Re: Re: Orderless indexed functions

*To*: mathgroup at smc.vnet.net*Subject*: [mg20919] Re: [mg20891] Re: [mg20874] Orderless indexed functions*From*: "N Monomachoff" <nmonomachoff at email.msn.com>*Date*: Wed, 1 Dec 1999 01:49:51 -0500 (EST)*References*: <819lat$3j9$4@dragonfly.wolfram.com>*Sender*: owner-wri-mathgroup at wolfram.com

If you can be sure that {n_Integer} never appears in your expressions, then by far the simplest method is SetAttributes[f,Orderless] and representing the indexed form by f[{n_},x___] It doesn't matter where in the function the 'index list' appears, since it can always be distinguished. The full complement of pattern matching is retained, with little overhead. For example, my definition f[n_][x___,f[0][y___]]:=f[n][x,y] is fully represented by f[{n_},x___,f[{0},y___]]:=f[{n},x,y] N Monomachoff nmonomachoff at msn.com Andrzej Kozlowski wrote in message <819lat$3j9$4 at dragonfly.wolfram.com>... >Just one more brief remark on this topic. It seems to me now that my usage >of the Notation package in this context was really a bit of a red herring. >One can get the same result more simply as follows: > >In[1]:= >f[i_] := ToExpression[ToString[f] <> ToString[i]] > >In[2]:= >f[3] >Out[2]= >f3 > >In[4]:= >Do[SetAttributes[Evaluate[f[i]], Orderless], {i, 1, 50}] > >In[5]:= >f[1][c, a, b] >Out[5]= >f1[a, b, c] > >As long as one needs to deal only with a finite (reasonably large) number of >functions indexed by integers this ought to be enough. > > >> >> This seems to be a very hard problem. All I can suggest at this time is the >> following. If you can be satisfied with defining only a finite number of >> f[n]'s with n an integer between, say, 0 and 100 (in principle you could use >> a very large number) you could do something like this: >> >> << Utilities`Notation` >> >> In[2]:= >> \!\(\* >> RowBox[{"Symbolize", "[", >> TagBox[\(f\__\), >> NotationBoxTag, >> TagStyle->"NotationTemplateStyle"], "]"}]\) >> >> In[3]:= >> f[i_] := ToExpression[ >> "f\[UnderBracket]Subscript\[UnderBracket]" <> ToString[i]] >> >> In[4]:= >> Do[SetAttributes[Evaluate[f[i]], Orderless], {i, 1, 100}] >> In[5]:= >> Do[f[i][x___, f[0][y___]] = f[i][x, y], {i, 1, 100}]; >> >> Now you get >> >> In[9]:= >> f[5][c, b, f[0][a]] >> Out[9]= >> f\[UnderBracket]Subscript\[UnderBracket]5[a, b, c] >> >> which seems to be right. You could also use the same method to define other >> values for f[i]. The only problem I see at the moment is that you can't get >> more than a finite number of f[i]'s. However, for some purposes this maybe >> enough. I can's see at this time how to overcome this restriction. >> >> Andrzej >> >> >> -- >> >> >>> From: "N Monomachoff" <nmonomachoff at email.msn.com> To: mathgroup at smc.vnet.net >>> Date: Sun, 21 Nov 1999 00:59:58 -0500 >>> To: "Andrzej Kozlowski" <andrzej at tuins.ac.jp>, <hay at haystack.demon.co.uk>, >>> <mathgroup at smc.vnet.net>, <hwolf at debis.com> >>> Subject: [mg20919] Re: [mg20891] Re: [mg20874] Orderless indexed functions >>> >>> Dear Andrzej Kozlowski, >>> >>> I want to thank you for the extraordinary efforts you've expended on this >>> problem. Had I appreciated how nontrivial my original question was, I >>> would have been much more explicit: >>> >>> Not only do I need pattern matching within the body of the function, I need >>> to pattern match the index itself, as in the following: >>> >>> f[n_][x___,f[0][y___]]:=f[n][x,y] >>> >>> The ability to do this is precisely why I chose the indexed form. >>> Unfortunately, none of the proposed solutions, not even your last, seems to >>> allow for such definitions. Apparently the Notation package is unable to >>> stretch this far. Even a definition as simple as: >>> >>> \!