Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: ,  examine a recently introduced type of eective reduction which applies solely to problems of equivalence or isomorphism: the \kernel reduction". Specically,  examine reductions among languages in the complexity class consisting of all languages induced by equivalence relations for which membership can be decided by a non-deterministic polynomial time Turing machine. This class is called NPEq; the denitions for PEq and coNPEq are analagous.  prove a general theorem which provides a problem which is hard under polynomial time kernel reductions for several classes of equivalence relations, including kPEq and PSPACEEq. In fact, such a problem is complete for PSPACEEq under polynomial time kernel reductions.  also show that if there is a complete problem under kernel reductions in NPEq, then that problem is also complete under many-one reductions in NP. Finally  use a proof of Ladner’s theorem to show that if PEq6 NPEq and there are problems in NPEq which are complete under polynomial time kernel reductions then there are NPEq-intermediary problems|problems which are in NPEq, but not complete under kernel reductions and not in PEq.
['Jeffrey Finkelstein', 'Benjamin J. Hescott']
I am writing a research paper on a idea Polynomial-time kernel reductions
[{"from": "human", "value": "I am writing a research paper on a idea Polynomial-time kernel reductions"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: ,  examine a recently introduced type of eective reduction which applies solely to problems of equivalence or isomorphism: the \\kernel reduction\". Specically,  examine reductions among languages in the complexity class consisting of all languages induced by equivalence relations for which membership can be decided by a non-deterministic polynomial time Turing machine. This class is called NPEq; the denitions for PEq and coNPEq are analagous.  prove a general theorem which provides a problem which is hard under polynomial time kernel reductions for several classes of equivalence relations, including kPEq and PSPACEEq. In fact, such a problem is complete for PSPACEEq under polynomial time kernel reductions.  also show that if there is a complete problem under kernel reductions in NPEq, then that problem is also complete under many-one reductions in NP. Finally  use a proof of Ladner\u2019s theorem to show that if PEq6 NPEq and there are problems in NPEq which are complete under polynomial time kernel reductions then there are NPEq-intermediary problems|problems which are in NPEq, but not complete under kernel reductions and not in PEq."}]