Control structures in programs and computational complexity

This thesis is concerned with analysing the impact of nesting (restricted) control structures in programs, such as primitive recursion or loop statements, on the running time or computational complexity. The method obtained gives insight as to why some nesting of control structures may cause a blow up in computational complexity, while others do not. The method is demonstrated for three types of programming languages. Programs of the first type are given as lambda terms over ground-type variables enriched with constants for primitive recursion or recursion on notation. A second is concerned with ordinary loop programs and stack programs, that is, loop programs with stacks over an arbitrary but fixed alphabet, supporting a suitable loop concept over stacks. Programs of the third type are given as terms in the simply typed lambda calculus enriched with constants for recursion on notation in all finite types. As for the first kind of programs, each program t is uniformly assigned a measure μ(t), being a natural number computable from the syntax of t. For the case of primitive recursion, it is shown that programs of μ-measure n + 1 compute exactly the functions in Grzegorczyk level n + 2. In particular, programs of μ-measure 1 compute exactly the functions in FLINSPACE , the class of functions computable in binary on a Turing machine in linear space. The same hierarchy of classes is obtained when primitive recursion is replaced with recursion on notation, except that programs of μ-measure 1 compute precisely the functions in FPTIME , the class of the functions computable on a Turing machine in time polynomial in the size of the input. Another form of measure μ is obtained for the second kind of programs. It is shown that stack programs of μ-measure n compute exactly the functions computable by a Turing machine in time bounded by a function in Grzegorczyk level n + 2. In particular, stack programs of μ-measure 0 compute precisely the FPTIME functions. Furthermore, loop programs of μ-measure n compute exactly the functions in Grzegorczyk level n + 2. In particular, loop programs of μ-measure 0 compute precisely the FLINSPACE functions. As for the third kind of programs, building on the insight gained so far, it is shown how to restrict recursion on notation in all finite types so as to characterise polynomial-time computability. The restrictions are obtained by using a ramified type structure, and by adding linear concepts to the lambda calculus. This gives rise to a functional programming language RA supporting recursion on notation in all finite types. It is shown that RA programs compute exactly the FPTIME functions.