Comparative schematology

While we may have the intuitive idea of one programming language having greater power than another, or of some subset of a language being an adequate 'core' for that language, we find when we try to formalize this notion that there is a serious theoretical difficulty. This lies in the fact that even quite rudimentary languages are nevertheless 'universal' in the following sense. If the language allows us to program with simple arithmetic or list-processing functions then any effective control structure can be simulated, traditionally by encoding a Turing machine computation in some way. In particular, a simple language with some basic arithmetic can express programs for any partial recursive function. Such an encoding is usually quite unnatural and impossibly inefficient. Thus, in order to carry on a practical study of the comparative power of different languages we are led to banish explicit functions and deal instead with abstract, uninterpreted programs or schemas. What follows is a brief report on some preliminary exploration in this area.