Senior%20Project%20

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Senior Project Computer Science -- 2005 Notions
of Smallness in Oracle Space Nikhil
Srivastava Advisors Brian Postow (CS), Alan
Taylor (Math)
Abstract We study the abundance of oracles A for
which PANPA under three different notions of
size.
2. Oracles Suppose we give a TM access to some
information for free viz. we allow it to query an
set X (an oracle) at no cost.. MX Turing
Machine M w/ access to oracle X PX Languages
recognized by deterministic MX NPX Languages
recognized by nondeterministic MX A A
PANPA B B PBNPB Question What happens
most of the time?
1. Preliminaries Language (infinite) Set of
binary strings Decision problem Turing Machine
(TM) Computer Program P Languages
recognized by poly deterministic TMs
Problems that can be solved in poly time NP
Languages recognized by poly non-deterministic
TMs Problems for which solutions can be
guessed and checked in poly time
3. Smallness

• Question Is A big or small?
• Two contexts
• All oracles ALL (uncountable)
• Computable Oracles REC (countable)

Languages in S
Language Characteristic Sequence real number
in 0,1 Class of Languages S subset of
0,1 Can use measure theory on the real line.
S
Measure Zero Squeezed into Small Spaces
Meager Full of Holes
Porous (Small Ramsey)
X is nowhere dense if ?J ? J? J such that J?
X?
Measure Zero. S can be approximated by
arbitrarily small intervals ?? gt0 ? Jn such
that S?? ?? Jn and ? Jnlt?
Given Y, Y infinite subsets of Y
Y
Y
?
?

• Intuition
• If S is small, SC is big.
• If SC is big, Y ? SC for lots of Y.

?
Porous. S is porous if ?X ? Y ? X such that Y
? S ?
Meager S can be approximated by nowhere dense
sets, in that S is a countable union of nowhere
dense sets. Effectively Meager. The above
definition is useless in REC, since REC is itself
countable. Solution replace ? with computable
function f. Now, f picks subintervals J
?
X
?
Y ? S ?
Effectively Measure Zero The above definition
useless in REC, since REC is countable. A
computable version of Measure Zero, studied by
Lutz, can be applied to REC and subclasses.
Y
Is A Porous in ALL? Is there a reasonable
definition of Effectively Porous? Is A
Effectively Porous in REC?

f
Bennett-Gill (1981) A is Measure Zero in ALL
Mehlhorn (1973) A is Effectively Meager in REC
Bigger Questions Is A small in any reasonable
topology? Can we characterize A and B any better?