'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ f(a(), a()) -> f(a(), b())
, f(a(), b()) -> f(s(a()), c())
, f(s(X), c()) -> f(X, c())
, f(c(), c()) -> f(a(), a())}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ f^#(a(), a()) -> c_0(f^#(a(), b()))
, f^#(a(), b()) -> c_1(f^#(s(a()), c()))
, f^#(s(X), c()) -> c_2(f^#(X, c()))
, f^#(c(), c()) -> c_3(f^#(a(), a()))}
The usable rules are:
{}
The estimated dependency graph contains the following edges:
{f^#(a(), a()) -> c_0(f^#(a(), b()))}
==> {f^#(a(), b()) -> c_1(f^#(s(a()), c()))}
{f^#(a(), b()) -> c_1(f^#(s(a()), c()))}
==> {f^#(s(X), c()) -> c_2(f^#(X, c()))}
{f^#(s(X), c()) -> c_2(f^#(X, c()))}
==> {f^#(c(), c()) -> c_3(f^#(a(), a()))}
{f^#(s(X), c()) -> c_2(f^#(X, c()))}
==> {f^#(s(X), c()) -> c_2(f^#(X, c()))}
{f^#(c(), c()) -> c_3(f^#(a(), a()))}
==> {f^#(a(), a()) -> c_0(f^#(a(), b()))}
We consider the following path(s):
1) { f^#(a(), a()) -> c_0(f^#(a(), b()))
, f^#(c(), c()) -> c_3(f^#(a(), a()))
, f^#(s(X), c()) -> c_2(f^#(X, c()))
, f^#(a(), b()) -> c_1(f^#(s(a()), c()))}
The usable rules for this path are empty.
We have oriented the usable rules with the following strongly linear interpretation:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
a() = [0]
b() = [0]
s(x1) = [0] x1 + [0]
c() = [0]
f^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
c_3(x1) = [0] x1 + [0]
We have applied the subprocessor on the resulting DP-problem:
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules:
{ f^#(a(), a()) -> c_0(f^#(a(), b()))
, f^#(c(), c()) -> c_3(f^#(a(), a()))
, f^#(s(X), c()) -> c_2(f^#(X, c()))
, f^#(a(), b()) -> c_1(f^#(s(a()), c()))}
Weak Rules: {}
Details:
We apply the weight gap principle, strictly orienting the rules
{f^#(s(X), c()) -> c_2(f^#(X, c()))}
and weakly orienting the rules
{}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(s(X), c()) -> c_2(f^#(X, c()))}
Details:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
a() = [0]
b() = [0]
s(x1) = [1] x1 + [4]
c() = [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [3]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(c(), c()) -> c_3(f^#(a(), a()))}
and weakly orienting the rules
{f^#(s(X), c()) -> c_2(f^#(X, c()))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(c(), c()) -> c_3(f^#(a(), a()))}
Details:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
a() = [0]
b() = [0]
s(x1) = [1] x1 + [0]
c() = [8]
f^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [8]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [7]
Finally we apply the subprocessor
We apply the weight gap principle, strictly orienting the rules
{f^#(a(), b()) -> c_1(f^#(s(a()), c()))}
and weakly orienting the rules
{ f^#(c(), c()) -> c_3(f^#(a(), a()))
, f^#(s(X), c()) -> c_2(f^#(X, c()))}
using the following strongly linear interpretation:
Processor 'Matrix Interpretation' oriented the following rules strictly:
{f^#(a(), b()) -> c_1(f^#(s(a()), c()))}
Details:
Interpretation Functions:
f(x1, x2) = [0] x1 + [0] x2 + [0]
a() = [0]
b() = [8]
s(x1) = [1] x1 + [0]
c() = [0]
f^#(x1, x2) = [1] x1 + [1] x2 + [1]
c_0(x1) = [1] x1 + [0]
c_1(x1) = [1] x1 + [0]
c_2(x1) = [1] x1 + [0]
c_3(x1) = [1] x1 + [0]
Finally we apply the subprocessor
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(a(), a()) -> c_0(f^#(a(), b()))}
Weak Rules:
{ f^#(a(), b()) -> c_1(f^#(s(a()), c()))
, f^#(c(), c()) -> c_3(f^#(a(), a()))
, f^#(s(X), c()) -> c_2(f^#(X, c()))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {f^#(a(), a()) -> c_0(f^#(a(), b()))}
Weak Rules:
{ f^#(a(), b()) -> c_1(f^#(s(a()), c()))
, f^#(c(), c()) -> c_3(f^#(a(), a()))
, f^#(s(X), c()) -> c_2(f^#(X, c()))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_0() -> 2
, a_1() -> 4
, a_2() -> 12
, b_0() -> 2
, b_1() -> 5
, b_2() -> 13
, s_0(2) -> 2
, s_1(4) -> 7
, s_2(12) -> 15
, c_0() -> 2
, c_1() -> 8
, c_2() -> 16
, f^#_0(2, 2) -> 1
, f^#_1(4, 4) -> 9
, f^#_1(4, 5) -> 3
, f^#_1(4, 8) -> 10
, f^#_1(7, 8) -> 6
, f^#_2(12, 13) -> 11
, f^#_2(12, 16) -> 17
, f^#_2(15, 16) -> 14
, c_0_1(3) -> 1
, c_0_2(11) -> 9
, c_1_0(1) -> 1
, c_1_1(6) -> 1
, c_1_1(6) -> 3
, c_1_2(14) -> 3
, c_1_2(14) -> 11
, c_2_0(1) -> 1
, c_2_1(10) -> 6
, c_2_2(17) -> 14
, c_3_0(1) -> 1
, c_3_1(9) -> 1}