Error generating localized variables (as constants)

Question

The usage message for Set reminds us that multiple assignments can easily be made across two lists, without having to rip anything apart. For example:

Remove[x1, x2, y1, y2, z1, z2];
{x1, x2} = {a, b}

Performs the assignment and returns:

{a, b}

Thread, commonly used to generate lists of rules, can also be called explicitly to achieve the same outcome:

Thread[{y1, y2} = {a, b}]
Thread[{z1, z2} -> {a, b}]

Gives:

{a, b}
{z1 -> a, z2 -> b}

However, employing this approach to generate localized constants generates an error. Consider this trivial example function:

Remove[f];
f[x_] :=
 With[{{x1, x2} = {a, b}},
  x + x1 + x2
  ]
f[z]

Here the error message:

With::lvset: "Local variable specification {{x1,x2}={a,b}} contains 
{x1,x2}={a,b}, which is an assignment to {x1,x2}; only assignments 
to symbols are allowed."

The error message documentation (ref/message/With/lvw), says in the 'More Information' section that, "This message is generated when the first element in With is not a list of assignments to symbols." Given this explanation, I understand the mechanics of why my assignment failed. Nonetheless, I'm puzzled and wondering if this is necessary restriction by WRI, or a minor design oversight that should be reported.

So here's my question:

Can anyone shed some light on this behavior and/or offer a workaround? I experimented with trying to force Evaluation, without luck, and I'm not sure what else to try.

Answer 1

What you request is tricky. This is a job for macros, as already exposed by the others. I will explore a different possibility - to use the same symbols but put some wrappers around the code you want to write. The advantage of this technique is that the code is transformed "lexically" and at "compile-time", rather than at run-time (as in the other answers). This is generally both faster and easier to debug.

So, here is a function which would transform the With with your proposed syntax:

Clear[expandWith];
expandWith[heldCode_Hold] :=
 Module[{with}, 
   heldCode /. With -> with //. {
       HoldPattern[with[{{} = {}, rest___}, body_]] :> 
              with[{rest}, body],
       HoldPattern[
         with[{
           Set[{var_Symbol, otherVars___Symbol}, {val_, otherVals___}], rest___}, 
           body_]] :>
              with[{{otherVars} = {otherVals}, var = val, rest}, body]
     } /. with -> With]

Note that this operates on held code. This has the advantage that we don't have to worry about possible evaluation o the code neither at the start nor when expandWith is finished. Here is how it works:

In[46]:= expandWith@Hold[With[{{x1,x2,x3}={a,b,c}},x+x1+x2+x3]]
Out[46]= Hold[With[{x3=c,x2=b,x1=a},x+x1+x2+x3]]

This is, however, not very convenient to use. Here is a convenience function to simplify this:

ew = Function[code, ReleaseHold@expandWith@Hold@code, HoldAll]

We can use it now as:

In[47]:= ew@With[{{x1,x2}={a,b}},x+x1+x2]
Out[47]= a+b+x

So, to make the expansion happen in the code, simply wrap ew around it. Here is your case for the function's definition:

Remove[f];
ew[f[x_] := With[{{x1, x2} = {a, b}}, x + x1 + x2]]

We now check and see that what we get is an expanded definition:

?f
Global`f
f[x_]:=With[{x2=b,x1=a},x+x1+x2]

The advantage of this approach is that you can wrap ew around an arbitrarily large chunk of your code. What happens is that first, expanded code is generated from it, as if you would write it yourself, and then that code gets executed. For the case of function's definitions, like f above, we cansay that the code generation happens at "compile-time", so you avoid any run-time overhead when usin the function later, which may be substantial if the function is called often.

Another advantage of this approach is its composability: you can come up with many syntax extensions, and for each of them write a function similar to ew. Then, provided that these custom code-transforming functions don't conlict with each other, you can simply compose (nest) them, to get a cumulative effect. In a sense, in this way you create a custom code generator which generates valid Mathematica code from some Mathematica expressions representing programs in your custom languuage, that you may create within Mathematica using these means.

EDIT

In writing expandWith, I used iterative rule application to avoid dealing with evaluation control, which can be a mess. However, for those interested, here is a version which does some explicit work with unevaluated pieces of code.

