Dear Noson - On 21/09/2015 14:42, Noson S. Yanofsky wrote:
Dear Steve,
Thanks for taking the interest in the paper.
Whether or not the colimit exists depends on which category you are in. In my paper the colimit is taken in the category of small categories where it does exist. Like you point out, (and David Roberts pointed out in a previous post) ...
(I don't seem to have David's post. Apologies if I'm repeating what he said.)
... you might want to do some of this stuff in other categories such as the effective topos etc where this colimit would not exist. I do not know if the category of small categories is "omniscient" or not, and if that is a good thing or a bad thing, but the result stands.
You seem to be shifting uncertainly between the category where colimits may or may not exist and a category (e.g. Set or a topos) that provides an ambient logic for your mathematics. Do you assume that 2 has all colimits? (Then the proof for the halting problem says take the colimit of the diagram H: ?? -> 2. You've parameterized that by x and y for program and data, but the essence of the argument can be seen even without those.) Or even those colimits of the specific form H? If so, then you are using a logical principle of your ambient mathematics, not pure category theory.
(It is not a typo. Since 2 is indiscreet 0-->1 once the output goes to 1 it must stay in 1. This is exactly what Halt(x,y,t) does. Once it halts it stays halted. But all the outputs can remain 0.)
"Indiscrete" means for each pair of objects x, y, there is a unique morphisms from x to y. From your definition on p.4, 2 is not indiscrete, since there is no morphism 1 -> 0. From the definition on p.14, ??_i is indiscrete. Suppose you have a functor H: ??_i -> 2. For any n you have H(0 -> n): H(0) -> H(n) and H(n -> 0): H(n) -> H(0). Hence, by antisymmetry in the poset 2, H(n) = H(0). What you are saying in words is captured by a functor from ?? (poset) to 2, not from ??_i to 2. Steve.
All the best, Noson
-----Original Message----- From: Steve Vickers [mailto:s.j.vickers@cs.bham.ac.uk] Sent: Monday, September 21, 2015 6:10 AM To: Noson S. Yanofsky Cc: categories@mta.ca Subject: Re: categories: Computability and Complexity of Categorical Structures
Dear Noson,
It seems to me it's an oversimplification to say that categories can solve the Halting Problem. There's a logical principle involved.
In your proof on p.23, the central step is that if H: ?? -> 2 is a functor (where ?? and 2 are the posets of natural numbers and {0,1}), then it has a colimit H': 1 -> 2. H(t) describes whether the computation has halted by step t. (By the way, you use ??_i, the indiscreet natural numbers, for the type of t. That's a typo, isn't it? Since 2 is antisymmetric, that would imply H is constant.)
Does the colimit exist? That's not a fact of pure category theory, but a logical principle of "omniscience". If you go beyond classical logic (and category theory facilitates this) then the colimit might not exist.
For example, it will not work in a topos. To find the colimit you will need to embed 2 in ??, the poset on the subobject classifier. There you still have the classical objects 0 and 1 (false and true), but also intermediate objects corresponding to truth values of other propositions such as "this computation halts". This switch from 2 to ?? corresponds to relaxing decidability to semidecidability - and semidecidability of the halting problem is trivial.
If my calculations are correct, you can see this in the topos of sheaves over the 1-point compactification of ??. This is the classifying topos for functors H, and the generic H has no colimit.
An alternative way to understand this is what you see in denotational semantics. Here, when a map to 2 takes the value 0 ("bottom"), that is interpreted as meaning that the assignment of result to argument gets ensnared in a divergence.
Regards,
Steve.
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