Hi All,
I have read (not in full details, though) the text "Mathematical foundations
of Joy" (See: http://www.latrobe.edu.au/philosophy/phimvt/joy/j02maf.html )
Here some small remarks.
1) At the beginning of the secion "Function composition and the identity
function" .
<quote>
If the programs P and Q denote the same function, then the functions P and Q
are identical. Two functions are identical if for all values in the
intersection of their domains they yield the same value.
...
The identity relation between functions is clearly reflexive, symmetric and
transitive.
</quote>
It seems that the second sentence must be:
Two functions are identical if and only if their domains are coincided and for
all values in their domains these functions yield the same value.
Otherwise, if we take the second sentence as the definition of "identical",
than this relation will not be transitive.
2) At the end of the secion "Function composition and the identity function" .
<quote>
It is appropriate to remark here that there is also a left zero element and
there is a right zero element.
Two such elements l and r satisfy the following for all programs P:
l P == l P r == r
Since function composition is not commutative, the two zero elements are not
identical. In Joy the left zero l is the abort operator, it ignores any
program following it.
The right zero r is the clearstack operator, it empties the stack and hence
ignores any calculations that might have been done before.
</quote>
I think in our case there is no such thing as right zero.
Assume that program P will be run forever for some initial state, than
clearly for that P we have
P r =/= r (clearstack operator on the left side will never get a chance to
execute)
Regards,
Michael Nedzelsky
Hi All,
I have read (not in full details, though) the text "Mathematical foundations
of Joy" (See: http://www.latrobe.edu.au/philosophy/phimvt/joy/j02maf.html )
Here some small remarks.
1) At the beginning of the secion "Function composition and the identity
function" .
<quote>
If the programs P and Q denote the same function, then the functions P and Q
are identical. Two functions are identical if for all values in the
intersection of their domains they yield the same value.
...
The identity relation between functions is clearly reflexive, symmetric and
transitive.
</quote>
It seems that the second sentence must be:
Two functions are identical if and only if their domains are coincided and for
all values in their domains these functions yield the same value.
Otherwise, if we take the second sentence as the definition of "identical",
than this relation will not be transitive.
2) At the end of the secion "Function composition and the identity function" .
<quote>
It is appropriate to remark here that there is also a left zero element and
there is a right zero element.
Two such elements l and r satisfy the following for all programs P:
l P == l P r == r
Since function composition is not commutative, the two zero elements are not
identical. In Joy the left zero l is the abort operator, it ignores any
program following it.
The right zero r is the clearstack operator, it empties the stack and hence
ignores any calculations that might have been done before.
</quote>
I think in our case there is no such thing as right zero.
Assume that program P will be run forever for some initial state, than
clearly for that P we have
P r =/= r (clearstack operator on the left side will never get a chance to
execute)
Regards,
Michael Nedzelsky
Hi All,
I have read (not in full details, though) the text "Mathematical foundations
of Joy" (See: http://www.latrobe.edu.au/philosophy/phimvt/joy/j02maf.html )
Here some small remarks.
1) At the beginning of the secion "Function composition and the identity
function" .
<quote>
If the programs P and Q denote the same function, then the functions P and Q
are identical. Two functions are identical if for all values in the
intersection of their domains they yield the same value.
...
The identity relation between functions is clearly reflexive, symmetric and
transitive.
</quote>
It seems that the second sentence must be:
Two functions are identical if and only if their domains are coincided and for
all values in their domains these functions yield the same value.
Otherwise, if we take the second sentence as the definition of "identical",
than this relation will not be transitive.
2) At the end of the secion "Function composition and the identity function" .
<quote>
It is appropriate to remark here that there is also a left zero element and
there is a right zero element.
Two such elements l and r satisfy the following for all programs P:
l P == l P r == r
Since function composition is not commutative, the two zero elements are not
identical. In Joy the left zero l is the abort operator, it ignores any
program following it.
The right zero r is the clearstack operator, it empties the stack and hence
ignores any calculations that might have been done before.
</quote>
I think in our case there is no such thing as right zero.
Assume that program P will be run forever for some initial state, than
clearly for that P we have
P r =/= r (clearstack operator on the left side will never get a chance to
execute)
Regards,
Michael Nedzelsky