Imai Laboratory Introduction
Studies on Computational Complexity
Time versus Space in the Computation Universe
Complexity Classes
• L = the set of languages computable in logarithm space.
• P = the set of languages computable in polynomial time.
• PSPACE = the set of language computable in polynomial space.
Function Complexity Classes
Reversal versus Access
• FP = the set of functions computable
in polynomial time.
• FPSPACE = the set of functions computable
in polynomial space.
P=PSPACE ⇔FP=FPSPACE
1
2
13
3
4
2
4
6
8
10
12
5
6
3
Recursion Theoretic Operators
•Reversal complexity is the total number of tape head reversals.
•Access complexity is the maximum number of accesses
among all tape cells.
Comp*(C) = the smallest class containing C
and closed under Comp
Space Bounded Reversal and Access Complexity Classes
Reversal
Access
PSPACE
exp
P versus PSPACE
FPSPACE-completeness
We introduce a notion of
FPSPACE-completeness and
show some function is in FP
iff P=PSPACE.
PSPACE
exp
L
poly
Theorem
FPSPACE=Comp*(BRec(FP))
Corollary
FP is closed under BRec
⇔P=PSPACE
1
3
5
7
9
11
P
L
poly
log
P
log
L
1
1
1
[1]
PSPACE
P
L
[2]
log
poly
Space
1
log
poly
Space
Random Combinatorial Structures
Probabilistic analyses
of reversal and access
complexity using the
Balls-into-Bins model.
Balls ⇒Time Complexity
Bins ⇒Space Complexity
To clarify the notion of efficient algorithms and the limit of computation,
we give structural analyses on the fundamental models of computation.
References
[1] Kenya Ueno: "Recursion Theoretic Operators for Function Complexity Classes,"
The 16th Annual International Symposium on Algorithms and Computation (ISAAC 2005),
Sanya, Hainan, China, December, 2005. (LNCS 3827, pp.748-756)
[2] Kenya Ueno: "Reversal versus Access: Complexity Classes and Random Combinatorial Structures,"
The First AAAC Annual Meeting (AAAC 2008), Hong Kong, China, April, 2008.