\(f\_n_ := h[n]\) >>> >>> doesn't work. On the other hand, I haven't delved into the deeper parts of >>> the Notation package, so perhaps I'm wrong. >>> >>> N Monomachoff >>> nmonomachoff at msn.com >>> >>> >>> >>>> From: Andrzej Kozlowski <andrzej at tuins.ac.jp> To: mathgroup at smc.vnet.net >>>> Date: Sat, 20 Nov 1999 17:28:49 +0900 >>>> To: Andrzej Kozlowski <andrzej at tuins.ac.jp>, <mathgroup at smc.vnet.net> >>>> Subject: [mg20919] Re: [mg20891] Re: [mg20874] Orderless indexed functions >>>> >>>> I think I have worked out the right solution to your problem, which will >>>> produce an indexed function with an Orderless attribute. This is what you >>>> have to do. First, you need to load the Notation package: >>>> >>>> In[2]:= >>>> << Utilities`Notation` >>>> >>>> (I was wrong to say you need ot move it inot the Autoload folder since the >>>> ExtraPackages folder is actually included in $Path ) >>>> >>>> Okay, now you symbolize "f\[UnderBracket]Subscript\[UnderBracket]i" using >>>> the Notation palette: >>>> >>>> In[4]:= >>>> \!\(\* >>>> RowBox[{"Symbolize", "[", >>>> TagBox[\(f\__\), >>>> NotationBoxTag, >>>> TagStyle->"NotationTemplateStyle"], "]"}]\) >>>> >>>> >>>> >>>> Once you have done this you need a function that will convert f[i] to >>>> "f\[UnderBracket]Subscript\[UnderBracket]i". The following will work: >>>> >>>> >>>> In[9]:= >>>> f[i_] := ToExpression[ >>>> "f\[UnderBracket]Subscript\[UnderBracket]" <> ToString[i]] >>>> >>>> e.g. >>>> >>>> In[10]:= >>>> f[3] >>>> Out[10]= >>>> f\[UnderBracket]Subscript\[UnderBracket]3 >>>> >>>> Now all you need to do is >>>> >>>> In[13]:= >>>> \!\(SetAttributes[f\_3, Orderless]\) >>>> >>>> and you have what you wanted: >>>> >>>> In[14]:= >>>> f[3][a, c, b] >>>> Out[14]= >>>> \!\(f\_3[a, b, c]\) >>>> >>>> Unlike with all the other solutions that have been proposed (if I didn't >>>> miss something) you get the benefit of correct behaviour with respect to >>>> pattern matching: >>>> >>>> In[15]:= >>>> F[f[3][a, b]] + F[f[3][a, c]] + F[f[3][b, d]] /. >>>> F[f[3][a, x_]] + F[f[3][c, y_]] -> p[x, y] >>>> Out[15]= >>>> \!\(F[f\_3[b, d]] + p[b, a]\) >>>> >>>> >>>> >>>> -- >>>> Andrzej Kozlowski >>>> Toyama International University >>>> JAPAN >>>> http://sigma.tuins.ac.jp >>>> >>>> >>>>> From: Andrzej Kozlowski <andrzej at tuins.ac.jp> To: mathgroup at smc.vnet.net >>>>> Date: Sat, 20 Nov 1999 01:07:05 -0500 (EST) >>>>> To: mathgroup at smc.vnet.net >>>>> Subject: [mg20919] [mg20891] Re: [mg20874] Orderless indexed functions >>>>> >>>>> I just noticed that there is a third way to do what you want, which is >>>>> possibly the simplest. Just define your function like this: >>>>> >>>>> f[1][x__]:= your definition; >>>>> f[1][x__] /; Not[OrderedQ[{x}]] := f[1][Apply[Sequence, Sort[{x}]]] >>>>> >>>>> However, I should also add that only the first of the three approaches I >>>>> have suggested (the one based on using the Notation package) actually >>> gives >>>>> you the full functionality of having the Orderless attribute. All the >>>>> functions defined in these ways will be "commutative", but they will not >>>>> behave like Orderless functions for the purpose of Pattern matching. For >>>>> example, consider the function f, which has a genuine Orderless >>> attribute: >>>>> >>>>> In[4]:= >>>>> SetAttributes[f, Orderless] >>>>> >>>>> Now consider the