I've been thinking for a while about different ways two Turing degrees might be "independent" of each other (from the point of view of computability theory). The simplest such notion would be to say that they have no information in common: $ d\_0\wedge d\_1= $ **0**. This is a very natural notion.


A different notion of independence that I've been playing around with, is the following: say that degrees $d\_0$, $d\_1$ are independent if $$ \forall A\in d\_0,\forall B\in d\_1, A\Delta B\equiv\_T A\oplus B$$ (where "$\Delta$" denotes symmetric difference). The intuition behind this definition is that two sets are independent if there is no way to lay sets equivalent to them over each other and "cancel out" any of the information contained in them. Note that this is equivalent to asking that $\forall A, C\in d\_0, \forall B, D\in d\_1, A\Delta B\equiv\_T C\Delta D$.


This is a somewhat odd notion of independence. For instance, a *strong minimal cover* of a degree $d$ is a degree $e>\_T d$ such that for all $a<\_T e$, $a\le\_T d$. Under this definition, if $e$ is a strong minimal cover of $d$ then $e$ and $d$ are independent: since for $A\in e, B\in d$, we have $(A\Delta B)\oplus B\equiv\_T A\oplus B\equiv\_T A>\_T B$, so $A\Delta B\not \le\_T B$, but since $A>\_T B$ we have $A\ge\_T A\Delta B$, so $A\Delta B\equiv\_T A\equiv\_T A\oplus B$ since $e$ is a strong minimal cover of $d$. Also, this version of independence is not obviously definable in the u.s.l. of Turing degrees. Still, I've found it interesting to play with, but I haven't made much progress towards understanding it, so my question is: has this (or anything like this) been looked at, and more generally, what are some sources on the possible degrees of the symmetric differences of sets from prescribed degrees?



Notion of independence of Turing degrees
Rather belatedly, I should probably mention that Uri Andrews, Peter Gerdes, Steffen Lempp, Joe Miller, and I have written a paper on this: [*Computability and the symmetric difference operator*](https://people.math.wisc.edu/%7Ejmiller/Papers/symdiff_deg.pdf) ([DOI](https://academic.oup.com/jigpal/article-abstract/30/3/499/6291047?redirectedFrom=fulltext)); see also [these slides of Peter](http://invariant.org/talks/sym-diff-talk.pdf).


In particular, we proved that there are incomparable r.e. degrees ${\bf a},{\bf b}$ with well-defined symmetric difference degree but for which there is *also* an r.e. ${\bf c}<\_T{\bf a\oplus b}$ such that ${\bf c\oplus a}\ge\_T{\bf b}$ and ${\bf c\oplus b}\ge\_T{\bf a}$. The general question of which nonzero r.e. degrees are one half of a "symmetric pair" is still open; Peter(?) conjectured that the answer is the *promptly simple* degrees, but I dont' know enough about that topic to have an opinion.


