Can Learn The Real Recursive Function Can Be Calculated

Computability and complexity theory over discrete structures have played an important role in mathematics and computer science, leading to the discovery, understanding and classification of decidable/undecidable problems, paving the way to the modern computer era, and affecting deeply our view of the world. Recent new paradigms of computation, based on biological and physical models, address in a radically new way questions of efficiency and challenge assumptions about the so-called Turing barrier.Most mathematical models in physics and engineering, however, are based on the real number concept. Thus, a computability theory and a complexity theory over the real numbers and over more general continuous data structures is needed. Unlike the well established classical theory over discrete structures, the theory of computation over continuous data is still in early stages of development, despite remarkable progress in recent years. Many important fundamental problems have not yet been studied, and presumably numerous unexpected and surprising results are waiting to be detected.The theory of computational learning developed in 1967, is applied in various area on computer science, such as proof checking. In this paper, we give a definition of learnable real functions based on computational learning. Here, we employ a strong definition of learnable real functions—Para-computability. Try to set up relation between this class of real learnable functions and the real recursive functions. The theory of analog computation, where the internal states of a computer are continuous rather than discrete, has enjoyed a recent resurgence of interest. Here, we explore recursion theory on the reals, the analog counterpart of recursive function theory. There, the discrete operations, such as primitive recursion is replaced by operations on continuous functions(differential recursion). We define recursive classes of real valued functions similarly to the classical approach in recursion theory and we study those classes to understand the computational power of operators that map real valued functions into real valued functions.We prove upper bounds (elementary computability) for continuous-time analog systems—the primitive real recursive functions, therefore, it can be characterized in terms of standard computational complexity classes. We propose a theory of real learnable functions. And here, we have a relation between para-computable functions and FDA\. We give the definition of learnable real numbers, and set up the relation with other hierarchy of real numbers.We also propose a definition of LGPAC, which based on GPAC-generable functions and taking of infinite limiting. We prove that LGPAC equals to Moore'sμ-hierarchy real recursive function and Mycka's L-hierarchy real recursive function. Our definition enjoy a realizable lowest hierarchy - GPAC and a basic operator in analysis - infinite limiting, therefore, we can remove Moore 'ambiguous operator—differential recursive.