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- Newsgroups: comp.theory
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- From: cap@ifi.unizh.ch (Clemens Cap)
- Subject: Higher recursion theory
- Message-ID: <1992Dec22.152623.5105@ifi.unizh.ch>
- Followup-To: poster
- Sender: cap@ifi.unizh.ch
- Organization: University of Zurich, Department of Computer Science
- Distribution: world
- Date: Tue, 22 Dec 92 15:26:23 GMT
- Lines: 75
-
-
- Desperately seeking
-
- ***************************************
- *help (examples, references, pointers)*
- ***************************************
-
- for higher order recursion theory.
-
-
- Introdution
- ************
-
- In classical recursion theory one studies questions like the following:
-
- A functional F is a function from partial recursive functions to partial
- recursive functions.
-
- A functional F is called effective, iff there exists an extensional, partial
- recursive function h, such that the effect functional F has on a function g
- can be described by an action of this function h on an arbitrary program q
- of this function g.
-
- A functional F is called monotonic, iff whenever for two functions f and g
- there is f < g, then also F(f) < F(g). ( < is the usual ordering of partial
- functions).
-
- A functional F is called compact, iff for every function f there exists
- a function u with finite domain and u < f, such that F(f) = F(u).
-
- Classical recursion theory now studies the following questions:
-
- (0) A functional is continuous, iff it is monotonic and compact.
- (1) A functional is effective, iff it is continuous.
- (2) A functional having the properties of (1) (and (2) always has a least
- fixed point function m (so F(m) = m), which can effectively be calculated
- either by inductive techniques or by applying a suitable recursive function
- to a program of the extensional function belonging to the functional, thus
- leading to a program of the least fixed point m.
-
- **********************
- Now comes my QUESTION:
- **********************
-
- A) Is there an analogon to this for functionals of higher type?
- For example, let us call a functional of type-2 an operation which takes
- a number of ordinary functionals (as defined above) as argument
- and yields an ordinary functional as result.
- What can be said about representations via programs, or extensional
- functions, about continuity, monotonicity and compactness ?
- Is there an analogon to above theorem from type-1 recursion theory ?
-
- B) G. Plotkin writes in his lecture notes on domains on page 6
- "And indeed at higher types continuity is a real restriction."
- I would like to understand the meaning of this sentence in the context of
- above lines and I am looking for a concrete example.
-
- C) I have studied Kechris and Moschovakis on "Recursion in Higher Types" in
- the Handbook of Mathematical Logic by Barwise, and found some pointers.
- However, I am interested essentially in "real computable and constructive"
- stuff -- everything should have some consequences for a programmer sitting
- in front of his Turing Machine. So also I am very happy for help along the
- line of "fully-abstract-categoric-domain-theoretical-stuff" I would even be
- more happy with help along the line of "apply-example-program-useful-recursive"
-
-
- Please reply to cap@ifi.unizh.ch
- I will post a summary to comp.theory, if I receive interesting replies.
-
- Happy Xmas to everyone on the net.
- --
- * Dr. Clemens H. CAP cap@ifi.unizh.ch (email)
- * Ass. Professor for Formal Methods in CS +(1) 257-4326 (office)
- * Dept. of Computer Science +(1) 322 02 19 (home)
- * University of Zurich +(1) 363 00 35 (fax)
-