recursive analysis in compiler


1. Recursive analysis, also called computable analysis, has been introduced by Turing[Tur36], Grzegorczyk [Grz55], Lacombe [Lac55].

2. It has shown to provide a very robust concept of computability that enables to discuss most arguments of mathematical analysis from the computability point of view: see e.g. monograph [Wei00].

3. In this framework, a function f : R –> R over the reals is considered as computable, if there is some computable functional, that maps any sequence quickly converging to some x to a sequence quickly converging to f(x), for all x.

4. That means that this notion of computability requests a priori to deal with functionals, or higher order Turing machines.

5. In a recent work [BH04a, BH05], extending some classes proposed by Campagnolo, Moore and Costa [CMC00, CMC02, Cam01] by a suitable limit schema, we proved that a particular subclass of computable functions over the reals can be characterized algebraically in a machine independent way. Indeed, elementary functions in the sense of recursive analysis were characterized as the smallest class of functions that contains some basic functions, and closed by composition, linear integration, and a simple limit schema.

6. Recursive analysis develops natural number computations into a framework appropriate for real numbers. This text is based upon primary recursive arithmetic and presents a unique combination of classical analysis and intuitional analysis. Written by a master in the field, it is suitable for graduate students of mathematics and computer science and can be read without a detailed knowledge of recursive arithmetic

7. Recursive analysis is a statistical method for multivariable analysis. Recursive partitioning creates a decision tree that strives to correctly classify members of the population by splitting it into sub-populations based on several dichotomous independent variables. The process is termed recursive because each sub-population may in turn be split an indefinite number of times until the splitting process terminates after a particular stopping criterion is reached.

8. Recursion occurs when a thing is defined in terms of itself or of its type. Recursion is used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition.


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