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plurals

PO
primarykey
Id
11150046
data
AcceptedAnswerId
0
AnswerCount
0
ClosedDate
CommentCount
0
CommunityOwnedDate
CreationDate
2012-06-22T04:09:08.670
FavoriteCount
0
LastActivityDate
2012-06-22T04:09:08.670
LastEditDate
LastEditorUserId
0
OwnerUserId
1161878
ParentId
11032481
PostTypeId
2
Score
3
ViewCount
0
LastEditorDisplayName
text
Body
Fixed point theory comes in flavors. Those for programming languages are studied under the heading of denotational semantics. They depend on values forming a structured countable set with special properties. <a href="http://en.wikipedia.org/wiki/Lattice_%28order%29" rel="nofollow">Lattices</a>and <a href="http://en.wikipedia.org/wiki/Complete_partial_order" rel="nofollow">Complete Partial Orders</a> are two examples. All these sets have a "bottom" element, which turns out to be the fixed point that means "no useful result". In fact, the only fixed point operators you're interested in with computer programs are least fixed point operators: those that find the unique minimum fixed point that's lowest in the structured set of values. (Note all integers are on the same "level" in these structured sets. Only the bottom element lives beneath. The rest of the layers are composed of more complex types like function and tuple types, i.e. structures.) If you have some discrete math, <a href="http://www.cl.cam.ac.uk/teaching/1112/DenotSem/lectures/lecture-4.pdf" rel="nofollow">this</a> lays it out pretty nicely. Tarsky's fixed point theorem actually says that every function that is monotone (or alternately continuous) has a fixed point. See the reference above for definitions. In operational computer programs, the bottom element corresponds to a non-terminating computation: an infinite recursion. The point of all this is that if you have a rigorous mathematical model of computation, you can start proving interesting things about type systems and program correctness. So it's not just an academic exercise. 
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1. POY combinator: Some functions do not have fixed points
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1. USGene
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