A minimal representation for continuous functions

This paper presents a variation of a celebrated result of Kawamura and Cook specifying the least set of information about a continuous function on the unit interval which is needed for fast function evaluation. To make the above description precise, one has to specify what is considered a ‘set of information’ about a function and what ‘fast’ means. Kawamura and Cook use secondorder complexity to define what ‘fast’ means but do not use the most general notion of a ‘set of information’ this framework is able to handle. Instead they additionally require the codes to be ‘length-monotone’. This paper changes the setting in that it removes the additional premise of length-monotonicity, and instead imposes further conditions the speed of the evaluation and the translations. The sense in which the evaluation has to be fast is given the name ‘hyper-linear’, for the translations another technical condition is necessary. Hyper-linear time computation is very restrictive: The running time is not only forbidden to contain iterations of the length functions, but also the length function is only allowed to be applied to a shift by a constant of the argument instead of an arbitrary polynomial. The paper proves that there is a minimal set of information necessary for hyper-linear evaluation and that the set of information obtained by Kawamura and Cook is polynomial-time translatable, but not polynomial-time equivalent to it. Indeed, the encoding constructed in this paper is not polynomial-time equivalent to any encoding only using length-monotone names.