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

CiE 2008 5

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

CiE 2008 7

The collection of queries made during the course of computation in s steps

Q(i,j)

CiE 2008 8

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 9

Let i and j be two programs (indexes of OTM)

CiE 2008 10

How to speed-up?

CiE 2008 11

Query-optimal OTM

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

CiE 2008 13

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, ?, ?

CiE 2008 14

Type-2 functional K

CiE 2008 15

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?