Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   study a reducibility that has been introduced by Klaus ihrauch or, more precisely, a natural extension of this reducibility for multi-valued functions on represented spaces.  call the corresponding equivalence classes ihrauch degrees and  show that the corresponding partial order induces a lor semi-lattice with the disjoint union of multi-valued functions as greatest lor bound operation.  show that parallelization is a closure operator for this semi-lattice and the parallelized ihrauch degrees even form a lattice with the product of multi-valued functions as greatest lor bound operation.  show that the Medvedev lattice and hence the Turing upper semi-lattice can both be embedded into the parallelized ihrauch lattice in a natural way. The importance of ihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the ihrauch reductions beten theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. This allows a new purely topological or computational approach to metamathematics that sheds new light on the nature of theorems. As crucial corner points of this classification scheme  study the limited principle of omniscience $\LPO$, the lesser limited principle of omniscience $\LLPO$ and their parallelizations.  show that parallelized $\LLPO$ is equivalent to ak Konig's Lemma and hence to the Hahn-Banach Theorem in this new and very strong sense.  call a multi-valued function akly computable if it is reducible to the ihrauch degree of parallelized $\LLPO$ and  present a new proof that the class of akly computable operations is closed under composition. This proof is based on a computational version of Kleene's ternary logic. Moreover,  characterize akly computable operations on computable metric spaces as operations that admit upper semi-computable compact-valued selectors and  show that any single-valued akly computable operation is already computable in the ordinary sense.
['Vasco Brattka', 'Guido Gherardi']
I am writing a research paper on a idea Weihrauch Degrees, Omniscience Principles and Weak Computability.
[{"from": "human", "value": "I am writing a research paper on a idea Weihrauch Degrees, Omniscience Principles and Weak Computability."}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:   study a reducibility that has been introduced by Klaus ihrauch or, more precisely, a natural extension of this reducibility for multi-valued functions on represented spaces.  call the corresponding equivalence classes ihrauch degrees and  show that the corresponding partial order induces a lor semi-lattice with the disjoint union of multi-valued functions as greatest lor bound operation.  show that parallelization is a closure operator for this semi-lattice and the parallelized ihrauch degrees even form a lattice with the product of multi-valued functions as greatest lor bound operation.  show that the Medvedev lattice and hence the Turing upper semi-lattice can both be embedded into the parallelized ihrauch lattice in a natural way. The importance of ihrauch degrees is based on the fact that multi-valued functions on represented spaces can be considered as realizers of mathematical theorems in a very natural way and studying the ihrauch reductions beten theorems in this sense means to ask which theorems can be transformed continuously or computably into each other. This allows a new purely topological or computational approach to metamathematics that sheds new light on the nature of theorems. As crucial corner points of this classification scheme  study the limited principle of omniscience $\\LPO$, the lesser limited principle of omniscience $\\LLPO$ and their parallelizations.  show that parallelized $\\LLPO$ is equivalent to ak Konig's Lemma and hence to the Hahn-Banach Theorem in this new and very strong sense.  call a multi-valued function akly computable if it is reducible to the ihrauch degree of parallelized $\\LLPO$ and  present a new proof that the class of akly computable operations is closed under composition. This proof is based on a computational version of Kleene's ternary logic. Moreover,  characterize akly computable operations on computable metric spaces as operations that admit upper semi-computable compact-valued selectors and  show that any single-valued akly computable operation is already computable in the ordinary sense."}]