# Hey Pentti, We Did It Again!: Differentiable vector-symbolic types that prove polynomial termination

arXiv: https://arxiv.org/abs/2510.16533 · HTML: https://mlsanity.com/paper/2510.16533/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 4 of its 31 references that are in the map.
- First posted on arXiv 18 Oct 2025.

## Builds on (most cited first, 4 of 4)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Scaling Scaling Laws with Board Games](https://mlsanity.com/paper/2104.03113/) (2021)
- [Homotopy Type Theory: Univalent Foundations of Mathematics](https://mlsanity.com/paper/1308.0729/) (2013)
- [Computing with Residue Numbers in High-Dimensional Representation](https://mlsanity.com/paper/2311.04872/) (2023)

## Abstract (arXiv)

We present a typed computer language, Doug, in which all typed programs may be proved to halt in polynomial time, encoded in a vector-symbolic architecture (VSA). Doug is just an encoding of the light linear functional programming language (LLFPL) described by (Schimanski2009, ch. 7). The types of Doug are encoded using a slot-value encoding scheme based on holographic declarative memory (HDM; Kelly, 2020). The terms of Doug are encoded using a variant of the Lisp VSA defined by (Flanagan, 2024). Doug allows for some points on the embedding space of a neural network to be interpreted as types, where the types of nearby points are similar both in structure and content. Types in Doug are therefore learnable by a neural network. Following (Chollet, 2019), (Card, 1983), and (Newell, 1981), we view skill as the application of a procedure, or program of action, that causes a goal to be satisfied. Skill acquisition may therefore be expressed as program synthesis. Using Doug, we hope to describe a form of learning of skilled behaviour that follows a human-like pace of skill acquisition (i.e., substantially faster than brute force; Heathcote, 2000), exceeding the efficiency of all currently existing approaches (Kaplan, 2020; Jones, 2021; Chollet, 2024). Our approach brings us one step closer to modeling human mental representations, as they must actually exist in the brain, and those representations' acquisition, as they are actually learned.