On ϵ-sensitive monotone computations
We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial f ∈ R [ x 1 , ⋯ , x n ] of degree d has an arithmetic circuit of size s then (...
Gespeichert in:
| Veröffentlicht in: | Computational complexity Jg. 29; H. 2 |
|---|---|
| 1. Verfasser: | |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Cham
Springer International Publishing
01.12.2020
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 1016-3328, 1420-8954 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Zusammenfassung: | We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial
f
∈
R
[
x
1
,
⋯
,
x
n
]
of degree
d
has an arithmetic circuit of size
s
then
(
x
1
+
⋯
+
x
n
+
1
)
d
+
ϵ
f
has a monotone arithmetic circuit of size
O
(
s
d
2
+
n
log
n
)
, for some
ϵ
>
0
. Second, if
f
:
{
0
,
1
}
n
→
{
0
,
1
}
is a Boolean function, we associate with
f
an explicit exponential-size matrix
M
(
f
) such that the Boolean circuit size of
f
is at least
Ω
(
min
ϵ
>
0
(
rk
+
(
M
(
f
)
-
ϵ
J
)
)
-
2
n
)
, where
J
is the all-ones matrix and
rk
+
denotes the nonnegative rank of a matrix. In fact, the quantity
min
ϵ
>
0
(
rk
+
(
M
(
f
)
-
ϵ
J
)
)
characterizes how hard is it to distinguish rejecting and accepting inputs of
f
by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar
ϵ
-sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1016-3328 1420-8954 |
| DOI: | 10.1007/s00037-020-00196-6 |