title:Machines That Invent Logic: Self-Discovering Symbolic
Abstractions from Unlabeled Primitives
author: Tofara Moyo
abstract
Wepresent a self-discovering abstraction engine that invents its own symbolic language from scratch—starting
only from six grounded, unlabeled primitive operations over the natural numbers: zero (returns 0), succ(successor), eq (equality test), add (addition), sub (truncated subtraction), and mod (modulo operation).
Critically, the system receives no semantic labels, no logical operators, no quantifiers, and no pre-definedcontrol flow; these primitives are provided solely as black-box functions with known arity but unknownmeaning. From input–output examples alone—such as a list of numbers with either true or false if theyare even or not—the engine autonomously discovers reusable computational patterns, assigns them internal identifiers (e.g., S_o), and recursively composes them into higher-order abstractions.
In a landmark demonstration, it rediscovers the predicate ”even” as the program eq(mod(x,succ(succ(zero()))),zero()) and generalizes it universally via behavioral validation up to large n, checking it on numbers it has
never seen.
Through this process, the system effectively derives fragments of first-order predicate
logic—including Boolean connectives, universal quantification, and mathematical induction—from pure arithmetic primitives and data.
The architecture rests on three pillars: (1) variational symmetry, which clusters programs by behavioral equivalence to reveal latent concepts; (2) typed program induction, which infers types from I/O behavior and promotes only those abstractions that yield compression under the
minimum description length principle; and (3) neural-symbolic compilation, which decouples internal reasoning (conducted entirely in the machine’s native symbolic language) from external communication (translated into human-understandable form only after discovery).
This work proves that logic and proof are not prerequisites for intelligence—but emergent consequences of structure, symmetry, and reuse. The framework is domain-agnostic and lays the foundation for machines that reason in concepts they invent
themselves.
(PDF) Machines That Invent Logic: Self-Discovering Symbolic Abstractions from Unlabeled Primitives
