After having searched a bit, it seems I can't find terminology or references for this outermost class, 2Σ* in blue -- see below. What is it describing?

enter image description here

  • 2
    $\begingroup$ It's the power set of all sentences over $\Sigma$. In other words, it is all possible languages built from the alphabet $\Sigma$, since a language is just a subset of $\Sigma^*$, and the power set describes all possible subsets of a set. $\endgroup$
    – rici
    Nov 18, 2021 at 18:11
  • 1
    $\begingroup$ Several comments on the diagram. 1. A language is either "recognizable" or "not recognizable"; there should be no room outside of these two boxes. 2. Whether $\text{NP} \neq \text{P}$ is not known/proved. 3. Whether $\text{PSPACE}\neq\text{EXPTIME}$ is not known/proved. 4. Any given language can be included in a "Turing degree." $\endgroup$
    – John L.
    Nov 18, 2021 at 18:58
  • $\begingroup$ @JohnL. it seems to be missing caveats like "contained in" or <= or ?=, rather than a distinct = or . I don't have a caption for this figure. $\endgroup$ Nov 19, 2021 at 14:19
  • $\begingroup$ Good point. However, had I drawn the diagram, I would have drawn the boundary of a box differently, such as dotted lines, if the box does not mean strict containment by the area outside of it. $\endgroup$
    – John L.
    Nov 19, 2021 at 15:52

1 Answer 1


$2^{\Sigma^{*}}$ means the powerset of the full language $\Sigma^{*}$. It means the set of all subsets of $\Sigma^{*}$, including the empty set and $\Sigma^{*}$ itself, i.e., all possible languages with alphabet $\Sigma$.

Here is the simple understanding. When we want to describe a language $X$ with alphabet $\Sigma$, for each string $w$ in $\Sigma^{*}$, there are $2$ choices, including $w$ in $X$ or excluding $w$ from $X$. All these choices are independent.


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