Talk:Parity function
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Machine learning
Parity functions have received some interest in (theoretical) machine learning as well; see e.g. John Langford's blog, and recall that the inability of linear models to learn XOR (parity of two inputs) was a major point in the book Perceptrons back in 1969, or at least in the controversy surrounding the book. I'd love to write a bit more about this, but I'm not too familiar with the subject. QVVERTYVS (hm?) 01:38, 4 January 2014 (UTC)
equivalence classes of the infinite parity function?
I'm having trouble making sense of the following paragraph:
- Assuming axiom of choice it can be easily proved that parity functions exist and there are many of them - as many as the number of all functions from to . It is enough to take one representative per equivalence class of relation defined as follows: if and differ at finite number of coordinates. Having such representatives, we can map all of them to 0; the rest of values are deducted unambiguously.
The first sentence is fine- we have lots of parity functions f. The second sentence introduces without naming it. But it has a name, its the
Anyway, there are still such equivalence classes; the cardinality of E is the continuum. Each equivalence class contains only a countable number of representatives (because m and n are countable). Next, for each pick one representative , and set . OK, so far. For all of the other , the value of is unambiguous (because x and y differ in only finitely many places, and there are only countably many different y grand total.) And this can be done for any other equivalence class . Yes, I can see how this is horribly non-constructive. To build E, I need to somehow pick uncountably many real numbers, and do so in such a way that no two of them differ by a dyadic fraction. Ouch. Anyway, I think that this is what the above paragraph was trying to telegraph. It took me a while to figure this out. That paragraph should be expanded into something less concise, and easier to understand. A reference would be nice, too.
Oh, duhh! The collection of equivalence classes is just the