any() and all() on empty list?

[Steve R. Hastings]

So, Python 2.5 will have new any() and all() functions.
http://www.python.org/dev/peps/pep-0356/
any(seq) returns True if any value in seq evaluates true, False otherwise.

all(seq) returns True if all values in seq evaluate true, False otherwise.

I have a question: what should these functions return when seq is an empty
list?

Here is Guido's original article where he suggested any() and all():
http://www.artima.com/weblogs/viewpost.jsp?thread=98196

He offered this sample code for the semantics of any() and all():

def any(S):
for x in S:
if x:
return True
return False

def all(S):
for x in S:
if not x:
return False
return True

And he pointed out how nifty it is to combine generator functions with
these two new functions:
any(x > 42 for x in S) # True if any elements of S are > 42
all(x != 0 for x in S) # True if all elements if S are nonzero

I'm completely on board with the semantics for any(). But all() bothers
me. If all() receives an empty list, it will return True, and I don't
like that. To me, all() should be a more restrictive function than any(),
and it bothers me to see a case where any() returns False but all()
returns True.

In the all() example, if there *are* no values in S, then none of the
values can be != 0, and IMHO all() should return False.

Therefore, I propose that all() should work as if it were written this way:

def all(S):
ret_val = False

for x in S:
ret_val = True
if not x:
return False

return ret_val
Comments?

P.S. I searched with Google, and with Google Groups, trying to find
anyplace this might have been discussed before. Apologies if this has
already been discussed and I missed it somehow.
--
Steve R. Hastings "Vita est"
st***@hastings. org http://www.blarg.net/~steveha

[Paul Rubin]

"Steve R. Hastings" <st***@hastings .org> writes:

In the all() example, if there *are* no values in S, then none of the
values can be != 0, and IMHO all() should return False.

That goes against the usual meaning of "all" in, say, mathematical logic.

Usually, "for all X in S, PRED(x) is true" means:
there does not exist X in S so that PRED(x) is false.

So, all(empty sequence) should be true.

[Paul McGuire]

"Paul Rubin" <http://ph****@NOSPAM.i nvalid> wrote in message
news:7x******** ****@ruckus.bro uhaha.com...

"Steve R. Hastings" <st***@hastings .org> writes:

In the all() example, if there *are* no values in S, then none of the
values can be != 0, and IMHO all() should return False.
That goes against the usual meaning of "all" in, say, mathematical logic.

Usually, "for all X in S, PRED(x) is true" means:
there does not exist X in S so that PRED(x) is false.

How do you get this "usually" stuff? I would agree that this is usually
implemented as a short-circuited loop through the list, that breaks out at
the first False value. But I would not be quick to equate "commonalit y of
implementation" with "meaning".
So, all(empty sequence) should be true.

"should be"? Or "usually turns out to be"?

To my mind, the *meaning* of all() is that every element in the list asserts
True. But this is with an initial assumption that all() is False, unless I
test every value and find them to be True. Since I assume False to begin
with, I get no values in the list to contradict the assumption, and so
all([]) returns False.

It would seem that the resolution rests on which initial condition we
choose, False or True. Perhaps we should consult a more formal mathematical
resource for this.

-- Paul
"If it was so, it might be; and if it were so, it would be; but as it isn't,
it ain't. That's logic."

[Tim Peters]

[Steve R. Hastings]

So, Python 2.5 will have new any() and all() functions.
http://www.python.org/dev/peps/pep-0356/
any(seq) returns True if any value in seq evaluates true, False otherwise..

all(seq) returns True if all values in seq evaluate true, False otherwise..

I have a question: what should these functions return when seq is an empty
list?
Here, from the current development trunk, is what they _do_ return:

Python 2.5a0 (trunk:43410M, Mar 28 2006, 16:42:49) ...
Type "help", "copyright" , "credits" or "license" for more information.

any([]) False all([])

True
Here is Guido's original article where he suggested any() and all():
http://www.artima.com/weblogs/viewpost.jsp?thread=98196

He offered this sample code for the semantics of any() and all():

def any(S):
for x in S:
if x:
return True
return False

def all(S):
for x in S:
if not x:
return False
return True

...
|
I'm completely on board with the semantics for any(). But all() bothers
me. If all() receives an empty list, it will return True,
Yes.
and I don't like that.
Tough ;-)
To me, all() should be a more restrictive function than any(),
and it bothers me to see a case where any() returns False but all()
returns True.
There are deeper principles at work: so that endcases work out as
smoothly as possible, a "reduction" function applied to an empty
collection always arranges to return the identity element for the
reduction operation. This is the reason that sum([]) returns 0, for
example: 0 is the identity element for addition, meaning that x+0=x
for all x.

