion diameter and $ε$-mixing time of the configuration spaces of hard disk and hard sphere systems of increasing size. Explicit geometries are constructed for the configuration spaces of these systems, and numerical evidence suggests that a discontinuity in the $ε$-mixi
Rapid progress in machine learning for natural language processing has the potential to transform debates about how humans learn language. However, the learning environments and biases of current artificial learners and humans diverge in ways that weaken the impact of the evidence obtained from learning simulations. For example, today's most effective neural language models are trained on roughly one thousand times the amount of linguistic data available to a typical child. To increase the relevance of learnability results from computational models, we need to train model learners without significant advantages over humans. If an appropriate model successfully acquires some target linguistic knowledge, it can provide a proof of concept that the target is learnable in a hypothesized human learning scenario. Plausible model learners will enable us to carry out experimental manipulations to make causal inferences about variables in the learning environment, and to rigorously test poverty-of-the-stimulus-style claims arguing for innate linguistic knowledge in humans on the basis of speculations about learnability. Comparable experiments will never be possible with human subjects due to
Michael Perryman has interviewed some of the scientists and project leaders in the Hipparcos and Gaia missions, the interviews with photos of the persons are given at his site: https://www.michaelperryman.co.uk . Michael has also written essays -- 84 to date ! -- about results from the Gaia mission and they are placed at his site. Three of the interviews are with me and transcriptions, co-authored with Michael, are provided below with the titles: #1. An interview about astronomy and astrometry up to 1980. #2. An interview about the revival of astrometry after 1980. #3. The billion-star astrometry after 1990. The third interview begins in 1990 when I had the first ideas for a Hipparcos successor. In 1992 I made a detailed design with direct imaging on CCD detectors in a satellite proposal called Roemer. In 1993 a supposedly better option was proposed with the acronym GAIA where the capital "I" stood for Interferometer. In 1998, however, interferometry was shown to be unsuited for the purpose and we returned to the original idea from 1992 for the further development. The name was later changed to Gaia -- for the sake of continuity.
This paper forms part of a wider campaign: to deny pointillisme. That is the doctrine that a physical theory's fundamental quantities are defined at points of space or of spacetime, and represent intrinsic properties of such points or point-sized objects located there; so that properties of spatial or spatiotemporal regions and their material contents are determined by the point-by-point facts. More specifically, this paper argues against pointillisme about the concept of velocity in classical mechanics; especially against proposals by Tooley, Robinson and Lewis. A companion paper argues against pointillisme about (chrono)-geometry, as proposed by Bricker. To avoid technicalities, I conduct the argument almost entirely in the context of ``Newtonian'' ideas about space and time, and the classical mechanics of point-particles, i.e. extensionless particles moving in a void. But both the debate and my arguments carry over to relativistic physics.