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Topic: AN ERROR IN PEANO ARITHMETIC! x X [ s Y ] [ + X Z ] <- x X Y Z
Replies: 3   Last Post: Nov 5, 2013 3:48 PM

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Graham Cooper

Posts: 4,293
Registered: 5/20/10
AN ERROR IN PEANO ARITHMETIC! x X [ s Y ] [ + X Z ] <- x X Y Z
Posted: Nov 5, 2013 6:57 AM
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MULTIPLICATION IN PEANO ARITHMETIC







REVISION: ADDITION



PROGRAM LIST

RULE 1 + X 0 X

RULE 2 + X [ s Y ] [ s Z ] <- + X Y Z





TRACE
+ [s[s 0]] [s[s 0]] A ?

TRY 1
RULE 2
+ X [ s Y ] [ s Z ]
| TAIL 1
| + X Y Z
| + { s s 0 } { s 0 } Z
| TRY 1
| RULE 2
| + X [ s Y ] [ s Z ]
| | TAIL 1
| | + X Y Z
| | + { s s 0 } { 0 } Z
| | TRY 1
| | RULE 1
| | + X 0 X
| | MATCH
| | TRUE 1
| MATCH
| TRUE 1
MATCH




+ [s[s 0]] [s[s 0]] A ?
A = s s s s 0



i.e. 2+2=4







___________________________________________


MULTIPLICATION



PROGRAM LIST
1 + X 0 X
2 + X [ s Y ] [ s Z ] <-
+ X Y Z
3 x X [ s 0 ] X
4 x X [ s Y ] [ + X Z ] <-
x X Y Z





TRACE
x [s 0] [s 0] A ?

TRY 1
RULE 3
x X [ s 0 ] X
MATCH


x [s 0] [s 0] A ?
A = s 0



i.e. 1 x 1 = 1







________________________________________


TRACE
x [s[s[s 0]]] [s[s[s[s 0]]]] A ?

TRY 1
RULE 4
x X [ s Y ] [ + X Z ]
| TAIL 1
| x X Y Z
| x { s s s 0 } { s s s 0 } Z
| TRY 1
| RULE 4
| x X [ s Y ] [ + X Z ]
| | TAIL 1
| | x X Y Z
| | x { s s s 0 } { s s 0 } Z
| | TRY 1
| | RULE 4
| | x X [ s Y ] [ + X Z ]
| | | TAIL 1
| | | x X Y Z
| | | x { s s s 0 } { s 0 } Z
| | | TRY 1
| | | RULE 3
| | | x X [ s 0 ] X
| | | MATCH
| | | TRUE 1
| | MATCH
| | TRUE 1
| MATCH
| TRUE 1
MATCH




x [s[s[s 0]]] [s[s[s[s 0]]]] A ?
A = + s s s 0 + s s s 0 + s s s 0 s s s 0



i.e. 3 x 4 = 3+3+3+3 ????









PEANO MULTIPLICATION is not a LOGIC INFERENCE RULE!

x X [ s Y ] [ + X Z ] <- x X Y Z



To use Peano's Rules of Multiplication you need a more powerful
expression reduction system like lambda calculus!

This is similar to a FINITE STATE MACHINE can do ADDITION
but a TURING MACHINE is needed to MULTIPY!




Herc
--
www.PrologDatabase.com



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