arXiv subjects

Tristan Stérin

Publications and source records attributed to Tristan Stérin.

9 recordsLinked to original sources

arXiv2026

Putnam 2025 Problems in Rocq using Opus 4.6 and Rocq-MCP

We report on an experiment in which Claude Opus~4.6, equipped with a suite of Model Context Protocol (MCP) tools for the Rocq proof assistant, autonomously proved 10 of 12 problems from the 2025 Putnam Mathematical Competition. The MCP tools, designed with Claude by analyzing logs from a prior experiment on miniF2F-Rocq, encode a "compile-first, interactive-fallback" strategy. Running on an isolated VM with no internet access, the agent deployed 141 subagents over 17.7 hours of active compute (51.6h wall-clock), consuming approximately 1.9 billion tokens. All proofs are publicly available.

cs.LG↗

arXiv2026

Determination of the fifth Busy Beaver value

The Busy Beaver value $S(n)$ is the maximum number of steps that an $n$-state 2-symbol Turing machine can perform from the all-zero tape before halting. $S$ was historically introduced by Tibor Radó in 1962 as one of the simplest examples of an uncomputable function. We prove that $S(5) = 47,176,870$ using the Coq proof assistant. The proof enumerates $181,385,789$ Turing machines with 5 states and, for each machine, decides whether it halts or not. Our result marks the first determination of a new Busy Beaver value in over 40 years and the first Busy Beaver value ever to be formally verified, attesting to the effectiveness of massively collaborative online research (bbchallenge$.$org).

cs.LO↗

arXiv2025

Turing machines deciders, part I

The Busy Beaver Challenge (or bbchallenge) aims at collaboratively solving the following conjecture: "$S(5) = 47{,}176{,}870$" [Radó, 1962], [Marxen and Buntrock, 1990], [Aaronson, 2020]. This conjecture says that if a 5-state Turing machine runs for more than 47,176,870 steps without halting, then it will never halt -- starting from the all-0 tape. Proving this conjecture amounts to deciding whether 181,385,789 Turing machines with 5 states halt or not -- starting from the all-0 tape [bbchallenge, 2025]. To do so, we write $\textit{deciders}$: programs that take as input a Turing machine and output either HALT, NONHALT, or UNKNOWN. Each decider is specialised in recognising a particular type of non-halting behavior. After two years of work, the Busy Beaver Challenge achieved its goal in July 2024 by delivering a proof of "$S(5) = 47{,}176{,}870$" formalised in Coq [bbchallenge, 2025]. In this document, we present deciders that were developed before the Coq proof and which were mainly not used in the proof; nonetheless, they are relevant techniques for analysing Turing machines. Part II of this work is the decider section of our paper showing "$S(5) = 47{,}176{,}870$" [bbchallenge, 2025], presenting the deciders that were used in the Coq proof.

cs.LO↗

arXiv2024

Hardness of busy beaver value BB(15)

The busy beaver value BB(n) is the maximum number of steps made by any n-state, 2-symbol deterministic halting Turing machine starting on blank tape. The busy beaver function $n \mapsto \text{BB}(n)$ is uncomputable and, from below, only 4 of its values, BB(1) ... BB(4), are known to date. This leads one to ask: from above, what is the smallest BB value that encodes a major mathematical challenge? Knowing BB(4,888) has been shown by Yedidia and Aaronson [28] to be at least as hard as solving Goldbach's conjecture, with a subsequent improvement, as yet unpublished, to BB(27) [4,1]. We prove that knowing BB(15) is at least as hard as solving the following Collatz-related conjecture by Erdős, open since 1979 [9]: for all n > 8 there is at least one digit 2 in the base 3 representation of $2^n$. We do so by constructing an explicit 15-state, 2-symbol Turing machine that halts if and only if the conjecture is false. This 2-symbol Turing machine simulates a conceptually simpler 5-state, 4-symbol machine which we construct first. This makes, to date, BB(15) the smallest busy beaver value that is related to a natural open problem in mathematics, bringing to light one of the many challenges underlying the quest of knowing busy beaver values.

cs.LO↗

arXiv2022

The Collatz process embeds a base conversion algorithm

The Collatz process is defined on natural numbers by iterating the map $T(x) = T_0(x) = x/2$ when $x\in\mathbb{N}$ is even and $T(x)=T_1(x) =(3x+1)/2$ when $x$ is odd. In an effort to understand its dynamics, and since Generalised Collatz Maps are known to simulate Turing Machines [Conway, 1972], it seems natural to ask what kinds of algorithmic behaviours it embeds. We define a quasi-cellular automaton that exactly simulates the Collatz process on the square grid: on input $x\in\mathbb{N}$, written horizontally in base 2, successive rows give the Collatz sequence of $x$ in base 2. We show that vertical columns simultaneously iterate the map in base 3. This leads to our main result: the Collatz process embeds an algorithm that converts any natural number from base 3 to base 2. We also find that the evolution of our automaton computes the parity of the number of 1s in any ternary input. It follows that predicting about half of the bits of the iterates $T^i(x)$, for $i = O(\log x)$, is in the complexity class NC$^1$ but outside AC$^0$. Finally, we show that in the extension of the Collatz process to numbers with infinite binary expansions ($2$-adic integers) [Lagarias, 1985], our automaton encodes the cyclic Collatz conjecture as a natural reachability problem. These results show that the Collatz process is capable of some simple, but non-trivial, computation in bases 2 and 3, suggesting an algorithmic approach to thinking about existence, prediction and structure of cycles in the Collatz process.

