Abstract

A key problem in implicit complexity is to analyse the impact on program run times of nesting control structures, such as recursion in all finite types in functional languages or for-do statements in imperative languages.

Three types of programs are studied. One type of program can only use ground type recursion. Another is concerned with imperative programs: ordinary loop programs and stack programs. Programs of the third type can use higher type recursion on notation as in functional programming languages.

The present approach to analysing run time yields various characterisations of fptime and all Grzegorczyk classes at and above the flinspace level. Central to this approach is the distinction between “top recursion” and “side recursion”. Top recursions are the only form of recursion which can cause an increase in complexity. Counting the depth of nested top recursions leads to several versions of “the μ-measure”. In this way, various recent solutions of key problems in implicit complexity are integrated into a uniform approach.

Keywords: Implicit computational complexity; Imperative programming languages; Grzegorczyk hierarchy; Higher type recursion; Lambda calculus; Polynomial time

MSC: 03D15; 03D20; 68N15; 68N18; 68N30; 68Q15; 68Q17; 68W40; 03-02