Other useful identities follow from this, and from the associativity
of most reduction operations. For example, sum(seq) = sum(seq[:i]) +
sum(seq[i:]) for any i >= 0, even if i is such that one (or both!) of
the slices on the right-hand side is empty. That wouldn't be true if
sum([]) were not 0, and arranging to make it true saves programmers
from having to deal with some otherwise special cases.

The reduction operation for any() is logical-or, and False is the
identity element for logical-or: x logical-or False = x for all
Boolean x.

Likewise the reduction operation for all() is logical-and, and True is
the identity element for that: x logical-and True = x for all Boolean
x.

Examples of other useful identities that follow from picking the
identity elements in the empty case, which hold even if `seq` is
empty:

any(seq) = not all(not x for x in seq)
all(seq) = not any(not x for x in seq)
In the all() example, if there *are* no values in S, then none of the
values can be != 0, and IMHO all() should return False.
That would break everything mentioned above. Think of it another way:
if all(seq) is false, shouldn't it be the case that you can point to
a specific element in seq that is false? After all (pun intended
;-)), if it's not the case that all x in seq are true, it must be the
case that some x in seq is false. But if seq is empty, there is no x
in seq that's either true or false, and in particular there's no x in
seq that's false. Since we can't exhibit an x in seq such that x is
false, saying that all(seq) is false would be very surprising to you
on some other day ;-)
Therefore, I propose that all() should work as if it were written this way:

def all(S):
ret_val = False

for x in S:
ret_val = True
if not x:
return False

return ret_val
Comments?

That won't happen, for three reasons: several were given above; this
is also the convention used for universal and existential quantifiers
applied to empty sets in mathematical logic (and for much the same
reasons there); and it matches prior art in the ABC language (which is
one of Python's predecessors, and which had direct syntactic support
for universal and existential quantifiers in Boolean expressions).

[Ben Finney]

"Steve R. Hastings" <st***@hastings .org> writes:

So, Python 2.5 will have new any() and all() functions.
http://www.python.org/dev/peps/pep-0356/
And there was much rejoicing.
any(seq) returns True if any value in seq evaluates true, False otherwise.
Yep.
all(seq) returns True if all values in seq evaluate true, False otherwise.
Not quite how I'd phrase it. I prefer, for symmetry with 'any':

all(seq) returns False if any value in seq evaluates false, True otherwise.
I have a question: what should these functions return when seq is an
empty list?

Here is Guido's original article where he suggested any() and all():
http://www.artima.com/weblogs/viewpost.jsp?thread=98196

He offered this sample code for the semantics of any() and all():

def any(S):
for x in S:
if x:
return True
return False

def all(S):
for x in S:
if not x:
return False
return True
I love the symmetry of these semantics, find them quite intuitive, and
therefore disagree with your interpretation of 'all()'.
I'm completely on board with the semantics for any(). But all() bothers
me. If all() receives an empty list, it will return True, and I don't
like that. To me, all() should be a more restrictive function than any(),
and it bothers me to see a case where any() returns False but all()
returns True.

-1.

You may as well argue that "any() should be a more restrictive
function than all(), and it bothers me to see a case where all()
returns False but any() returns True."
It seems clear to me that an empty argument list fails a check for
"any True?", because that's the same as a check for "all False?". The
only reasonable alternative would be a special case for an empty
argument list, and that's too ugly.

It seems clear to me that an empty argument list passes a check for
"all True?", because that's the same as a check for "any False?". The
only reasonable alternative would be a special case for an empty
argument list, and that's too ugly.

To imagine that one of these "should be a more restrictive function"
would belie their simple, elegant, and (to me) obvious definitions. I
disagree with your interpretation.

--
\ "My house is made out of balsa wood, so when I want to scare |
`\ the neighborhood kids I lift it over my head and tell them to |
_o__) get out of my yard or I'll throw it at them." -- Steven Wright |
Ben Finney

[Paul McGuire]

"Steve R. Hastings" <st***@hastings .org> wrote in message
news:pa******** *************** *****@hastings. org...

So, Python 2.5 will have new any() and all() functions.
http://www.python.org/dev/peps/pep-0356/
any(seq) returns True if any value in seq evaluates true, False otherwise.

all(seq) returns True if all values in seq evaluate true, False otherwise.

