• To: mathgroup at yoda.physics.unc.edu
• From: bert at netcom.com (Roberto Sierra)
• Date: Wed, 10 Nov 1993 17:04:32 -0800

```Gigi Ventura (VENTURA at pd.infn.it) and Giacomo Torzo
(TORZO at mvxpd5.pd.infn.it) pointed out the following 'strange'
behavior of MMA:

>In[1]:=
>        g1[x_]:=+x
>In[2]:=
>        g1[2,3,4]
>Out[2]=
>        9
>In[3]:=
>        g2[x_]:=-x
>In[4]:=
>        g2[2,3,4]
>Out[4]=
>        -24

which can be 'corrected' by the following:

>In[5]:=
>        g3[x_]:=-(+x)
>In[6]:=
>        g3[2,3,4]
>Out[6]=
>        -9

First of all, Giacomo appears to have mistyped Gigi's original
definitions by leaving out an underscore.  No doubt he meant

g1[x__]:=+x
g2[x__]:=-x
g3[x__]:=-(+x)

The reason that g2 behaves the way it does is not 'strange' at all,
but results from two fairly obscure internal aspects of Mathematica

First of all, +x really means Plus[x], -x really means Times[-1,x], and
-(+x) really means Times[-1,Plus[x]].  Though there is an operator Minus[x],
it expands automatically to Times[-1,x], and also only accepts a single
argument (thus Minus[a,b,c] would be illegal, unlike Plus[a,b,c]).  Note
that some other functions expand automatically, for example Sqrt[x]
expands out to Power[x,Rational[1,2]] wherever it appears.

Second, when a repeated pattern like x__ is used, a rather strange
internal object representation is used to pass multiple arguments
along to the 'right hand' side of the definition.  Argument sequences
like 2,3,4 are passed using Sequence objects, as in Sequence[2,3,4].
Unfortunately, the Sequence object is an undocumented internal object
that few people know about.  Its basic function appears to be to allow
argument sequences to be 'spliced' into other function calls.  For example,

Foo[a,Sequence[b,c,d],e]

is exactly equivalent to typing

Foo[a,b,c,d,e]

Thus, with g1[x__] defined as +x (or Plus[x]), the x argument is
passed to Plus as a Sequence object, so that g1[2,3,4] is evaluated as
Plus[Sequence[2,3,4]], which reduces to Plus[2,3,4], which reduces to 9.

With g2[x__] defined as -x (or Times[-1,x]), the x argument is also
passed as a Sequence object, so that g1[2,3,4] is exactly equivalent to
Times[-1,Sequence[2,3,4]], or Times[-1,2,3,4], or -24.  Though this is
not what one would expect *mathematically* -x to do, this is what one
would expect MMA to do from a *programmatic* standpoint, considering
that the *pattern* x corresponds to a sequence of arguments to be directly
spliced into the right hand equation, -x, which happens to be represented
internally as Times[-1,x].

My experience is that you need to be careful when you use a repeated
pattern which can match multiple arguments and not blindly paste them
into right hand expressions.  Typically, by introducing a 'fence' object
-- something that will convert volatile Sequence objects into something
safer, like a List -- you can work with multiple arguments with impunity.
Once converted to a List, you can manipulate the list to produce whatever
results are desired.  Thus, a 'clean' (though ugly) way to compute the
desired result might be

g4[x__] := - (Plus @@ {x})

This results in the following evaluation of g4[2,3,4]

Times[-1, Apply[Plus, List[Sequence[2,3,4]] ]]
Times[-1, Apply[Plus, List[2,3,4] ]]
Times[-1, Plus[2,3,4] ]
Times[-1, 9]
-9

Of course, Plus is also 'safe' to splice an argument sequence directly
into, so a simpler way to do the same thing would be

g3[x__] := -Plus[x]

This looks odd, but is exactly equivalent to Gigi's original definition
of g3, namely

g3[x__]:=-(+x)

which also produces the desired result, -9.

Hope this clears up the mystery.  Mathematica is strange, but not
*that* strange when you get to know it.

\\|//                         "Television is a medium -- it is
- -                           neither rare nor well done."
o o                                            -- Ernie Kovacs
J   roberto sierra
O   tempered microdesigns    NOTICE:
\_/  san francisco, ca        The ideas and opinions expressed
bert at netcom.com          herein are not those of the author.

```

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