Vol. 1 No. 4 (2026)

Articles

Failure-State Curriculum Optimization for Long-Horizon Algebraic Reasoning

Published 2026-08-15

  • Charlotte M. Thompson
    • ,
  • Kayla J. Scott

Charlotte M. Thompson
Department of Advanced Computing, Viterbi School of Engineering, University of Southern California, Los Angeles, California, USA

Kayla J. Scott
Department of Advanced Computing, Viterbi School of Engineering, University of Southern California, Los Angeles, California, USA

Language, Knowledge and Intelligent Systems, Volume 1, Issue 4, 2026 cover

Abstract

Long-horizon algebraic reasoning remains a formidable challenge for contemporary artificial intelligence models, often characterized by cascading errors and an inability to recover from intermediate missteps. This paper introduces a novel paradigm, Failure-State Curriculum Optimization, designed to systematically expose reasoning agents to intermediate failure states and train them to formulate recovery trajectories. By dynamically generating a curriculum of progressively complex algebraic problems and intentionally perturbing intermediate derivation steps, the proposed framework enables models to learn robust error-correction mechanisms. We hypothesize that traditional reinforcement learning and supervised fine-tuning approaches overfit to idealized, error-free reasoning paths, rendering them fragile when confronted with complex, multi-step tasks. Through extensive empirical analysis across multiple algebraic benchmarks, this study demonstrates that incorporating failure-state exposure into the training curriculum significantly improves overall reasoning accuracy and resilience. The methodology integrates automated failure generation, difficulty scoring, and adaptive curriculum pacing to ensure optimal learning efficiency. Our findings reveal that models trained with Failure-State Curriculum Optimization exhibit a remarkable capacity to identify, isolate, and correct intermediate logical fallacies without requiring human-annotated recovery trajectories. This research bridges the gap between brittle step-by-step reasoning and human-like cognitive resilience, offering a scalable solution for complex mathematical problem-solving in artificial intelligence systems.

Keywords

Curriculum Learning, Algebraic Reasoning, Error Recovery, Cognitive Resilience

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