Reflective Programs in Tree Calculus

Tree calculus is seeded by a single operator, whose self-application gives rise to pow- erful reflective programs including a size program that can compute its own size, an equality program that can decide its own equality, and a self-evaluator that can evaluate itself. This is achieved without any of the usual outside machinery, such as the Gödel numbering of Turing machines, the quotation of lambda-abstractions or serialisation of programs. The resulting theory of computation is both simple and powerful: simple because it is based on three equations only; powerful because one program can query the internal structure of another, using a mix of extensional and intensional techniques.