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