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functional style - Why are we so sure about Map results?


I do abuse it often but when I have faced this feature first time I was really suprised:


i = 0;
(i++; # - i) & /@ Range[5]



{0, 0, 0, 0, 0}



What am I asking about is how do we know Map will do 'mapping' position after position? Of course it might look logical but it is not stated, and then, since I know nothing about memory allocation etc., I could expect strange results. To be more precise, I could expect them only in cases where mapping funcion is changing during mapping.


I have failed to find a word in documentation, only related but not important in this case:



Leaves are visited before roots




Have I missed something?




Answer



I'm afraid my comment was too obscure to be noticed. Further, I disagree with one premise somewhere in the commentary, and I wish to make a fuller explanation to see if I understand correctly or incorrectly. Finally, I think the question is answered in the documentation on the Standard Evaluation Procedure:




  • Evaluate the head of the expression.

  • Evaluate each element in turn.

  • Apply transformations associated with the attributes Orderless, Listable, and Flat.

  • Apply any definitions that you have given.

  • Apply any built-in definitions.

  • Evaluate the result.




The first two implies that parts 0, 1, 2,... will be evaluated in order (unless one cavils that "in turn" does not imply order).


When Map[f, {1, 2, 3}] is evaluated, we get, after Map, f, and {1, 2, 3} are evaluated,


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

Next this List is evaluated, with f[1], f[2], f[3] being evaluated in turn. Thus with the OP's function, the side effect on i is defined. Trace will show that what happens conforms to standard evaluation (of course).


Note that the mapping part (applying f to each element of {1, 2, 3}) might be done in any order. What matters is that the intermediate list {f[1], f[2], f[3]} is then evaluated in a defined order (left-to-right).


So, I think this behavior is defined by the documentation.




Edit: An additional Reference


In the tutorial on Evaluation, it states in "an expression like h[Subscript[e, 1], Subscript[e, 2]\[Ellipsis]], Mathematica evaluates "each element Subscript[e, i] in turn." I think the "in turn" with the reference to the subscript i must mean in the natural order of 1, 2,.... There always seem to be questions of interpretation in documentation, but if writers did not mean that, I think they could be fairly criticized for misleading users, which I doubt they are.


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