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View Code? Open in Web Editor NEWOn the computational complexity of equivalence relations under kernel reductions
License: Other
On the computational complexity of equivalence relations under kernel reductions
License: Other
Does this help decrease the complexity of deciding whether a given set of pairs is an equivalence relation? Can we use this to do something with the complement of an equivalence relation?
Check that the PSPACEEq-complete problem is not kernel-sparse. This would be a contradiction with prop:density
in generalcompleteness.tex
.
Are polynomially clocked Turing machines necessary in our phrasing of thm:generalcompleteness
in generalcompleteness.tex
?
Show that a Sigma_kEq-complete problem implies a Sigma_k+1Eq-complete problem.
Change L(P_i)
and L(Q_i)
to A_i
and B_i
, respectively, for the sake of brevity in the proof of thm:diag
in intermediary.tex
.
Use real definitions for classes of "structures" in the introductory paragraphs of npeqcompleteness.tex
.
Can we use NP disjoint pairs to construct equivalence relations which have similar properties?
See also the more recent paper concerning propositional proof systems as well as canonical pairs for NP disjoint pairs.
npeqcompleteness.tex
discusses C-complete properties for arbitrary structures. We have in mind graphs when proving things in this section. Do there exist graph properties which induce complete problems for arbitrary complexity classes in the polynomial hierarchy?
If there exists a kernel-complete equivalence relation for a complexity class, must that class be closed under complement? Under polynomially-bounded universal quantification? Both?
This is the converse of thm:generalcompleteness
in generalcompleteness.tex
.
The next question would be this: if there is a kernel-complete equivalence relation for NPEq, does NP equal coNP?
This problem was suggested by Josh.
Show that given an equivalence relation with an infinite number of equivalence classes, we can construct a new equivalence relation, polynomial time kernel equivalent to it, with an infinite number of finite equivalence classes.
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