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MathGroup Archive 2005

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Re: simultaneous ... and ___

  • To: mathgroup at smc.vnet.net
  • Subject: [mg60920] Re: simultaneous ... and ___
  • From: Maxim <ab_def at prontomail.com>
  • Date: Mon, 3 Oct 2005 04:06:26 -0400 (EDT)
  • References: <dhj4ll$smo$1@smc.vnet.net><dhlc4l$cuj$1@smc.vnet.net> <dhnso6$1eu$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

On Sun, 2 Oct 2005 05:55:18 +0000 (UTC), borges2003xx at yahoo.it  
<borges2003xx at yahoo.it> wrote:

> sorry but for a refuse (copy and paste ) i write
> m1[{___, x1__ .., x2___ ..., x3___ ..., ___, x3___ ..., x2___ ...,
>       x1__ .., ___}] := {}
> /; (Length[{x1}] + Length[{x1}] + Length[{x1}]) >= 15
> m1[l1_] := l1
>
> the real version is
> m1[{___, x1__ .., x2___ ..., x3___ ..., ___, x3___ ..., x2___ ...,
>       x1__ .., ___}] := {}
> /; (Length[{x1}] + Length[{x2}] + Length[{x3}]) >= 15 (or 5 or...)
> m1[l1_] := l1
>
> i apologies
> i use triple ___ for x2 and x3 cause i don't want to be filled always
> with something
>

The key point seems to be that Repeated cannot repeat sequences  
(BlankSequence or another Repeated):

In[1]:= MatchQ[{a, b, a, b}, {(x__)..}]

Out[1]= False

In principle there could be a match if x were set to Sequence[a, b]. But  
Mathematica pattern matching doesn't do that (string patterns are  
different in that aspect). The pattern (x__).. can be matched only if x  
can be set to a sequence of length 1:

In[2]:= MatchQ[{a, a}, {(x__)..} /; Length@ {x} == 1]

Out[2]= True

In[3]:= MatchQ[{a, a}, {(x__)..} /; Length@ {x} == 2]

Out[3]= False

As we can see, x can be set to a, which provides a match, but not to  
Sequence[a, a]. In my opinion, this gives the user the wrong impression  
that sequences can be repeated, when in fact this isn't quite so.

This leads to certain inconsistencies:

In[4]:= MatchQ[{a, a}, {x__, (x__)..}]

Out[4]= False

In[5]:= MatchQ[{a, a}, {x__, (y__)..} /; {x} === {y}]

Out[5]= True

It is hard to explain why the patterns x and y cannot have the same name,  
if they can match identical subexpressions (this also shows that it is not  
clear how this construct works even with one-element sequences).

One might try to use (x_).. inside a Flat function (say, Plus) to make  
Mathematica automatically group a sequence inside Plus so that it can be  
matched by x_ instead of x__. But that doesn't work either:

In[6]:= MatchQ[Hold[a + b + a + b], Hold[Plus[x_, x_]]]

Out[6]= True

In[7]:= MatchQ[Hold[a + b + a + b], Hold[Plus[(x_)..]]]

Out[7]= False

Mathematica rewrites the expression as Plus[Plus[a, b], Plus[a, b]] only  
in the first example.

One way around this is to use the sequence x___, y___ and add a test to  
verify that y comprises repetitions of x. Also nested repeats of that kind  
can be done via string patterns:

In[8]:=
StringCases[
   StringJoin[ToString /@ {0,1,1,2,2,2,3,4,3,3,3,3,2,1,1,1,1,8}],
   x1__..~~x2__..~~x3__..~~___~~x3__..~~x2__..~~x1__.. /;
       StringLength[x1~~x2~~x3] >= 4 :>
     {x1, x2, x3}]

Out[8]=
{{"11", "2", "3"}}

This correctly determines that x1 can be set to "11" (repeated once before  
x2 and twice after x2), and so on. I used __.. to avoid trivial matches,  
e.g., all elements of the pattern except one set to empty strings.

Maxim Rytin
m.r at inbox.ru


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