Re: Scoped variables

*To*: mathgroup at smc.vnet.net*Subject*: [mg13924] Re: Scoped variables*From*: "Allan Hayes" <hay at haystack.demon.cc.uk>*Date*: Mon, 7 Sep 1998 01:22:49 -0400*References*: <6ssuub$m87@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Ersek, Ted R wrote in message <6ssuub$m87 at smc.vnet.net>... >Consider the lines below. You get nothing very interesting at Out[1]. > >In[n]:= >Clear[a]; >x=2; >{a_->x^2, x_->x^2, x_}//InputForm > >Out[1]= >{a_ -> 4, x_ -> 4, x_} > > >It works a little different if you do the same thing inside With (see >below). > >In[2]:= >With[{x=7},{a_->x^2, x_->x^2, x_}]//InputForm > >Out[2]= >{a$_ -> 49, x_ -> 4, Pattern[7, _]} > > >I understand there were extensive discussions on this sort of thing in >this group in recent years. However, I can't find anything, mostly >because I haven't figured out how to use the MathGroup search engine. >Anyway I figured some members would still be interested. > >Ted Ersek > > Ted: Scoping is a mechanism for avoiding clashes of variable If we enter f[y_] = Function[x,y+x] then we might expect that f[p] (p not equal to x) would be the function Function[x,p+x] that adds p to its input but that f[x] would be Function[x,x+x], which is the function that multiplies its input by 2. We are saved from this by a renaming mechanism that gives In[1]:= f[p] Out[1] Function[x$,p+x$] In[2]:= f[x] Out[2]= Function[x$,x+x$] Here is an attempt to describe what seems to go on (some of the terms are used in my own way). I hope that it is not too sketchy as to be useful and I would welcome observations and suggestions for improvement. Definitions The expressions outlined on the right below are called scoping constructs; the displayed x scopes all x in body. In any expression, if a symbol scopes or is scoped then it is bound,otherwise it is free. The targets of a variable such as the displayed x are defined in the right column. A target of the expression is any target of such an x. Outline expression Targets of x (.. x:pat ..)->(or:>)body all x in body (.. x:pat ..)= (or:=)body ,, (.. x:pat ..)/; body ,, Function[{..x..}, body] all x free in body With[{..x=.,or:=,..}, body] ,, Module[{..x,or x=., or x:=.,..}, body] ,, Evaluation When the following are used in an evaluation, each scoping construct in body that contains a target of the whole expression in its body has each of its bound symbols v that is not a target of the expression replaced by v$. The evaluation then proceeds Replacement by {(..)->(or:>) body,...} A call to stored (..)=(or:=) body (..)/; body Function[{..}, body][..] With[{..}, body] Module[{..}, body] Applying these ideas to your example In[3]:= With[{x=7},{a_->x^2, x_->x^2, x_}]//InputForm Out[3]//InputForm= {a$_ -> 49, x_ -> x^2, Pattern[7, _]} we see that x in {x=7) has as target the x in a_->x^2 and the x in x_, but not the x's in x_->x^2, which are not free. So a is replaced by a$ and then, by the way that With evaluates, each free x in {a$_ -> x^2, x_ -> x^2, Pattern[x, _]} is replaced by 7. The evaluation then continues to give the output shown above. Here are some more examples. In[4]:= Function[x,{a_->x^2, x_->x^2, x_}][7]//InputForm Out[4]//InputForm= {a$_ -> 49, x_ -> x^2, Pattern[7, _]} In[5]:= Module[{x=7},{a_->x^2, x_->x^2, x_}]//InputForm Out[5]//InputForm= {a$_ -> 49, x_ -> x^2, x$5_} (Module works via an assignment x$5 = 7 and Pattern has the attribute HoldFirstm so x$5_ does not become 7_ ) In[6]:= 7/.x_->{a_->x^2, x_->x^2, x_} Out[6]= {a$_ -> 49, Pattern[7, _] -> 49, Pattern[7, _]} (we also get several warnings) Allan ------------------------------------------------------------- Allan Hayes Training and Consulting Leicester UK http://www.haystack.demon.co.uk hay at haystack.demon.co.uk voice: +44 (0)116 271 4198 fax: +44(0)116 271 8642

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