proof assistants

Mathematician Kevin Buzzard explains why he believes the Lean theorem prover will shape the future of mathematics, and describes digitizing his university curriculum in Lean with undergraduates, plus the cultural challenges.

Vladimir Voevodsky's IAS lecture starts from Godel's second incompleteness theorem to explore whether first-order arithmetic could be inconsistent. It motivates his vision of univalent foundations and computer-checked proofs.

The first lecture of MIT 6.042J Mathematics for Computer Science, taught by Tom Leighton, introducing mathematical proofs. It covers propositions, implication and basic proof methods.

David Christiansen gives a Strange Loop 2018 introduction to dependent types, where one language serves for both compile-time and run-time code and programs unite with proofs. It draws on his book The Little Typer.

A short South Park clip from the 'Post COVID: The Return of COVID' special in which Stan relies on his Alexa. It plays on voice assistants responding to their wake word.

Veritasium's explainer on Godel's incompleteness theorems and the limits of mathematical proof, tracing set theory, Hilbert's program and the road to Turing and modern computing.

WIRED has UCLA professor Amit Sahai explain zero-knowledge proofs to a child, a teen, a college student, a grad student and an expert. Each round goes deeper into the cryptography and why it matters.

A Saturday Night Live sketch for the Amazon Echo Silver, a smart speaker for the elderly, in which older users misunderstand Alexa. It parodies voice assistants.

Aditya Siram introduces ATS, a statically typed ML-like language that compiles to C and offers type-safe pointer arithmetic and a full proof language. The Strange Loop talk shows off C interop, concurrency and verification features with non-trivial examples.

Andrej Bauer's IAS talk introduces constructive mathematics and what it feels like to do mathematics without excluded middle. It is framed as a tour for classical mathematicians.

Gernot Heiser presents seL4, a microkernel with a mathematical proof connecting security properties to the binary code, and explains what its open-sourcing means for users. It covers its roles as a microkernel, separation kernel and hypervisor.

3Blue1Brown explains the math and mechanics behind Bitcoin visually: ledgers, digital signatures, proof of work and the blockchain, building up a decentralized payment system from first principles.

A Google Tech Talk by Steve Weis that introduces modern cryptography and its use in practice and at Google. It covers proofs of security and security definitions and a special topic. HN commenters treat it as a talk on verifiable electronic voting.

Mark Rober builds a machine that plays the Cyclone arcade game with perfect timing and discovers the game is rigged so that perfect play still loses. The video explains the physics and electronics tricks behind carnival-style skill games.

Engineering Explained follows up on Tesla's claim that a Cybertruck towing a Porsche 911 could beat the 911 alone over a quarter mile. He walks through the MotorTrend drag race and explains why the claim did not hold up.

A two-minute clip from Silicon Valley season 1, episode 7 featuring the human heater idea that is rejected in a startup competition scene.

Apple's 1987 Knowledge Navigator concept video imagines a conversational voice assistant, multi-touch tablet and video calling used by a professor. It was made years before the graphical web existed.

Louis Rossmann reviews the Lenovo ThinkPad P50, covering build, thermals and throttling, then spills liquid on it to test the keyboard drainage and spill protection.

Microsoft's Brian Beckman gives an hour-long, whiteboard-style walk through monads, building from function composition and monoids using school-level math. It is aimed at practicing programmers rather than category theorists.

Meredith Patterson and Sergey Bratus argue that networked software stays insecure because protocols and inputs are too complex to recognize safely. They propose using formal language theory and grammars to build secure, verifiable input handling.

Philip Wadler's Strange Loop talk on Propositions as Types, the link between logic and computation (the Curry-Howard correspondence). He traces its history through Church, Godel and Turing and shows how it shaped theorem provers and programming languages.

Vsauce's roughly 24 minute explainer on the Banach-Tarski paradox, building from infinity and set theory to the decomposition of a sphere into pieces that form two spheres.

Leslie Lamport argues that programmers, like architects, should write rigorous specifications before coding, especially for concurrent and distributed systems. The talk introduces the case for writing specs (the TLA+ mindset) as a way of thinking above the code.

Vsauce's Michael Stevens takes viewers past infinity using transfinite numbers, ordinals and cardinals, with input from mathematician Hugh Woodin. It is a fast-paced, mind-bending tour of different sizes of infinity.

Jay Parlar's Strange Loop talk shows how model checkers such as Alloy can find complex bugs in a system's design without running or reading any code. He walks through modeling architectures and checking every possible sequence of operations, including a real security bug and a UI feature-interaction bug.

Edwin Brady, creator of Idris, presents Idris 2 and demonstrates type-driven development, where the compiler assists by filling holes and suggesting implementations from types.

Numberphile's James Grime explains Cantor's result that some infinities are bigger than others, including why the reals outnumber the integers. It is a short, accessible introduction to countable and uncountable infinity.

Guy Steele's PPoPP 2017 keynote surveying the history of computer science metanotation, including BNF, inference rules, and overline notation. He argues the field's notations are inconsistent and fragmented.

Edwin Brady presents Idris 2, a rewrite of the Idris dependently typed language, and shows type-driven development in practice. The demos show the compiler filling in whole functions from their types.

Hillel Wayne's PyCon 2018 talk on testing beyond pytest and mypy. It introduces property-based testing with the Hypothesis library and design-by-contract with dpcontracts, and shows how to combine them with an existing test suite.

Hillel Wayne's Strange Loop talk introducing TLA+ as a specification language for finding concurrency bugs in designs. It covers stalling, race conditions and dropped messages, with PlusCal examples.

The first lecture of an MIT course on Douglas Hofstadter's Godel, Escher, Bach, covering themes like self-reference, recursion and incompleteness. The full lecture series is on YouTube.

Markus Triska tours Prolog's most distinctive features, including its use as a database and query language, rule processing, constraint solving and compile-time rewriting. It also covers concise meta-interpreters.

Philip Wadler's Lambda Days talk on propositions as types, the link between logic and computation that underlies theorem provers and programming language design.

A Clojure/conj 2016 talk demoing Barliman, a prototype smart editor that does program synthesis. Given a partial program and tests, it completes the program using a relational interpreter written in miniKanren.

Nadia Polikarpova's Strange Loop talk on program synthesis driven by refinement types, as in Synquid. It shows how rich type specifications can be turned into verified, executable code.

Tikhon Jelvis's Compose Conference talk on using the Z3 SMT solver to analyze and verify programs, including checking invariants with bounded loops. He demonstrates verifying programs in a small language by compiling them to Z3.

Veritasium shows how Hilbert's infinite hotel can still run out of room, using infinite buses of guests. It explains the surprising answer in a short animated segment.

Robert Moss's JuliaCon 2015 talk describes using Julia as a specification language for ACAS X, the next-generation airborne collision avoidance system that succeeds TCAS.

An AdaCore webinar in which NVIDIA engineers explain why and how they adopted Ada and SPARK for security-critical firmware. It covers the business case and the engineering experience of the transition.