Friday, August 15, 2014

Span(Graph) II: Circuits with feedback

Previous Span(Graph) post; next Span(Graph) post.

As an introduction to ${\bf Span(Graph)}$ I will describe how to extend the category of straight-line circuits to allow circuits with state and feedback. But before doing that I would like to point out a couple of things about straight-line circuits.

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Wednesday, August 13, 2014

Lex total categories and Grothendieck toposes II

I said in the first post in this series that totally complete categories have a strong adjoint functor theorem.

Here is the adjoint functor theorem: if $A$ and $B$ are locally small categories, $A$ is totally cocomplete and $F: A\to B$ is a functor which preserves colimits of discrete fibrations with small fibres then $F$ has a right adjoint.

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Saturday, August 09, 2014

The sleeping beauty problem: how some statisticians calculate probability

This problem was drawn to my attention by calculations of physicists and philosophers (see my last post), but I see that some eminent statisticians (Jeffrey S. Rosenthal)  are also convinced that the answer is $1/3$.

What I think is the real problem is that statisticians immediately start calculating with conditional probability, Bayes formula etc without modelling the problem mathematically. It is also important to generalize a problem to avoid being confused by particular details.
In any case, I want to clarify what I said in the last post by considering  a simple example.

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Tuesday, August 05, 2014

The sleeping beauty problem: how some philosphers and physicists calculate probability

Recently there have been surprising discussions and disputes amongst philosophers and physicists about an elementary probability problem called the Sleeping Beauty problem. The following remarks are, as usual in this blog, the result of discussions with Nicoletta Sabadini.

The problem goes as follows: A beauty is told that the following procedure will be carried out. On Sunday a fair coin will be tossed without her knowing the result. She will go to sleep. Then on Monday one of two possibilities will occur.

In the case that the toss of the coin resulted in tails she will be wakened and asked her opinion of the probability that the result of coin was heads. She will then have her memory of what happened on Monday erased and will be put to sleep. On Tuesday  (again in the case of tails, without a further toss of the coin) she will be wakened and asked her estimate of the result of the coin toss being heads.

In the case of heads, on Monday she will be asked her estimate of the probability that the result of the coin toss was heads. In that case she will not be asked again.

It seems clear intuitively that, when this procedure is carried out, in all three responses  she has learnt nothing about the result of the coin toss, and that she should answer in each case $1/2$.

Strangely a considerable number of philosophers and physicists make an elementary error in the calculation of probabilities and believe that she should answer $1/3$.

Update: I see also that a recipient of the COPSS Presidents' Award (the Nobel Prize of Statistics, according to Wikipedia), Jeffrey S. Rosenthal, believes the $1/3$ answer.

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Friday, August 01, 2014

Span(Graph) I

Next Span(Graph) post.


I have decided to write a series of tutorial posts about Span(Graph) since I think this work has been unfairly neglected. I am simultaneously introducing a little TeX into the blog, at the encouragement of Keith Harbaugh.

We begin with a simpler category as an algebra of straight line circuits which we will use as motivation for then in a later post introducing ${\bf Span}({\bf Graph})$.
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Thursday, July 24, 2014

Lex total categories and Grothendieck toposes

This is the first of a series of connected posts. The next post is here.

In the very valuable volume Sketches of an Elephant, A Topos Theory Compendium, Volume 1, Peter Johnstone mentions a lecture of Andre Joyal at the 1981 Cambridge Summer Meeting where he listed seven different descriptions of 'what a topos is like'. His fifth one was

(v) 'A topos is a totally cocomplete object in the meta-2-category of  finitely complete categories'.

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Monday, July 14, 2014

The algebra of processes XI

Previous post in this series is here.

We have been thinking about categorical algebra for computer science now for 25 years. I want to discuss the problems of thinking seriously about two distinct disciplines, and how one might avoid the danger of falling between two stools.
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