Clear[expandWithAlt];
expandWithAlt[heldCode_Hold] :=
 Module[{myHold},
    SetAttributes[myHold, HoldAll];
    heldCode //. HoldPattern[With[{Set[{vars__}, {vals__}]}, body_]] :>
     With[{eval = 
              (Thread[Unevaluated[Hold[vars] = Hold[vals]], Hold] /.
                   Hold[decl___] :> myHold[With[{decl}, body]])},
       eval /; True] //. myHold[x_] :> x]

I find it considerably more complicated than the first one though.

Answer 2

The tricky issue is to keep the first argument of Set unevaluated. Here is my suggestion (open to improvements of course):

  SetAttributes[myWith, HoldAll];
    myWith[{s : Set[a_List, b_List]}, body_] :=
     ReleaseHold@
      Hold[With][
       Table[Hold[Set][Extract[Hold[s], {1, 1, i}, Hold], 
         Extract[Hold[s], {1, 2, i}]], {i, Length@b}], Hold@body]
    x1 = 12;
    Remove[f];
    f[x_] := myWith[{{x1, x2} = {a, b}}, x + x1 + x2]
    f[z]

results in

a+b+z

Inspired by halirutan below I think his solution, made slightly more safely, is equivalent to the above:

SetAttributes[myWith, HoldAll];
myWith[{Set[a : {__Symbol}, b_List]} /; Length[a] == Length[b], 
  body_] := 
 ReleaseHold@
  Hold[With][
   Replace[Thread[Hold[a, b]], Hold[x_, y_] :> Hold[Set[x, y]], 1], 
   Hold@body]

Answer 3

The tutorial "LocalConstants" says

The way With[{x=Subscript[x, 0],...},body] works is to take body, and replace every occurrence of x, etc. in it by Subscript[x, 0], etc. You can think of With as a generalization of the /. operator, suitable for application to Mathematica code instead of other expressions.

Referring to this explanation it seems obvious that something like

x + x1 + x2 /. {x1, x2} -> {a, b}

will not work as it might be expected in the With notation.

Let's assume you really want to hack around this. With[] has the attribute HoldAll, therefore everything you give as first parameter is not evaluated. To make such a vector-assignment work you would have to create

With[{x1=a, x2=b}, ...]

from the vector-notation. Unfortunately,

Thread[{a, b} = {1, 2}]

does not work because the argument to Thread is not held and the assignment is evaluated before Thread can do anything.

Lets fix this

SetAttributes[myThread, HoldFirst];
myThread[Set[a_, b_]] := mySet @@@ Transpose[{a, b}]

gives

In[31]:= myThread[{a, b, c} = {1, 2, 3}]
Out[31]= {mySet[a, 1], mySet[b, 2], mySet[c, 3]}

What looks promising at first, just moved the problem a bit away. To use this in With[] you have to replace at some point the mySet with the real Set. Exactly then, With[] does not see the list {a=1, b=2, c=3} but, since it has to be evaluated, the result of all assignments

In[32]:= With[
 Evaluate[myThread[{a, b, c} = {1, 2, 3}] /. mySet :> Set], a + b + c]

During evaluation of In[32]:= With::lvw: Local 
variable specification {1,2,3} contains 1, which is not an assignment to a symbol. >>

Out[32]= With[{1, 2, 3}, a + b + c]

There seems to be not easy way around this and there is a second question here: If there is a way around this restriction, is it as fast as With would be or do we lose the speed advantage compared to Module? And if speed is not so important, why not using Module or Block in the first place?

Answer 4

You could use Transpose to shorten Rolfs solution by 100 characters:

SetAttributes[myWith, HoldAll];
myWith[{Set[a_List, b_List]}, body_] := 
 ReleaseHold[Hold[With][Hold[Set[#1, #2]] & @@@ Transpose[{a, b}],
   Hold@body
   ]]

@Heike, yep the above breaks if either variable has already a value. What about this:

SetAttributes[myWith, HoldAll];
myWith[{Set[a_List, b_List]}, body_] := 
 ReleaseHold@
  Hold[With][Thread[Hold[a, b]] /. Hold[p__] :> Hold[Set[p]], 
   Hold@body]