[racket] arity of + versus <=

On Fri, Oct 28, 2011 at 12:53 PM, Robby Findler
<robby at eecs.northwestern.edu> wrote:
> On Fri, Oct 28, 2011 at 11:44 AM, Carl Eastlund <cce at ccs.neu.edu> wrote:
>> On Fri, Oct 28, 2011 at 12:07 PM, John Clements
>> <clements at brinckerhoff.org> wrote:
>>>
>>> On Oct 28, 2011, at 8:12 AM, Joe Marshall wrote:
>>>
>>>> On Wed, Oct 26, 2011 at 8:32 PM, Dan Grossman <djg at cs.washington.edu> wrote:
>>>>> Very minor point, but is there a rationale beyond historical precedent
>>>>> for + and * to allow any number of arguments but, =, <=, <, >, >= to
>>>>> require at least two arguments?
>>>>
>>>> 0 is the additive identity. 1 is the multiplicative identity.
>>>> What is the equality identity?
>>>
>>> No, I don't buy that. operators in \alpha X \alpha -> \beta can never have identities, but that doesn't mean they can't be generalized.
>>>
>>> I can definitely imagine that you would choose to disallow unary use of comparison operations to prevent a certain class of programming errors, but it seems pretty clear to me that the generalization of, e.g., <= is "is every sequential pair of items in the argument list related by the given operator."
>>>
>>> Am I missing something here?
>>>
>>> John
>>
>> Furthermore, that generalization is useful, as it makes (apply <= xs)
>> into a simple implementation of "is xs monotonically non-decreasing?",
>> just as (apply + xs) implements "the sum of the elements of xs".  If
>> <= must accept 2 or more arguments (or even 1 or more), that does not
>> work for all lists.  Personally, I'd prefer if <= and friends were
>> generalized.  It seems more in tune with Racket's permissive Scheme
>> heritage -- if append accepts "improper" lists, + mixes precise
>> numbers with floating point, and all values act as booleans, why can't
>> <= accept 0 or 1 arguments?
>
> Does this rationale also suggest that it is fine that (cdr '()) = '()?
>
> Robby

No.  I don't see any sense in which that's a generalization of cdr.
In fact it's quite non-uniform; for instance, applying cdr to a
non-empty list produces a list one shorter.  If we want to extend cdr
to length 0 lists, we need to somehow produce a list of length -1.  We
restrict cdr from empty lists precisely because it can't be
generalized.  Or at least that's how I see it.

--Carl