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PHP: a fractal of bad design (fuzzy notepad)PHP: a fractal of bad design (fuzzy notepad)Posted Apr 17, 2012 17:32 UTC (Tue) by nybble41 (subscriber, #55106)In reply to: PHP: a fractal of bad design (fuzzy notepad) by butlerm Parent article: PHP: a fractal of bad design (fuzzy notepad)
No, I'm pretty sure they meant "transitive". Commutative is (A == B) <=> (B == A). Transitive is (A == B) && (B == C) <=> (A == C).
> To be fair, == isn't transitive in Javascript either.
The first line is backwards, strictly speaking, but ('' == 0) is also true. Substituting that form, we have A = '', B = 0, and C = '0'. (A == B) and (B == C) are thus both true, but (A == C) is false, so the relation is not transitive.
Equality may not be commutative in all cases, either, but for that you would need different examples. All of the cases listed here are commutative.
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PHP: a fractal of bad design (fuzzy notepad) Posted Apr 18, 2012 15:08 UTC (Wed) by k3ninho (subscriber, #50375) [Link]
Commutivity is more-often defined in terms of the sequence you use operators. It's not obvious, but Equality is the operator in your example. An alternative example: Commutative means that, for functions A:x->x, B:x->x: A(B(x)) == B(A(x)).
K3n.
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 18, 2012 19:43 UTC (Wed) by dtlin (✭ supporter ✭, #36537) [Link] Indeed. The word for a == b ⇔ b == a is reflexive.
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 18, 2012 20:18 UTC (Wed) by nybble41 (subscriber, #55106) [Link]
> Indeed. The word for a == b iff b == a is reflexive.
The answer is "both". The equality _relation_ is reflexive. The equality _operation_ is commutative.
> Commutative means that, for functions A:x->x, B:x->x: A(B(x)) == B(A(x)).
This doesn't fit any definition of "commutative" I was able to find. Every case I could locate involved the order of _operands_ to a single _binary_ function. Of course, you can turn you example into something like that, though with slightly different types, using higher-order functions (in pseudo-Haskell):
f1, f2 :: (a -> b) -> b
but that is equivalent to the much simpler form:
(==) B A == (==) A B
or:
(B == A) == (A == B)
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 19, 2012 22:55 UTC (Thu) by mmorrow (subscriber, #83845) [Link]
>> Commutative means that, for functions A:x->x, B:x->x: A(B(x)) == B(A(x)).
> This doesn't fit any definition of "commutative" I was able to find.
The elusive binary operator here is function composition ;)
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 19, 2012 23:14 UTC (Thu) by mmorrow (subscriber, #83845) [Link]
> The elusive binary operator here is function composition ;)
A reference for the interested being:
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 20, 2012 0:03 UTC (Fri) by nybble41 (subscriber, #55106) [Link]
>>> Commutative means that, for functions A:x->x, B:x->x: A(B(x)) == B(A(x)).
> The elusive binary operator here is function composition ;)
I thought you were presenting a general version of the commutative property, not a specialized version for function composition. That explains why I couldn't reconcile your example with other commutative operators (equality, addition, multiplication) without making "x" the operator and "A" and "B", in essence, the parameters.
Writing "(A . B)(x) == (B . A)(x)" or "A . B == B . A" would make the commutative part of the formula a bit more visible--not that it was really hidden. I just wasn't looking at it from the right perspective.
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 20, 2012 0:29 UTC (Fri) by mmorrow (subscriber, #83845) [Link]
> I thought you were..
(I wasn't the author of the original comment.)
> Writing "(A . B)(x) == (B . A)(x)" or "A . B == B . A" would make the commutative part of the formula a bit more visible..
It definitely would have made it clearer.
> ..a general version of the commutative property, not a specialized version for function composition..
Ah, but actually the case where a binary operator is commutative over its entire domain is the special case (in the mathematical sense, blah).
It's just that in programming we don't often (explcitly) deal with non-commutative binary operators, and even less often with non-commutative operators which are commutative when restricted to a subset of their domain.
Here's a quick roughly-phrased example:
Consider a rubik's cube. Think of a "move" as a function which maps a cube configuration to another configuration. This is a non-commutative group with elements these cube-config-maps and binary operator function composition. Now, for any two moves f and g which don't "interfere" (e.g. rotate top, rotate bottom), (f . g) == (g . f).
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 20, 2012 1:13 UTC (Fri) by mmorrow (subscriber, #83845) [Link]
Here's a better example:
The "elements" are C stmts thought of as mappings of memory configurations, and the binary operator is `;`. This is a monoid with identity element the empty C stmt.
Define,
modify(S) := the set of memory locations statement S *may* modify.
So now, `;` is commutative for every pair of statements S,T which *must not* (i.e. cannot under any possible dynamic execution path) modify one or more of the same memory locations.
(S ; T)===(T ; S) <==> modify(S)/\modify(T)==empty_set
(where "===" := equivalence wrt effect on memory)
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 20, 2012 1:37 UTC (Fri) by mmorrow (subscriber, #83845) [Link]
Actually that's not quite correct, it's more like:
(S ; T)===(T ; S)
or something along these lines, but the idea is clear.
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 20, 2012 8:19 UTC (Fri) by mpr22 (subscriber, #60784) [Link] Programmers tell their programs to subtract, divide, and/or shift fairly frequently.
PHP: a fractal of bad design (fuzzy notepad) Posted Apr 19, 2012 15:01 UTC (Thu) by yaap (subscriber, #71398) [Link]
What you wrote is commutativity not reflexivity. Reflexivity for equality is the fact that "a == a" is true for all a.
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