  Prompting techniques have significantly enhanced the capabilities of Large
Language Models (LLMs) across various complex tasks, including reasoning,
planning, and solving math word problems. However, most research has
predominantly focused on language-based reasoning and word problems, often
overlooking the potential of LLMs in handling symbol-based calculations and
reasoning. This study aims to bridge this gap by rigorously evaluating LLMs on
a series of symbolic tasks, such as addition, multiplication, modulus
arithmetic, numerical precision, and symbolic counting. Our analysis
encompasses eight LLMs, including four enterprise-grade and four open-source
models, of which three have been pre-trained on mathematical tasks. The
assessment framework is anchored in Chomsky's Hierarchy, providing a robust
measure of the computational abilities of these models. The evaluation employs
minimally explained prompts alongside the zero-shot Chain of Thoughts
technique, allowing models to navigate the solution process autonomously. The
findings reveal a significant decline in LLMs' performance on context-free and
context-sensitive symbolic tasks as the complexity, represented by the number
of symbols, increases. Notably, even the fine-tuned GPT3.5 exhibits only
marginal improvements, mirroring the performance trends observed in other
models. Across the board, all models demonstrated a limited generalization
ability on these symbol-intensive tasks. This research underscores LLMs'
challenges with increasing symbolic complexity and highlights the need for
specialized training, memory and architectural adjustments to enhance their
proficiency in symbol-based reasoning tasks.