cs.DM↗

arXiv2021

Small tile sets that compute while solving mazes

We ask the question of how small a self-assembling set of tiles can be yet have interesting computational behaviour. We study this question in a model where supporting walls are provided as an input structure for tiles to grow along: we call it the Maze-Walking Tile Assembly Model. The model has a number of implementation prospects, one being DNA strands that attach to a DNA origami substrate. Intuitively, the model suggests a separation of signal routing and computation: the input structure (maze) supplies a routing diagram, and the programmer's tile set provides the computational ability. We ask how simple the computational part can be. We give two tiny tile sets that are computationally universal in the Maze-Walking Tile Assembly Model. The first has four tiles and simulates Boolean circuits by directly implementing NAND, NXOR and NOT gates. Our second tile set has 6 tiles and is called the Collatz tile set as it produces patterns found in binary/ternary representations of iterations of the Collatz function. Using computer search we find that the Collatz tile set is expressive enough to encode Boolean circuits using blocks of these patterns. These two tile sets give two different methods to find simple universal tile sets, and provide motivation for using pre-assembled maze structures as circuit wiring diagrams in molecular self-assembly based computing.

cs.ET↗

arXiv2020

Binary expression of ancestors in the Collatz graph

The Collatz graph is a directed graph with natural number nodes and where there is an edge from node $x$ to node $T(x)=T_0(x)=x/2$ if $x$ is even, or to node $T(x)=T_1(x)=\frac{3x+1}{2}$ if $x$ is odd. Studying the Collatz graph in binary reveals complex message passing behaviors based on carry propagation which seem to capture the essential dynamics and complexity of the Collatz process. We study the set $\mathcal{E} \text{Pred}_k(x)$ that contains the binary expression of any ancestor $y$ that reaches $x$ with a limited budget of $k$ applications of $T_1$. The set $\mathcal{E} \text{Pred}_k(x)$ is known to be regular, Shallit and Wilson [EATCS 1992]. In this paper, we find that the geometry of the Collatz graph naturally leads to the construction of a regular expression, $\texttt{reg}_k(x)$, which defines $\mathcal{E} \text{Pred}_k(x)$. Our construction, is exponential in $k$ which improves upon the doubly exponentially construction of Shallit and Wilson. Furthermore, our result generalises Colussi's work on the $x = 1$ case [TCS 2011] to any natural number $x$, and gives mathematical and algorithmic tools for further exploration of the Collatz graph in binary.

cs.DM↗

arXiv2020

scadnano: A browser-based, scriptable tool for designing DNA nanostructures

We introduce $\textit{scadnano}$ (https://scadnano.org) (short for "scriptable cadnano"), a computational tool for designing synthetic DNA structures. Its design is based heavily on cadnano, the most widely-used software for designing DNA origami, with three main differences: 1. scadnano runs entirely in the browser, with $\textit{no software installation}$ required. 2. scadnano designs, while they can be edited manually, can also be created and edited by a $\textit{well-documented Python scripting library}$, to help automate tedious tasks. 3. The scadnano file format is $\textit{easily human-readable}$. This goal is closely aligned with the scripting library, intended to be helpful when debugging scripts or interfacing with other software. The format is also somewhat more expressive than that of cadnano, able to describe a broader range of DNA structures than just DNA origami.

cs.ET↗

arXiv2020

Limitations on counting in Boolean circuits and self-assembly

In self-assembly, a $k$-counter is a tile set that grows a horizontal ruler from left to right, containing $k$ columns each of which encodes a distinct binary string. Counters have been fundamental objects of study in a wide range of theoretical models of tile assembly, molecular robotics and thermodynamics-based self-assembly due to their construction capabilities using few tile types, time-efficiency of growth and combinatorial structure. Here, we define a Boolean circuit model, called $n$-wire local railway circuits, where $n$ parallel wires are straddled by Boolean gates, each with matching fanin/fanout strictly less than $n$, and we show that such a model can not count to $2^n$ nor implement any so-called odd bijective nor quasi-bijective function. We then define a class of self-assembly systems that includes theoretically interesting and experimentally-implemented systems that compute $n$-bit functions and count layer-by-layer. We apply our Boolean circuit result to show that those self-assembly systems can not count to $2^n$. This explains why the experimentally implemented iterated Boolean circuit model of tile assembly can not count to $2^n$, yet some previously studied tile system do. Our work points the way to understanding the kinds of features required from self-assembly and Boolean circuits to implement maximal counters.

cs.ET↗