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Admissibility in Practise Arno Pauly and Shi Shu Computer - - PowerPoint PPT Presentation
Admissibility in Practise Arno Pauly and Shi Shu Computer - - PowerPoint PPT Presentation
Admissibility in Practise Arno Pauly and Shi Shu Computer Laboratory, University of Cambridge CCC 2015 What is admissibility about? Common idea: Admissibility is (just) the criterion for which representations are correct. This is not
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Another idea
◮ Common idea: Topology is the fundament for computability
- n spaces of cardinality c
◮ This is not entirely true, as above ◮ But even more: Topology is a natural consequence of
computability theory, not a prerequisite
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Defining continuity
Definition
Continuity is computability relative to an arbitrary oracle.
◮ The UTM-theorem gives us function spaces C(−, −) ◮ together with Sierpinski space S = ({⊤, ⊥}, δS) where
δS(p) = ⊥ iff p = 0N
◮ we get a notion of open sets O(X) ∼
= C(X, S).
◮ the preimage of open sets under continuous functions is
- pen –
◮ but the converse fails in general.
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Defining admissibility following Schröder
Definition
We call X admissible, if the canonic map κX : X → C(C(X, S), S) is computable invertible. Note: Admissibility has algorithmic content!
Theorem
For functions with admissible codomain, (realizer) continuity and topological continuity coincide.
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Closed and compact sets
Definition
A set is called closed, if its complement is open: Thus A(X) ∼ = O(X).
Definition
We define the compact sets K(X) by identifying A ⊆ X with {U ∈ O(X) | A ⊆ U} ∈ O(O(X)).
◮ ∩ : A(X) × K(X) → K(X) is computable. ◮ X is admissible, iff {x} → x :⊆ K(X) → X is well-defined
and computable.
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Using admissibility
- 1. Conclude that the solution set S is closed based on its
definition (this often involves the Hausdorff condition).
- 2. Obtain some compact candidate set K (either from the
situation, or by assumption - this often involves some bounds).
- 3. Compute S ∩ K as a compact set
- 4. Use domain-specific reasoning or assumption to conclude
that the solution s is unique.
- 5. As we have {s} available as a compact set, if our target
space is admissible, we can compute s.
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What we implemented in ML
◮ Exact reals with conversion to and from floating points. ◮ Addition and multiplication of reals. ◮ Sierpinski space, binary ∧ and ∨, countable ◮ isNotEqual and isPositive ◮ O(−), A(−), K(−) with the suitable operations ◮ [−1, 1] ∈ K(R) ◮ {x} → x :⊆ K(R) → R
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Compactness
◮ [−1, 1] ∈ K(R) is realized by some function c : C(R, S) → S ◮ We can only evaluate the input function on "points". ◮ Trick: Use disguised reals such as . − 1 0 DoNotLook ◮ Start with list . DoNotLook and evaluate input on all points.
If a point gets accepted, remove it. If the exception gets raised, add all "successor points" to the list. If the list ever gets empty, accept the input function.
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Admissibility
◮ Every real in (n − 1, n + 1) has a name with n before the
digit: So search for an open interval covering the compact singleton.
◮ Then submit the resulting interval in (n − 1, n),
(n − 0.5, n + 0.5), (n, n + 1) and search for one covering the compact singleton to determine the next digit, etc.
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What we used it for
◮ We can define x → 0.5x : [−2, 2] → [−1, 1] by letting 0.5x
be the unique y ∈ [−1, 1] such that Y + y = x.
◮ Unfortunately, this currently takes about 2 minutes to
compute 4 digits. . .
◮ Maybe some hopes for optimization and use for problems
where numerical algorithms are not yet available.
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The sources
- S. Shu.
An Exact Real Package in ML. Part II Project, University of Cambridge, 2015.
- A. Pauly.
On the topological aspects of the theory of represented spaces. Computability (accepted for publication, cf arXiv:1204.3763).
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Further reading
- M. Schröder.
Admissible Representations for Continuous Computations. PhD Thesis, FernUniversität Hagen, 2002.
- A. Bauer.
Sometimes all functions are continuous. http://math.andrej.com/2006/03/27/ sometimes-all-functions-are-continuous/
- M. Escardó.