Machines That Invent Logic: Self-Discovering Symbolic Abstractions from Unlabeled Primitives

We present 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-defined control flow; these primitives are provided solely as black-box functions with known arity but unknown meaning. From input-output examples alone-such as a list of numbers with either true or false if they are 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.Eventually the system forms proof strategies and can even have and use concepts for runtime complexity. The architecture rests on two pillars: (1) variational symmetry, which clusters programs by behavioral equivalence to reveal latent concepts and (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 . 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.