On the computational complexity of the languages of general symbolic dynamical systems and beta-shifts
We consider the computational complexity of languages of symbolic dynamical systems. In particular, we study complexity hierarchies and membership of the non-uniform class P/poly . We prove: 1. For every time-constructible, non-decreasing function t ( n ) = ω ( n ) , there is a symbolic dynamical sy...
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| Vydáno v: | Theoretical computer science Ročník 410; číslo 47; s. 4878 - 4891 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Oxford
Elsevier B.V
06.11.2009
Elsevier |
| Témata: | |
| ISSN: | 0304-3975, 1879-2294 |
| On-line přístup: | Získat plný text |
| Tagy: |
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| Shrnutí: | We consider the computational complexity of languages of symbolic dynamical systems. In particular, we study complexity hierarchies and membership of the non-uniform class
P/poly
. We prove:
1.
For every time-constructible, non-decreasing function
t
(
n
)
=
ω
(
n
)
, there is a symbolic dynamical system with language decidable in deterministic time
O
(
n
2
t
(
n
)
)
, but not in deterministic time
o
(
t
(
n
)
)
.
2.
For every space-constructible, non-decreasing function
s
(
n
)
=
ω
(
n
)
, there is a symbolic dynamical system with language decidable in deterministic space
O
(
s
(
n
)
)
, but not in deterministic space
o
(
s
(
n
)
)
.
3.
There are symbolic dynamical systems having hard and complete languages under
≤
m
l
o
g
s
- and
≤
m
p
-reduction for every complexity class above
LOGSPACE
in the backbone hierarchy (hence,
P
-complete,
NP
-complete,
coNP
-complete,
PSPACE
-complete, and
EXPTIME
-complete sets).
4.
There are decidable languages of symbolic dynamical systems in
P/poly
for every alphabet of size
|
Σ
|
≥
1
.
5.
There are decidable languages of symbolic dynamical systems not in
P/poly
iff the alphabet size is
>
1
.
For the particular class of symbolic dynamical systems known as
β
-shifts, we prove that:
1.
For all real numbers
β
>
1
, the language of the
β
-shift is in
P/poly
.
2.
If there exists a real number
β
>
1
such that the language of the
β
-shift is NP-hard under
≤
T
p
-reduction, then the polynomial hierarchy collapses to the second level. As NP-hardness under
≤
m
p
-reduction implies hardness under
≤
T
p
-reduction, this result implies that it is unlikely that a proof of existence of an NP-hard language of a
β
-shift will be forthcoming.
3.
For every time-constructible, non-decreasing function
t
(
n
)
≥
n
, there is a real number
1
<
β
<
2
such that the language of the
β
-shift is decidable in time
O
(
n
2
t
(
log
n
+
1
)
)
, but not in any proper time bound
g
(
n
)
satisfying
g
(
4
n
)
=
o
(
t
(
n
)
/
1
6
n
)
.
4.
For every space-constructible, non-decreasing function
s
(
n
)
=
ω
(
n
2
)
, there is a real number
1
<
β
<
2
such that the language of the
β
-shift is decidable in space
O
(
s
(
n
)
)
, but not in space
g
(
n
)
where
g
is any function satisfying
g
(
n
2
)
=
o
(
s
(
n
)
)
.
5.
There exists a real number
1
<
β
<
2
such that the language of the
β
-shift is recursive, but not context-sensitive. |
|---|---|
| Bibliografie: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
| ISSN: | 0304-3975 1879-2294 |
| DOI: | 10.1016/j.tcs.2009.06.037 |