Recognizing a formal language does not imply the ability to construct the computation that decides it. We operationalize this distinction as an executable benchmark for language-model reasoning: given a formal-language specification, an LLM must synthesize a deterministic or nondeterministic Turing machine that decides the language. We introduce TMEval, a benchmark of 253 language specifications across 7 structural categories spanning context-free and non-context-free phenomena. Each language includes 600 oracle-labeled membership tests, and every output is evaluated as an executable artifact rather than a textual response. Our pipeline enforces a strict output contract, validates machine construction, executes each buildable machine, and assigns one of four outcomes: static_fail, build_fail, semantic_mismatch, or exact_pass. Across eight LLMs, 2,024 model–language candidates, and up to 1,214,400 behavioral evaluations, only 11.12% exactly pass all tests. More than half fail before execution, over one-third implement the wrong language, and 75 languages remain unsolved by all systems. Failure depends less on the context-free/non-context-free boundary than on the operational structure required to realize the specification. TMEval reveals a substantial gap between formal-language recognition and machine-checked symbolic construction. TMEval is available at https://turing-machineeval.github.io/.

Artificial intelligence has long been evaluated by its ability to perform computation and structured reasoning. While recent large language models (LLMs) demonstrate strong capabilities in language generation, code synthesis, and structured problem solving, it remains unclear whether they can construct formal computational mechanisms from high-level specifications. In this work, we investigate whether modern LLMs can synthesize pushdown automata (PDAs) from language descriptions expressed in natural language or mathematical form. Because PDA construction requires explicit reasoning about states, transitions, and stack-based memory, it provides a controlled test of structured symbolic reasoning. We introduce the task of pushdown automaton synthesis and develop an automated evaluation framework that verifies generated PDAs through structural buildability checks and execution-based validation. Using a benchmark of context-free languages with labeled test cases, we evaluate contemporary LLMs and analyze their ability to construct and execute formal computational machines. Our results show that current state-of-the-art LLMs struggle to reliably construct executable PDAs, indicating that synthesizing computational mechanisms requiring coordinated reasoning over state transitions and stack-based memory manipulation remains challenging for present-day models. All code and data are available at: https://github.com/Dip-to/Pushdown-Automaton-Synthesis-Dataset.git

Interpreting visual scenes with high-level reasoning is essential for many real-world applications, such as autonomous systems andcontent moderation, but training and maintaining Vision–Language Models (VLMs) remains resource-intensive and opaque. In this work, we present CAPSTONE, a lightweight, modular framework designed for industrial settings. Instead of relying on multimodal training or fine-tuning large models, CAPSTONE transforms outputs from off-the-shelf vision models into structured text prompts that can be interpreted by a frozen Large Language Model (LLM). This plug-and-play architecture enables reasoning over visual input without access to raw pixels, dramatically reducing computational cost and complexity. On the POPE dataset, our system, using a 7B LLM, outperforms several fully trained VLMs in zero-shot evaluations, while on the VSR benchmark, the 4B model achieves competitive results, together demonstrating strong generalization without retraining. CAPSTONE offers a scalable and interpretable alternative for companies looking to integrate visual reasoning capabilities without the burden of full-scale VLM pipelines.