Teaching a Minimalist Machine to Discover Recursive Programs for Arithmetic

2026-10-05

AI Summary

This study addresses how machines can acquire complex recursive concepts from minimal experience. To this end, it proposes a minimalist inductive program synthesis framework that simulates the sequential introduction of concepts in elementary arithmetic curricula. Grounded in Prolog relational clauses, the framework performs Horn clause search and knowledge graph learning under a fixed schema. Its core innovation lies in eliminating dedicated meta-rules, allowing recursive structures to emerge naturally through predicate reuse while generating transparent reasoning traces. The proposed model successfully learns recursive programs for addition, subtraction, multiplication, and division from scratch, demonstrating human-like capabilities in concept acquisition and generalization.

Abstract

Humans can often acquire and synthesize complex, recursive concepts from minimal experience. Leveraging cognitive insights, we propose the Minimalist Machine, a framework for inductive program synthesis designed to model such conceptual learning. The system uses a compact relational subset of Prolog: Programs are searched within a fixed schema of body-free facts and two-body conjunctive Horn clauses. Recursion is not defined by a dedicated metarule. Instead, it emerges when a target predicate is reused inside the body of a learned clause. Inspired by a primary school curriculum, the model is taught through a human-curated, sequential introduction of new concepts in arithmetic. Starting from initially empty knowledge base, it first acquires simple structural predicates, then successor-based state transformations, and finally recursive programs for addition, subtraction, multiplication, and division. Ultimately, this approach yields the fully transparent, inductive reasoning trace necessary for human-like conceptual learning.

Problem

• inductive program synthesis
• recursive programs
• conceptual learning
• minimal experience
• arithmetic

Innovation

• Inductive Program Synthesis
• Minimalist Machine
• Emergent Recursion
• Relational Prolog Subset
• Conceptual Learning