CiE 2008 1
Computability in Europe 2008Athens Greece
School of Information Technology
Illinois State University
Normal, IL 61790, USA
Chung-Chih LiJune 18
Query-Optimal Oracle Turing Machines forType-2 Computations
CiE 2008 2
The original question:
Yes/No/Don’t know
Do we have a speed-up theorem in type-2computation in terms of query complexity?
Answer:
Depends on the definition of Query-optimal programs
CiE 2008 3
What is speed-up theorem
First remarked by Gödel in 1936
Blum’s Speed-up Theorems (1967,1971)
Operator Speed-up Theorems (Meyer et al, 1972, 1973)
CiE 2008 4
Type-2 computable =def Oracle TM computable
(N N) N N
Type-2 Speed-up Theorem (Li, 2007: CiE)
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
Type-2: ?
i.e., compact sets in the discrete topology
Compact sets in topology ???
Baire Topology doesn’t work
T(F) (Li 2004)
CiE 2008 6
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 the limit
The topology defined by the queries made
in the limit by programs i and j: Q (i, j)
CiE 2008 12
If we choose each definition of the query-optimal 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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1. Does every computable functional always have a query-optimal OTM for it?
2. If the answer to the first question is negative, then can we construct the query-speed-able functional?
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?
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
yes, yes, ?
no, no, ?
absolute, strong, weak
no, no, no
no, ?, ?