What can I use as the second argument to Return in my own functions?

Question

Having found out about the second argument to Return I played a bit with it, and the following tests all resulted in Return::nofunc:

f[x_]:=x
f[Return[3,f]]

SetAttributes[g,HoldAll];g[x_]:=x
g[Return[3,g]]

h[x_]:=f[x]
h[Return[3,f]]

SetAttributes[i,HoldAll];i[x_]:=f[x]
i[Return[3,f]]

On the other hand, the following works fine:

Module[{i},Return[3,Module]]

Does that mean the second argument of Return can only be used with a predefined list of symbols? If so, is there an easy way to find out that list? Otherwise, what am I doing wrong (and how would I do it right)?

Answer 1

Ok, my view on this is consistent with all of the cases presented so far...

The symbols allowed in Return are those which are responsible for a transformation in the partial evaluation of the expression so far. Of course, I'm referring only to the evaluation of the branch that includes the Return. There's a subtlety but I'll mention it later

Let's clarify this with your examples:

f[x_]:=x
f[Return[3,f]]

f has no attributes so, Return is run before any transformation is made. Doesn't work

h[x_]:=f[x]
h[Return[3,f]]

Again, Return[3, f] is executed first.

SetAttributes[i,HoldAll];
i[x_]:=f[x]
i[Return[3,f]]

Here, i is HoldAll so now we start talking. The i[Return[3,f]]->f[Return[3, f]] is first made. i was the symbol responsible. Now, we run f[Return[3, f]], but f has no attributes, so the Return[3, f] is run, and again, can't find f. Could have found i however.

Last one (actually second on your list, but I thought I'd leave it for last). Here comes the subtlety: Actually, not all symbols responsible for transformations are available. When there's a transformation associated with a symbol, say, s1, and then there's another one AT THE SAME LEVEL, associated with the symbol s2, then s1 is shadowed by s2 and no longer available

SetAttributes[g,HoldAll];
g[x_]:=x
g[Return[3,g]]

g is HoldAll, so that's a good start. First step: g[Return[3, g]]->Return[3, g] and g is responsible for the transformation... So, it would seem that we are succeeding. HOWEVER, the very last transformation from Return[3, g] to whatever, makes the symbol Return shadow g. So, this could be fixed by just making the Return work at a lower level

SetAttributes[g,HoldAll];
g[x_]:=# &[x]
g[Return[3,g]]

It is interesting to note that Return[2, Return] works, and would have worked in the previous example too.

Ok, now let's go to the cases where it actually works

Module[{i},Return[3,Module]]

Module is HoldAll. First, there's a transformation associated to Module. If that transformation returned Return[3, Module], then this wouldn't work. But this works. So, I choose to believe that despite what Trace shows, Module evaluated the Return expression internally before returning, just like our g[x_]:=# &[x] example.

@belisarius on the comments:

f[x_] := g[Return[x, f]];
f[3]

f[3]->g[Return[3, f]]. f was responsible. Now, Return[3, f] is evaluated (at a lower level in the expression tree) and success.

Bonus example:

SetAttributes[f, HoldAll];
g /: f[i_, g] := {i}

Now

f[Return[2, g], g]

This actually works! Why?

f[Return[2,g], g] is turned into {Return[2, g]} BECAUSE OF g. Now, Return is evaluated (at a lower level) and that's the way the cookie crumbles

Summary

Imagine the expressions as trees, and the evaluation procedure as a succession of trees, and in each transformation, draw a sign pointing at the new sub-tree, with the symbol responsible written on it. If you ever have to transform that same subtree again, the previous sign is lost. When it's time for Return to evaluate, you go up the tree and see the signs in the path to the top. Those are the ones you can use.