Abstract
We extend the framework for complexity of operators in analysis devised by Kawamura and Cook (2012) to allow for the treatment of a wider class of representations. The main novelty is to endow represented spaces of interest with an additional function on names, called a parameter, which measures the complexity of a given name. This parameter generalises the size function which is usually used in second-order complexity theory and therefore also central to the framework of Kawamura and Cook. The complexity of an algorithm is measured in terms of its running time as a second-order function in the parameter, as well as in terms of how much it increases the complexity of a given name, as measured by the parameters on the input and output side.As an application we develop a rigorous computational complexity theory for interval computation. In the framework of Kawamura and Cook the representation of real numbers based on nested interval enclosures does not yield a reasonable complexity theory. In our new framework this representation is polytime equivalent to the usual Cauchy representation based on dyadic rational approximation. By contrast, the representation of continuous real functions based on interval enclosures is strictly smaller in the polytime reducibility lattice than the usual representation, which encodes a modulus of continuity. Furthermore, the function space representation based on interval enclosures is optimal in the sense that it contains the minimal amount of information amongst those representations which render evaluation polytime computable.
Highlights
Computable analysis is an extension of the theory of computation over the natural numbers to continuous data, such as real numbers and real functions, based on the Turing machine model of computation
A real function is polytime computable with respect to the parametrised interval representation if and only if it is polytime computable in the usual sense. While this result might suggest that nothing much is gained from this new definition, we show that the natural uniform complexity structure on the space of real functions viewed as a parametrised space is different from the complexity structure induced by the Kawamura-Cook representation, and that the complexity induced by the natural parametrised space structure corresponds more closely to the complexity of operators in practical implementations
The basic feasible functionals satisfy the so-called Ritchie-Cobham property: the running time of an oracle machine which computes a basic feasible functional can be bounded by a basic feasible functional [Meh76]
Summary
Computable analysis is an extension of the theory of computation over the natural numbers to continuous data, such as real numbers and real functions, based on the Turing machine model of computation. An important step in this process, and something that opened the field up for applications, was the characterization of the basic feasible functionals by means of resource bounded oracle Turing machines due to Kapron and Cook [KC96] Based on this characterization, Kawamura and Cook introduced a framework for complexity of operators in analysis [KC12] that generalizes the definition of feasibly computable functions of Friedman and Ko to a wider class of spaces, including the aforementioned examples. Kawamura and Cook introduced a framework for complexity of operators in analysis [KC12] that generalizes the definition of feasibly computable functions of Friedman and Ko to a wider class of spaces, including the aforementioned examples This kicked off a series of investigations [FGH14, FZ15, Ste, SS17, and many more]. The representation for real numbers used in Appendix A coincides with the one used by Muller to model the behaviour of iRRAM [Mul01]
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