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pattern matching - Variable scoping problem when mapping over delayed replacement


This is something that got me curious while I was playing around with Mathematica. Consider the following (contrived) example:


Clear[x,y];

{{1, 2}, {3, 4}, {5, 6}} /. {x_, y_} :> {x, #} & /@ {y, y^2, y^3}
(* {{{1, 2}, {3, 4}, {5, 6}}, {{1, 4}, {3, 16}, {5, 36}}, {{1, 8}, {3, 
64}, {5, 216}}} *)

No problems there.


Now suppose y has been defined globally, so


y = 10;
{{1, 2}, {3, 4}, {5, 6}} /. {x_, y_} :> {x, #} & /@ {y, y^2, y^3}
(* {{{1, 10}, {3, 10}, {5, 10}}, {{1, 100}, {3, 100}, {5, 100}}, {{1, 
1000}, {3, 1000}, {5, 1000}}} *)


The problem is that y gets evaluated to 10 before getting mapped to the delayed rule's rhs.


I realised I could do


{{1, 2}, {3, 4}, {5, 6}} /. {x_, y_} :> {x, 
 ReleaseHold[#]} & /@ Thread[Hold[{y, y^2, y^3}]]
(* {{{1, 2}, {3, 4}, {5, 6}}, {{1, 4}, {3, 16}, {5, 36}}, {{1, 8}, {3, 
64}, {5, 216}}} *)

which looks kinda complicated, but works.


So, is there any better way to "shield" the y in the list being mapped over, from the global variable y?



I sort of randomly tried
Module[{y}, {{1, 2}, {3, 4}, {5, 6}} /. {x_, y_} :> {x, #} & /@ {y, y^2, y^3}]


but in this case the temporary variable's name(?) got substituted


(* {{{1, y$119596}, {3, y$119596}, {5, y$119596}}, {{1, y$119596^2}, {3, 
   y$119596^2}, {5, y$119596^2}}, {{1, y$119596^3}, {3, 
   y$119596^3}, {5, y$119596^3}}} *)

Answer



You have already seen that there are a number of ways to skin this cat but I'd like to add some comments of my own.


Other approaches


When possible I prefer to avoid these situations in the first place, instead using something like:



{{1, 2}, {3, 4}, {5, 6}} /. {x_, y_} :> Thread@{x, {y, y^2, y^3}} // Thread


{{{1, 2}, {3, 4}, {5, 6}}, {{1, 4}, {3, 16}, {5, 36}}, {{1, 8}, {3, 64}, {5, 216}}}

Here:



  1. Pattern names x and y remain local

  2. Pattern matching is done only once, rather than three times



You can also avoid interactions with global symbols either by using Block or Formal Symbols. Formal Symbols have the Protected attribute and exist expressly for avoiding Imperial Global entanglements. Block requires foreknowledge of the names of the troublesome symbols, and must be updated any time they are to avoid introducing a bug. Example of Block usage:


Block[{y},
 {{1, 2}, {3, 4}, {5, 6}} /. {x_, y_} :> {x, #} & /@ {y, y^2, y^3}
]


Substitutions inside RuleDelayed


Regarding substitutions into the right-hand side of RuleDelayed there is an important caveat: Mathematica transparently renames variables in nested scoping constructs, including RuleDelayed. Observe:


Cases[Hold[y, y^2, y^3], foo_ :> {x_, y_} :> {x, foo}]



{{x$_, y$_} :> {x$, y}, {x$_, y$_} :> {x$, y^2}, {x$_, y$_} :> {x$, y^3}}

Note that the pattern names on the LHS have changed to x$ and y$ and the RHS does not match. For this reason the best way to make such substitutions is often using a pure function with Slot or SlotSequence (# and ##) parameters which does not cause such renaming to take place. However, pure functions by default will evaluate arguments so you must guard against this. One way is to use Hold and ReleaseHold as you did. Another is to use Unevaluated on each individual argument:


List @@ ({{1, 2}, {3, 4}, {5, 6}} /. {x_, y_} :> {x, #} & /@ 
   Unevaluated /@ Hold[y, y^2, y^3])

You can also use an undocumented syntax of Function that combines the unnamed Slot parameters with the ability to specify function Attributes:


Function[Null, body, attributes]


Null can be implicit, therefore:


Function[, {{1, 2}, {3, 4}, {5, 6}} /. {x_, y_} :> {x, #}, HoldAll] /@ 
  Unevaluated[{y, y^2, y^3}]

Note that in this case Unevaluated is used to prevent evaluation not in our Function (as above), but rather in Map (/@).


It is possible to wrestle Mathematica into making the substitution without renaming as described in the linked answer above, e.g.:


rls = Cases[Hold[y, y^2, y^3], foo_ :> RuleDelayed @@ Hold[{x_, y_}, {x, foo}]]


{{x_, y_} :> {x, y}, {x_, y_} :> {x, y^2}, {x_, y_} :> {x, y^3}}


Combined with the syntax expr /. {{rules1}, {rules2}, {rules3}} to apply multiple rule lists we may write:


{{1, 2}, {3, 4}, {5, 6}} /.
 Cases[Hold[y, y^2, y^3], foo_ :> {RuleDelayed @@ Hold[{x_, y_}, {x, foo}]}]


{{{1, 2}, {3, 4}, {5, 6}}, {{1, 4}, {3, 16}, {5, 36}}, {{1, 8}, {3, 64}, {5, 216}}}

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