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I have been working on picking expressions apart using Head and Part and encountered a little mystery. Consider the canonical example

a + b + c

which has FullForm

Plus[a, b, c]

We expect, then, (a+b+c)[[0]] === Plus === Head[a+b+c] to be True, and it is. But then, we would expect (a+b+c)[[1;;3]] would be {a, b, c}, wouldn't we? But it isn't. The following is true:

(a + b + c)[[1;;3]] === (a + b + c)

Somehow, [[1;;3]], which is supposed to pick off elements 1 through 3 of its argument and put them in a List, doesn't get rid of the Head, which is element 0!

The questions are, then (and I will be grateful for hints and answers!)

  1. Why doesn't (a + b + c)[[1;;3]] get rid of (a + b + c)[[0]], the Head?

  2. What is the right way to get rid of the Head?

  3. (a + b + c)[[0;;3]] produces 0. I would expect it to produce {Plus, a, b, c}. Instead, it produces 0. This just deepens the mystery for me! Why?

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Point #1

Part always wraps element sequences with the original head of the expression.

expr = Hold[1 + 1, 2 + 2, 3 + 3, 4 + 4, 5 + 5];

expr[[{2, 3}]]
Hold[2 + 2, 3 + 3]

For this purpose a single part e.g. 1 is not a sequence but {1} and 1 ;; 1 are:

expr[[1]]

expr[[{1}]]

expr[[1 ;; 1]]
2

Hold[1 + 1]

Hold[1 + 1]

This applies at every level of the extraction:

exp2 = g[h[1, 2], i[3, 4]];

exp2[[  2 , 1  ]]
exp2[[ {2}, 1  ]]
exp2[[  2 ,{1} ]]
exp2[[ {2},{1} ]]
3

g[3]

i[3]

g[i[3]]

I used this nontrivially for Elegant manipulation of the variables list.

Point #2

Consider instead using Extract which wraps sequences in List:

Extract[expr, {{2}, {3}}]
{4, 6}

The third argument of Extract can be used to specify a function to apply to individual elements before they are evaluated:

Extract[expr, {{2}, {3}}, HoldForm]
{2 + 2, 3 + 3}

If you want all parts you can also use Level:

Level[(a + b + c), {1}, Heads -> True]
{Plus, a, b, c}

Or Cases:

Cases[(a + b + c), _, Heads -> True]
{Plus, a, b, c}

Or Replace/ReplaceAll:

(a + b + c) /. head_[body___] :> {head, body}
{Plus, a, b, c}

Point #3

The last point is more tricky and I had to check it myself. There is a behavior that I also did not expect:

Range[5][[0 ;; 5]]
{}

What I expected was an error as seen here:

Range[5][[0 ;; 4]]

and here:

Range[5][[0 ;; 6]]

When Span is used in Part[x, 0 ;; n] where n is the length of x, Part returns the head of the expression. Therefore (a + b + c)[[0;;3]] returns Plus[] and Plus[] evaluates to 0.


I believe Span behaves this way because of how it handles non-positive values, and zero-length spans. Consider:

Range[10][[-2 ;; 10]]
{9, 10}

You can see that it wraps around. Now consider:

Range[10][[5 ;; 4]]
{}

An empty span returns the head of the expression with no arguments.

Using 0 ;; n where n is the last element in the list, or 0 ;; -1, is also an empty span wrapping around the open end of the list.

Range[10][[0 ;; -1]]
{}
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  • 2
    $\begingroup$ Ah, I see that you can answer #1 as well, not just Leonid :) $\endgroup$ – Verbeia Feb 29 '12 at 6:26
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    $\begingroup$ Regarding point #3, you might find this discussion relevant (if not too enlightening). $\endgroup$ – Szabolcs Feb 29 '12 at 11:47
  • $\begingroup$ @Szabolcs That's important information. I cannot recall a situation where it would affect me, but it would be bloody annoying to track it down if it did. $\endgroup$ – Mr.Wizard Feb 29 '12 at 16:36
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    $\begingroup$ And another place for the all-powerful Cases: Cases[{a + b + c}, head_[args___] -> {head, args}] produces {{Plus, a, b, c}} $\endgroup$ – Reb.Cabin Mar 8 '12 at 23:04
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    $\begingroup$ @Reb doing that route you don't even need Cases: a + b + c /. head_[args___] :> {head, args} $\endgroup$ – Mr.Wizard Mar 9 '12 at 0:06
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Perhaps only the developers (or Leonid) can answer #1 and #3 and "why".

The quick answer to #2 is Apply, remembering that List is a Head like any other:

List @@ (a + b + c)

{a, b, c}

On #3, I can't explain why ;; syntax gives 0 when

(a + b + c)[[Range[0, 3]]]

gives

a + b + c + Plus

So you could try

List @@ ((a + b + c)[[Range[0, 3]]])

{a, b, c, Plus}

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  • $\begingroup$ There is something broken in the way the site previews answers. I have been amending my answer for the last twenty minutes and only now do I see that you also posted an answer. $\endgroup$ – Mr.Wizard Feb 29 '12 at 6:48
  • $\begingroup$ @Mr.Wizard You don't get notified of a new answer when editing your already existing answer... $\endgroup$ – rm -rf Feb 29 '12 at 8:15
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    $\begingroup$ The List @@ answer has a deep, hackish charm, coming closest to answering "... and everything but Head.". The Extract and Level answers have more versatility and clearly enable complete analysis and compilation of expressions. A metacircular evaluator for MMA would not seem too remote a possibility. $\endgroup$ – Reb.Cabin Feb 29 '12 at 8:17
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    $\begingroup$ In some sense, Sequence@@(a+b+c) comes closer to "everything but Head" than List@@(a+b+c): While List is just another head (with a special input/display syntax associated with it), Sequence is the closest Mathematica has to a headless expressions: If you insert it anywhere (except functions with attribute HoldAllComplete) it just gets interpolated. $\endgroup$ – celtschk May 21 '12 at 19:35
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    $\begingroup$ @celtschk For completeness the attribute SequenceHold also prevents the flattening of Sequence expressions, just as HoldAllComplete does, while allowing other evaluation. $\endgroup$ – Mr.Wizard Aug 14 '17 at 6:13

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