Wednesday, 26 October 2005

Skolem's paradox

I was cruising the net looking for information about skolemization when I stumbled on a paradox that I'd never seen before. Skolem's paradox is one of the most astounding paradoxes that I've ever seen. Here are a few links I dug up on the subject: The Löwenheim-Skolem Theorem MOORE ON SKOLEM'S PARADOX Reflections on Skolem's Paradox I'm don't totally understand all the implications of the paradox, but I do understand some of them and I find it both fascinating, and in some ways disheartening.

Tuesday, 18 October 2005

Wish List

I've often had questions that I wanted answers to and didn't know who to ask.  Sometimes the questions are unanswerable unless some mighty alien intelligence were to come and give them to me.  Often times however, they are questions that should in principle be answerable.  Here are some questions from my current wish list.

1. Is it the case that the operational procedures that are used to "prove" queries in logic programming languages are exactly program extraction?  Am I missing something in this picture?

2. The Curry-Howard correspondence gives rise to a simple logic as the type calculus in functional programming languages.  In a language like Haskell this means that you write mostly in the functional programming language and most of logical implications are infered using Hindley-Milner type inference.  If we go to the other extreme we see things like Coq that allow us to do program extraction (extraction of a functional program) from a logical specification.  My question is if it is possible to do something more like what Haskell does.  Namely, allow the user to occasionally describe the functional program associated with the specification.  This would be like writing types and having program inference, rather than writing programs with type inference.

3. I've spent a lot of time thinking about transactional logic since it seems critical that we deal with schema evolution and  encorporation of new facts.  How exactly does the Curry-Howard correspondence relate to schema evolution?

4. In the same vein as 3.  Is it possible to have schema evolution of types in a functional programming language by using atomic transactional code insertion.  Clearly without some sort of atomism we will be able to arive at inconsistent/type-unsafe intermediate stages.

5.  What logical type calculus is sufficent to capture the Deutch-Josza algorithm if we assume that the usual matrix algebra is the appropriate combinitor calculus for implementing  quantum algorithms (ie. what is the explicit Curry-Howard correspondence).

Anyone sensing a theme here?  I'm totally in love with the CH correspondence.  If you don't know about it.  You aught to go look it up on wikipedia and read a bit!