Xtk: A Rule-Based Expression Rewriting Toolkit for Symbolic Computation

We present Xtk (Expression Toolkit), a powerful and extensible system for symbolic expression manipulation through rule-based term rewriting. Xtk provides a simple yet expressive framework for pattern matching, expression transformation, and symbolic computation. The system employs an Abstract Syntax Tree (AST) representation using nested Python lists, enabling intuitive expression construction while maintaining formal rigor. We demonstrate that Xtk’s rule-based approach is Turing-complete and show its applicability to diverse domains including symbolic differentiation, algebraic simplification, theorem proving via tree search algorithms, and expression optimization. The toolkit includes an extensive library of predefined mathematical rules spanning calculus, algebra, trigonometry, and logic, along with an interactive REPL for exploratory computation. We present the theoretical foundations of the system, describe its implementation architecture, analyze its computational complexity, and provide comprehensive examples demonstrating its practical applications.