I have a question: what should these functions return when seq is an empty
list?

Here is my attempt at a more formal approach to this question, rather than
just using our intuition. Unfortunately, following this process proves my
earlier post to be wrong, but, oh well...

Consider two sets A and B where A+B is the union of the two sets.

if any(A+B) = True -> any(A) or any(B) = True
but we cannot assert either any(A)=True or any(B)=True.

if any(A+B) = False -> any(A) = False and any(B) = False.
if all(A+B) = True -> all(A)=True and all(B)=True
if all(A+B) = False -> all(A)=False or all(B)=False
but we cannot assert either all(A)=False or all(B)=False.
Now instead of B, lets add the empty set 0 to A. We want to come up logic
such that adding the empty set does not change the values of all() or any(),
since A+0=A.

any(A+0) = any(A) or any(0)

any(0) must be False, so that if any(A) is True, any(A+0) is True, and if
any(A) is False, any(A+0) is False.

all(A+0) = all(A) and all(0)

if all(A) is True, all(A+0) is True.
Therefore, all(0) is True.

-- Paul

[Steve R. Hastings]

Thank you very much for explaining this. And so thoroughly!

Of course I withdraw all objections. :-)
--
Steve R. Hastings "Vita est"
st***@hastings. org http://www.blarg.net/~steveha

[Paul Rubin]

"Paul McGuire" <pt***@austin.r r._bogus_.com> writes:

Usually, "for all X in S, PRED(x) is true" means:
there does not exist X in S so that PRED(x) is false.

How do you get this "usually" stuff? I would agree that this is usually
implemented as a short-circuited loop through the list, that breaks out at
the first False value. But I would not be quick to equate "commonalit y of
implementation" with "meaning".

See <http://en.wikipedia.or g/wiki/For_all>:

Generally, then, the negation of a propositional function's universal
quantification is an existential quantification of that propositional
function's negation; symbolically,

\lnot\ \forall{x}{\in} \mathbf{X}\, P(x) \equiv\
\exists{x}{\in} \mathbf{X}\, \lnot\ P(x)

[Sybren Stuvel]

Paul McGuire enlightened us with:

That goes against the usual meaning of "all" in, say, mathematical
logic.

Usually, "for all X in S, PRED(x) is true" means: there does not
exist X in S so that PRED(x) is false.
How do you get this "usually" stuff?

From its mathematical definition.
I would agree that this is usually implemented as a short-circuited
loop through the list, that breaks out at the first False value.
Implementation is irrelevant when it comes to the definition of a
mathematical operator.
But I would not be quick to equate "commonalit y of implementation"
with "meaning".
Which is good.
Perhaps we should consult a more formal mathematical resource for
this.

In mathematics, 'for all x in A, f(x) is True' is true when A is
empty. You can either look it up on trust someone who studied
mathematics (me) on it.

Sybren
--
The problem with the world is stupidity. Not saying there should be a
capital punishment for stupidity, but why don't we just take the
safety labels off of everything and let the problem solve itself?
Frank Zappa

[Ron Adam]

Paul McGuire wrote:

"Paul Rubin" <http://ph****@NOSPAM.i nvalid> wrote in message
news:7x******** ****@ruckus.bro uhaha.com... To my mind, the *meaning* of all() is that every element in the list asserts
True. But this is with an initial assumption that all() is False, unless I
test every value and find them to be True. Since I assume False to begin
with, I get no values in the list to contradict the assumption, and so
all([]) returns False.

Looking at in a different way... If we consider also having a function
none() (for comparison), would it be consistent with all()?

def none(S):
for x in S:
if x: return False
return True
any([]) -> False

none([]) -> True (same as 'not any(S)')

all([]) -> True ? False
I think I agree that all() should have an initial presumption of being
False.
Looking at it in yet another way... (yes, not as efficient)

def all(S):
S_ = [x for x in S if x]
return S_ == S

def any(S):
S_ = [x for x in S if x]
return S_ != []

def none(S):
S_ = [x for x in S if x]
return S_ == []
In this view and empty set can be True for all(). Is it posible
'all([])' is undefined? Here, none() and all() return contradicting
values. So maybe the correct version may be...

def all(S):
if S == []: return False
for x in S:
if x return True
return False

I think a few valid actual use case examples could clear it up. What
makes the most sense?

Cheers,
Ron