Computability in Europe 2008
Athens Greece
Query-Optimal Oracle Turing Machines for
Type-2 Computations
Chung-Chih Li
June 18
School of Information Technology
Illinois State University
Normal, IL 61790, USA
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The original question:
Do we have a speed-up theorem in type-2
computation in terms of query complexity?
Answer:
Yes/No/Don’t know
Depends on the definition of
Query-optimal programs
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What is speed-up theorem
First remarked by Gödel in 1936
Blum’s Speed-up Theorems (1967,1971)
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Operator Speed-up Theorems
(Meyer et al, 1972, 1973)
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Type-2 computable =def Oracle TM computable
(N  N)  N  N
Type-2 Speed-up Theorem (Li, 2007: CiE)
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This type-2
asymptotic notion
is a trouble
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Type-2 asymptotic notions
This notion allows exception on a small amount of inputs
Type-1: Finite sets
i.e., compact sets in the discrete
topology
Type-2: ?Compact sets in topology T
???
(F) (Li 2004)
Baire Topology doesn’t work 
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Prerequisites for having the complexity
theorems
Abstract Complexity Measure (Blum 1967)
Queries complexity is not such kind
Queries are considered as a resource in the study
of computability (not complexity)
Beigel, Gasarch, et al. around 1990’s
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The collection of queries made during
the course of computation in s steps
Q(i,j)
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The collection of queries made during
the course of computation in the limit
The topology defined by the queries made
in the limit by programs i and j: Q (i, j)
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Let i and j be two programs (indexes of OTM)
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How to speed-up?
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Query-optimal OTM
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If we choose each definition of the queryoptimal OTM and if we can argue that
there is a computable type-2 functional
that does not have a query-optimal OTM
for it, then we can claim that the speed-up
theorem holds in the corresponding sense
of the query-optimal OTM
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absolute, strong, weak
1. Does every computable functional
always have a query-optimal OTM for it?
no,
no,
?
yes,
yes,
?
3. If a given F does have a query-optimal
OTM for it, then can we uniformly
construct a query-optimal program for F
from an arbitrary OTM that computes F?
no,
no,
no
4. Suppose there is no query-optimal OTM
for F. Can we effectively construct an
infinite sequence of query-sped-up
version of OTM’s for F
no,
?,
?
2. If the answer to the first question is
negative, then can we construct the
query-speed-able functional?
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Type-2 functional K
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Conclusion and Future
•
The speed-up theorem, if holds, is very different from the
original speed-up theorem.
•
Do we have a speed-up theorem under the weakest notion of
query-optimal OTM?
•
Can we ask the same question in classical complexity
theorem under query-complexity?
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