The Illusion of Procedural Reasoning: Measuring Long-Horizon FSM Execution in LLMs

Large language models (LLMs) have achieved remarkable results on tasks framed as reasoning problems, yet their true ability to perform procedural reasoning, executing multi-step, rule-based computations remains unclear. Unlike algorithmic systems, which can deterministically execute long-horizon symbolic procedures, LLMs often degrade under extended reasoning chains, but there is no controlled, interpretable benchmark to isolate and measure this collapse. We introduce Finite-State Machine (FSM) Execution as a minimal, fully interpretable framework for evaluating the procedural reasoning capacity of LLMs. In our setup, the model is given an explicit FSM definition and must execute it step-by-step given input actions, maintaining state consistency over multiple turns. This task requires no world knowledge, only faithful application of deterministic transition rules, making it a direct probe of the model's internal procedural fidelity. We measure both Turn Accuracy and Task Accuracy to disentangle immediate computation from cumulative state maintenance. Empirical results reveal systematic degradation as task horizon or branching complexity increases. Models perform significantly worse when rule retrieval involves high branching factors than when memory span is long. Larger models show improved local accuracy but remain brittle under multi-step reasoning unless explicitly prompted to externalize intermediate steps. FSM-based evaluation offers a transparent, complexity-controlled probe for diagnosing this failure mode and guiding the design of inductive biases that enable genuine long-horizon procedural competence. By grounding reasoning in measurable execution fidelity rather than surface correctness, this work helps establish a rigorous experimental foundation for understanding and improving the algorithmic reliability of LLMs.

A Distinct Unsupervised Reference Model From The Environment Helps Continual Learning

The existing continual learning methods are mainly focused on fully-supervised scenarios and are still not able to take advantage of unlabeled data available in the environment. Some recent works tried to investigate semi-supervised continual learning (SSCL) settings in which the unlabeled data are available, but it is only from the same distribution as the labeled data. This assumption is still not general enough for real-world applications and restricts the utilization of unsupervised data. In this work, we introduce Open-Set Semi-Supervised Continual Learning (OSSCL), a more realistic semi-supervised continual learning setting in which out-of-distribution (OoD) unlabeled samples in the environment are assumed to coexist with the in-distribution ones. Under this configuration, we present a model with two distinct parts: (i) the reference network captures general-purpose and task-agnostic knowledge in the environment by using a broad spectrum of unlabeled samples, (ii) the learner network is designed to learn task-specific representations by exploiting supervised samples. The reference model both provides a pivotal representation space and also segregates unlabeled data to exploit them more efficiently. By performing a diverse range of experiments, we show the superior performance of our model compared with other competitors and prove the effectiveness of each component of the proposed model.