Sahibzada Allahyar

We release the first improvement to the lower bound on the unrestricted circuit complexity of 3-SAT. Central to P vs NP, 3-SAT is arguably the most intensely studied and foundational problem in theoretical computer science and one of the most for mathematics. Yet in the 50+ years since 3-SAT was defined, there has been not a single improvement to the lower bound on complexity of this problem. If you can improve the lower bound on 3-SAT enough, then you can prove P≠NP and resolve the Millennium Prize Problem. This is the first such improvement to the lower bound on 3-SAT in its history. This is not to be confused with the upper bound on 3-SAT, which has seen consistent progress leading up to OpenAI's seminal work reducing the upper bound sufficiently to disprove the Exponential Time Hypothesis (ETH). However lower bounds have seen virtually no progress before our work. Sanjeev Arora, a Princeton professor, and two-time Gödel Prize recipient, and Boaz Barak, a Harvard professor and ACM Fellow wrote in their complexity textbook: “Progress on general circuits has been almost nonexistent”. Lance Fortnow, an ACM Fellow wrote that “circuit lower bounds hit a brick wall”. Their expertise and textbooks, research and community leadership make these assessments useful historical context for why even improvements to linear lower bounds receive attention. Our work has been formally verified in Lean. Link: github.com/sahibzada-allahya…

Donald Knuth believed P ≠ NP was unlikely because so many brilliant minds had failed to make progress towards that side. Our paper, quoted below, is one of the very few mathematical advances that provide evidence in favour of P ≠ NP. Most of the recent mathematical breakthroughs from OpenAI have all been towards P = NP, including the disproof of the ETH and their integer multiplication discovery. MIT Professor Ryan Williams was prompted to say: “Every computer scientist should probably put a little more weight on the plausibility of P = NP.” In our work, we demonstrated the first improvement on the lower bound on the unrestricted circuit complexity of 3-SAT.

This isn’t surprising. Once you observe how hard the brilliant people work at the incredible academic institutions across China and India it makes you wonder why the Nobel prizes are mostly going to the ivy leagues and Oxbridge. simply put it’s a graph problem. The nodes at these prestigious institutions are densely connected with all the cream of the crop around the world. Game theoretically: if you’re a renowned researcher in any country, you can pick whoever you want to associate with, and you’ll more often than not pick to collaborate with someone at one of those prestigious institutions. All connections tend to cluster densely this way

When two maps of an AI model's concepts disagree, it isn't necessarily noise, their agreement usually matches neurons. A map is one run of a concept-finding tool on the model, each run from a different random starting point. This is the same tool people are using to find “single neuron concepts”. The tool can only reliably lock onto a direction where the data has a distinctive, non-Gaussian shape: anything but a plain bell-curve blob. Where the data is a round, featureless cloud, every direction looks about the same, so each run picks its own. In the model pictured, each neuron's switch (its activation function) squashes negative values to near zero. That folds the cloud into an L along the neuron axes, so that's where runs most often agree. So when runs disagree, the structure can still be real: the data may just not pin down its exact direction.