following example of pattern matching: >>>>> >>>>> In[5]:= >>>>> F[f[a, b]] + F[f[a, c]] + F[f[b, d]] /. F[f[a, x_]] + F[f[c, y_]] -> p[x, >>> y] >>>>> Out[5]= >>>>> F[f[b, d]] + p[b, a] >>>>> >>>>> If we use the definition above we won't get this behaviour: >>>>> >>>>> In[6]:= >>>>> F[f[1][a, b]] + F[f[1][a, c]] + F[f[1][b, d]] /. >>>>> F[f[1][a, x_]] + F[f[1][c, y_]] -> p[x, y] >>>>> Out[6]= >>>>> F[f[1][a, b]] + F[f[1][a, c]] + F[f[1][b, d]] >>>>> >>>>> The second method I gave in my first reply is even worse, as you can >>> check >>>>> yourself. So if you need your functions to be really orderless for the >>>>> purpose of pattern matching I can only suggest using subscripts and the >>>>> Notation package. >>>>> >>>>> >>>>>> From: Andrzej Kozlowski <andrzej at tuins.ac.jp> To: mathgroup at smc.vnet.net >>>>> To: mathgroup at smc.vnet.net >>>>>> Date: Fri, 19 Nov 1999 15:12:49 +0900 >>>>>> To: <nmonomachoff at email.msn.com>, <mathgroup at smc.vnet.net> >>>>>> Subject: [mg20919] [mg20891] [mg20874] Orderless indexed functions >>>>>> >>>>>> I can think of two approaches. If you are willing to abandon the >>> notation >>>>>> f[k] and instead use subscripts than probably the simplest way to get >>> what >>>>>> you want is to use the Notation package, which you can find in the >>>>>> ExtraPackages folder in the AddOns folder (you will have ot move it to >>>>>> somewhere like the AutoLoad folder to make things work as in y example >>>>>> below) >>>>>> >>>>>> In[23]:= << Utilities`Notation` >>>>>> >>>>>> In[27]:= Symbolize[NotationBoxTag[\(f\__\)]] >>>>>> >>>>>> In[28]:= SetAttributes[f\[UnderBracket]Subscript\[UnderBracket]2, >>> Orderless] >>>>>> >>>>>> In[29]:= f\[UnderBracket]Subscript\[UnderBracket]2[b, c, a] >>>>>> Out[29]= f\[UnderBracket]Subscript\[UnderBracket]2[a, b, c] >>>>>> >>>>>> If, on the other hand, you insist on the f[k] notation or you find the >>>>>> Notation package just too much bother (as I do) you could try something >>> like >>>>>> this. >>>>>> Suppose you want to define a function of the form: f[1][x__]:=B[x] >>> (where B >>>>>> is the body of your function) and give it the attribute Orderless. The >>>>>> following code will do it for you: >>>>>> >>>>>> f[1][x__] := Unevaluated[(Function[y, B[Sequence @@ y], >>>>>> {Orderless}])[x]] /. y -> Table[Unique[z],{Length[{x}]}] >>>>>> >>>>>> e.g. >>>>>> >>>>>> In[5]:= >>>>>> f[1][b, c, a, d] >>>>>> Out[5]= >>>>>> B[a, b, c, d] >>>>>> >>>>>> >>>>>> -- >>>>>> >>>>>> >>>>>>> From: "N Monomachoff" <nmonomachoff at email.msn.com> To: mathgroup at smc.vnet.net >>>>> To: mathgroup at smc.vnet.net >>>>>>> Date: Thu, 18 Nov 1999 20:46:23 -0500 (EST) >>>>>>> To: mathgroup at smc.vnet.net >>>>>>> Subject: [mg20919] [mg20891] [mg20874] Orderless indexed functions >>>>>>> >>>>>>> Hello, >>>>>>> >>>>>>> I need to use 'indexed functions' like f[0][x__] and f[1][x__]. >>>>>>> It's important that the functions be Orderless, but >>>>>>> >>>>>>> SetAttributes[f[0],Orderless] >>>>>>> >>>>>>> isn't accepted because f[0] isn't a symbol, and >>>>>>> >>>>>>> SetAttributes[f,Orderless] >>>>>>> >>>>>>> doesn't affect the ordering in the second set of brackets. >>>>>>> How does one make a nonsymbolic head orderless? >>>>>>> >>>>>>> Thanks for any help. >>>>>>> >>>>>>> N Monomachoff >>>>>>> nmonomachoff at msn.com >>>>>>> >>>>>>> >>>>>>> >>>>>> >>>>> >>>>> >>>> >>>> >>> >>> >>> >>> >>> > > > > >

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**Re: Orderless indexed functions**

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