A Formal Proof of PAC Learnability for Decision Stumps

We present a formal proof in Lean of probably approximately correct (PAC) learnability of the concept class of decision stumps. This classic result in machine learning theory derives a bound on error probabilities for a simple type of classifier. Though such a proof appears simple on paper, analytic and measure-theoretic subtleties arise when carrying it out fully formally. Our proof is structured so as to separate reasoning about deterministic properties of a learning function from proofs of measurability and analysis of probabilities.Comment: 13 pages, appeared in Certified Programs and Proofs (CPP) 202

Formal verification of higher-order probabilistic programs

Probabilistic programming provides a convenient lingua franca for writing succinct and rigorous descriptions of probabilistic models and inference tasks. Several probabilistic programming languages, including Anglican, Church or Hakaru, derive their expressiveness from a powerful combination of continuous distributions, conditioning, and higher-order functions. Although very important for practical applications, these combined features raise fundamental challenges for program semantics and verification. Several recent works offer promising answers to these challenges, but their primary focus is on semantical issues. In this paper, we take a step further and we develop a set of program logics, named PPV, for proving properties of programs written in an expressive probabilistic higher-order language with continuous distributions and operators for conditioning distributions by real-valued functions. Pleasingly, our program logics retain the comfortable reasoning style of informal proofs thanks to carefully selected axiomatizations of key results from probability theory. The versatility of our logics is illustrated through the formal verification of several intricate examples from statistics, probabilistic inference, and machine learning. We further show the expressiveness of our logics by giving sound embeddings of existing logics. In particular, we do this in a parametric way by showing how the semantics idea of (unary and relational) TT-lifting can be internalized in our logics. The soundness of PPV follows by interpreting programs and assertions in quasi-Borel spaces (QBS), a recently proposed variant of Borel spaces with a good structure for interpreting higher order probabilistic programs

Formalising Ordinal Partition Relations Using Isabelle/HOL

This is an overview of a formalisation project in the proof assistant Isabelle/HOL of a number of research results in infinitary combinatorics and set theory (more specifically in ordinal partition relations) by Erd\H{o}s--Milner, Specker, Larson and Nash-Williams, leading to Larson's proof of the unpublished result by E.C. Milner asserting that for all $m \in \mathbb{N}$, \omega^\omega\arrows(\omega^\omega, m). This material has been recently formalised by Paulson and is available on the Archive of Formal Proofs; here we discuss some of the most challenging aspects of the formalisation process. This project is also a demonstration of working with Zermelo-Fraenkel set theory in higher-order logic

Undergraduate Admissions Decisions of Selective Institutions: The Impact of Social Media Information

This causal comparative study examines the impact of decisions made by college admissions personnel at colleges and universities ranked as Highly Competitive, Highly Competitive Plus, Most Competitive, Very Competitive, and Very Competitive Plus by Barron’s Profiles of American Colleges (2018). Admissions representatives were asked to evaluate social media content of hypothetical applicants to their institution then complete a trait inference task based on the Deese-Roediger-McDermott false recognition paradigm. A total of 413 institutions were invited to participate in the online activity to establish the effect of online impression formation by admissions personnel and its impact on admissions decisions. The survey was completed by 44 institutional admissions representatives (n = 44). Admissions decisions results were then compared for effects of the treatment utilizing two one-way ANOVAs. A Welch’s t-test was then utilized to compare decisions between institutions with a self-reported policy regarding inclusion of social media in admissions decisions and those without such a policy in place. Results found significance on the false recognition paradigm, but not on admissions decisions based on the social media posts nor when institutions were classified by the presence of an institutional policy regarding its use in the admissions process. Thus, it was determined this sample of admissions personnel made spontaneous trait inferences from social media posts of hypothetical applicants. Suggestions for future research are included

Formalization of Normal Random Variables

Engineering systems often have components that exhibit random behavior. This randomness in many cases is normally distributed. To verify such systems, proba- bilistic analysis is used. Such engineering systems have applications in domains like transportation, medicine and military. Despite the safety-critical nature of these ap- plications, most of the analysis is done using informal techniques like simulation and paper-and-pencil analysis, and thus cannot be completely relied upon. The unreliable results produced by such methods may result in heavy financial loss or even the loss of a human life. To overcome the limitation of traditional methods, we propose to conduct the analysis of such systems within the trusted kernel of a higher-order-logic theorem prover HOL4. The soundness and the deduction style of the theorem prover guarantee the validity of the analysis and the results of this type of analysis are generic and valid for any instance of the system. For this purpose, we provide HOL4 formalization of Lebesgue measure and normal random variables along with the proof of their classical properties. We also ported the theory of Gauge integral and other required foundational concepts from HOL Light and Isabelle/HOL theorem provers. To illustrate the usefulness of our formalization, we conducted the formal analysis of two applications, i.e., error probability of binary transmission in the presence of Gaussian noise and probabilistic clock synchronization in wireless sensor networks

Scaling Up Automated Verification: A Case Study and a Formalization IDE for Building High Integrity Software

Component-based software verification is a difficult challenge because developers must specify components formally and annotate implementations with suitable assertions that are amenable to automation. This research investigates the intrinsic complexity in this challenge using a component-based case study. Simultaneously, this work also seeks to minimize the extrinsic complexities of this challenge through the development and usage of a formalization integrated development environment (F-IDE) built for specifying, developing, and using verified reusable software components. The first contribution is an F-IDE built to support formal specification and automated verification of object-based software for the integrated specification and programming language RESOLVE. The F-IDE is novel, as it integrates a verifying compiler with a user-friendly interface that provides a number of amenities including responsive editing for model-based mathematical contracts and code, assistance for design by contract, verification, responsive error handling, and generation of property-preserving Java code that can be run within the F-IDE. The second contribution is a case study built using the F-IDE that involves an interplay of multiple artifacts encompassing mathematical units, component interfaces, and realizations. The object-based interfaces involved are specified in terms of new mathematical models and non-trivial theories designed to encapsulate data structures and algorithms. The components are designed to be amenable to modular verification and analysis