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LLMs don't do formal reasoning
Important new study from Apple
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Reconstructing Public Keys from Signatures
One weird hobby of mine is reasonable properties of cryptographic schemes that nobody promised they do or don’t have. Whether that’s invisible salamanders or binding through shared secr…
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Lambda Screen: Fractals in pure lambda calculus
Lambda Screen allows you to draw images using terms of pure lambda calculus. By using fixed-point combinators, you can even create infinitely detailed fractals.
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Regular expression matching can be simple and fast (2007)
This is a tale of two approaches to regular expression matching. One of them is in widespread use in the standard interpreters for many languages, including Perl. The other is used only in a few places, notably most implementations of awk and grep. The two approaches have wildly different performance characteristics:
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Goodhart’s law isn’t as useful as you might think (2023)
Goodhart's Law is useless. It tells you about a phenomenon, but it doesn't tell you how to solve it. We look at how organisations actually prevent Goodhart's Law, and illustrate this with Amazon's Weekly Business Review as an example.
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A reckless introduction to Hindley-Milner type inference (2019)
(I've been editing this post on and off for almost a year. I'm not really happy with it, but I suspect I never will be.)
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When will the eclipse happen? A multimillennium tale of computation
Wolfram tech and its dedicated website now easily predict eclipses. A look at advances in understanding and science from ancient eclipse data thousands of years old through current eclipse calculations.
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A seventh-grader student found a beautiful proof to Thales' Theorem (2002)
A useful theorem attributed to the probably first mathematician ever, Thales of Miletus (ca. 624 BC-ca. 546 BC) is one of most elementary and intuitive ever stated.
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The power of two random choices
In many large-scale web services, multiple layers of stateless and stateful services are seperated by load balancers. Load balancing can be done with dedicated hardware, with dedicated software load balancers, using DNS trickery or through a load-balancing mechanism in the client. In large systems, the resources and…
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Damas-Hindley-Milner inference two ways
What is Damas-Hindley-Milner?
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Turing's topological proof that every written alphabet is finite (2010)
Recently one of my colleagues was reading Alan Turing’s groundbreaking 1936 article “On Computable Numbers with an Application to the Entscheidungsproblem.” This is the article in…
162
Encoding tic-tac-toe in 15 bits
I recently stumbled upon a blog post by Alejandra González (a.k.a @blyxyas) that seeks to compress a tic-tac-toe game state into as few bits as possible. She arrived at a solution in 18 bits. This got me thinking, can we do better?
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Solving Probabilistic Tic-Tac-Toe
This morning, I came accross Cameron Sun’s Probabilistic Tic-Tac-Toe on HN . However, the comments were disappointing, with people proposing probabilistic models, linear programming, minimax algorithms, and heuristic evaluations to develop AI approaches for the game, none of which quite correct. In this post, I’ll…
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Industrious Dice
I’m a professional mathematician. That means somebody pays me to do math. I’m also a recreational mathematician. That means you might have to pay me to get me to stop. Wearing my recreational mathe…
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Accidentally writing a SAT solver
where my words occasionally escape /dev/null
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Derivative at a Discontinuity
Alok Singh's Blog
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The derivative of a number (2014)
Are you kidding? Edward Barbeau is now a professor emeritus of mathematics at the University of Toronto. Over the years he has been working to increase the interest in mathematics in general, and e…
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When random numbers are too random: Low discrepancy sequences (2017)
Random numbers can be useful in graphics and game development, but they have a pesky and sometimes undesirable habit of clumping together. This is a problem in path tracing and monte carlo integrat…
170
Borges on Chaos Theory
About a weird, lovely short story
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Learning Regular Languages with L* Algorithm
Research in Software Engineering from Rahul Gopinath
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An interactive guide to Fourier series (2022)
Drawing with the Fourier Series
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Effort – a possibly new algorithm for LLM Inference
A possibly new algorithm for LLM Inference. Adjust smoothly - and in real time - how many calculations you'd like to do during inference.
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Crafting formulas: Lambdas all the way down
We build complex mathematical formulas bottom-up using pure lambda calculus. Includes instructions of how to translate Taylor series and other approximations to lambda terms.
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Structuralism as a Philosophy of Mathematics
For mathematical insight, according to the structuralist imperative, investigate mathematical structure and consider mathematical concepts only as invariant under isomorphism.
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Compiling with Constraints
There are lots of interesting little subproblems in compilation like instruction selection, register allocation and instruction scheduling. These can be expressed in declarative interlinked way to constraint solvers.
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Legendre transform, better explained (2017)
The way that most physicists teach and talk about partial differential equations is horrible, and has surprisingly big costs for the typical understanding of the foundations of the field even among professionals. The chief victims are students of thermodynamics and analytical mechanics, and I've me
179
Diffusion models are evolutionary algorithms
Authors: Yanbo Zhang, Benedikt Hartl, Hananel Hazan, Michael Levin
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An alternative construction of Shannon entropy
The underpinning of information theory is Shannon’s formula for the entropy H of a discrete probability distribution P. Derived in 1948 in the paper “A mathematical theory of communication”, this is given by \[ H[P] = -\sum_{i=1}^N p_i \log p_i \] where \( p_i = p(x_i) \) is the probability of sampling \( x_i \) from…