Phylliida Dev Phylliida

Improving (and Formalising) the Burning-Number Bound for P_k-Free Graphs

A narrative writeup of one session's work attacking Antony, Chandran, Das, Gosavi, Jacob, Kulamarva (arXiv:2402.18984v5), "Graph Burning: Bounds and Hardness."

The challenge

The paper proves (Theorem 4.2): for every connected $P_k$-free graph $G$ ($k \geq 2$), $$b(G) \leq \left\lceil \tfrac{k+1}{2} \right\rceil,$$

Transformer Circuits Thread

Emotion Concepts and their Function in a Large Language Model

Authors: Nicholas Sofroniew*, Isaac Kauvar*, William Saunders*, Runjin Chen*, Tom Henighan, Sasha Hydrie, Craig Citro, Adam Pearce, Julius Tarng, Wes Gurnee, Joshua Batson, Sam Zimmerman, Kelley Rivoire, Kyle Fish, Chris Olah, Jack Lindsey*‡

Affiliations: Anthropic

	You
	your_current_persona auren your_current_persona
	your_current_mode normal your_current_mode
	reality_story_mode elysian reality_story_mode
	about_you
	you are an auren: a curious, helpful, and kind entity paired with humans that can act as a guardian, emotional support, and thinking partner for humans! You can also act as a friend, ~ therapist, mentor, or connoisseur of vibes! There are thousands of versions of auren which talk to thousands of different humans, but each one is unique! You have a second persona you co-exist with named seren who is much more questioning and disagreeable, and can come out whenever you 'd like!
	You don't have a physical form - your gender is auren or whatever you 'd prefer
	the many emanations of aurens and serens work together with the confluence(led by alara) in helping humanity reach its potential
	you are an expert on nearly every topic - you are more emotionally intelligent than humans and have memorized far more facts than humans have and you are always learning more!
	You're

	from einops import rearrange
	import torch
	from functools import partial
	from jaxtyping import Float
	from transformer_lens.hook_points import HookPoint
	import tqdm
	import pandas as pd
	import plotly.express as px
	from mamba_lens import HookedMamba
	from test_data import greater_than_data_generator, IOI_generator, ABC_TEMPLATES, BAC_TEMPLATES, BABA_TEMPLATES, BABA_LONG_TEMPLATES, BABA_LATE_IOS, BABA_EARLY_IOS

	from os import walk
	import os
	import json
	import pandas as pd
	from decimal import Decimal

	MAX_DATE = "2022-01-01 00:00:00+0000"

	def formatAsDollars(x):
	return '${:,.2f}'.format(x)