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From: playboy@wacsvax.UUCP
Newsgroups: net.math.symbolic
Subject: Responce to Fatemans article "Eleven Proofs of sin^2+cos^2=1"
Message-ID: <258@wacsvax.OZ>
Date: Thu, 30-Jan-86 04:29:14 EST
Article-I.D.: wacsvax.258
Posted: Thu Jan 30 04:29:14 1986
Date-Received: Sat, 1-Feb-86 05:49:09 EST
Organization: Comp Sci, Uni. Western Australia
Lines: 81
Keywords: Pattern Matching

In his article "Eleven Proofs of sin^2+cos^2=1", Fateman [1] points out
"a deficiency of our current way of putting together algebra systems as
`encodings' of mathematical information" by showing that MACSYMA can't
`express' the identity

   2          2
Sin (x)  + Cos (x)   = 1.

It is possible with current pattern matching techniques [2]   to
`express' this, and much more complex identities, in an efficient and
simple fashion. This idea has been taken up by Inference Corp. and used
to extend the scope of the SMP pattern matcher.  (To clear the record,
it was possible to implement this particular identity in earlier
releases.)


Here is an example of an SMP implementation of the rule taken from the
SMP library file XSimp.

/* Part of the XSimp file */ 

/*: CSsmp[$expr]
        simplifies Sin[x]^2 + Cos[x]^2 to 1 in $expr */
        
        CSsmp[$expr] :: Si[$expr,\ Cos[$x]^2 + Sin[$x]^2 -> 1, \
                           $$a*Cos[$x]^2 + $$a*Sin[$x]^2 --> $$a, \
                           Cos[$x]^2/$b + Sin[$x]^2/$b -> 1/$b, \
                           ($$a*Cos[$x]^2)/$b + ($$a*Sin[$x]^2)/$b -> $$a/$b]
 
We now give an example of this.

SMP 1.5.0
Thu Jan 30 15:57:54 1986


#I[1]::  <XSimp

#I[2]::  cs : Sin[x]^2+Cos[x]^2 + a Sin[y]^2+b Sin[3z]^2+a Cos[y]^2+b Cos[3 z]^2

		 2           2  	  2            2	 2         2
#O[2]:   a Cos[y]  + a Sin[y]  + b Cos[3z]  + b Sin[3z]  + Cos[x]  + Sin[x]

#I[3]::  CSsmp[%]

#O[3]:   1 + a + b

#I[4]::  <end>

It has long been realised [3] the advantage of pattern matching
is information is `expressed' not `encoded'. This greatly simplifies the
implementation of identities in a CAS. We here have been using pattern matching
on very complex problems with great success for over a year. It is time that
the main stream computer scientists caught up with this idea.

Disclaimer: 
I do not wish to fuel the bitter debate between different current
computer algebra groups. This is not intended as a product comparison
or a slight of Richard Fateman, MACSYMA or any other person, group or
CAS. I do not work for Inference Corp. I am a satisfied customer who thinks SMP
is a wonderful product.


1.	Fateman R J. Eleven proofs of sin^2+cos^2=1,
	SIGSAM Bull. 19, 2 (May 1985), 25-28

2.	McIsaac K Pattern Matching alegbraic Identities,
	SIGSAM Bull. 19, 2 (May 1985), 4-13

3.	Lafferty E L. Hypergeometric Function Reduction - An Adventure 
	in Pattern Matching, MACSYMA Users Conference,
	Snowbird, 1979, 466-481


--------------------------------------------------------------------------------

Kevin McIsaac			ACSnet:	playboy@wacsvax
Dept. Physics			UUCP:	seismo!munnari!wacsvax.oz!playboy
Uni. of Western Australia          	decvax!mulga!wacsvax.oz!playboy
Nedlands, 6009
AUSTRALIA 			ARPA:	munnari!wacsvax.oz!playboy@seismo.arpa
					decvax!mulga!wacsvax.oz!playboy@Berkeley