'Weak Dependency Graph [60.0]'
------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1))))))))))
, 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))}
Details:
We have computed the following set of weak (innermost) dependency pairs:
{ 0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))
, 0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))
, 0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
The usable rules are:
{ 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1))))))))))
, 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))}
The estimated dependency graph contains the following edges:
{0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))}
==> {0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
{0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))}
==> {0^#(1(2(1(x1)))) ->
c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))}
{0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))}
==> {0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
{0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))}
==> {0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))}
{0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
==> {0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
{0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
==> {0^#(1(2(1(x1)))) ->
c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))}
{0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
==> {0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
{0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
==> {0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))}
{0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))}
==> {0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
{0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))}
==> {0^#(1(2(1(x1)))) ->
c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))}
{0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))}
==> {0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
{0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))}
==> {0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))}
{0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
==> {0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
{0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
==> {0^#(1(2(1(x1)))) ->
c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))}
{0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
==> {0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
{0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))}
==> {0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))}
We consider the following path(s):
1) { 0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))
, 0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
The usable rules for this path are the following:
{ 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1))))))))))
, 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))}
We have applied the subprocessor on the union of usable rules and weak (innermost) dependency pairs.
'Weight Gap Principle'
----------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1))))))))))
, 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))
, 0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))
, 0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
Details:
'fastest of 'combine', 'Bounds with default enrichment', 'Bounds with default enrichment''
------------------------------------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1))))))))))
, 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))
, 0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))
, 0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
Details:
The problem was solved by processor 'Bounds with default enrichment':
'Bounds with default enrichment'
--------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(x1))))))))))
, 0(1(2(1(x1)))) -> 1(2(1(1(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))
, 0^#(1(2(1(x1)))) -> c_0(0^#(1(2(0(1(2(x1)))))))
, 0^#(1(2(1(x1)))) ->
c_3(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1))))))))))))))))
, 0^#(1(2(1(x1)))) -> c_2(0^#(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))
, 0^#(1(2(1(x1)))) -> c_1(0^#(1(2(0(1(2(0(1(2(x1))))))))))}
Details:
The problem is Match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ 0_1(6) -> 20
, 0_1(9) -> 8
, 0_1(12) -> 14
, 0_1(15) -> 14
, 0_1(18) -> 17
, 0_2(30) -> 29
, 0_2(33) -> 32
, 0_2(36) -> 35
, 0_2(42) -> 41
, 0_2(45) -> 44
, 0_2(48) -> 47
, 0_2(51) -> 50
, 0_2(57) -> 56
, 0_2(60) -> 59
, 0_2(63) -> 62
, 0_2(66) -> 65
, 0_2(69) -> 68
, 0_2(75) -> 92
, 0_2(78) -> 77
, 0_2(84) -> 56
, 0_2(87) -> 86
, 0_2(90) -> 89
, 1_0(2) -> 2
, 1_0(3) -> 2
, 1_1(7) -> 6
, 1_1(10) -> 9
, 1_1(13) -> 12
, 1_1(14) -> 25
, 1_1(16) -> 15
, 1_1(17) -> 25
, 1_1(19) -> 18
, 1_1(20) -> 25
, 1_1(23) -> 8
, 1_1(25) -> 24
, 1_2(26) -> 20
, 1_2(28) -> 27
, 1_2(29) -> 28
, 1_2(31) -> 30
, 1_2(32) -> 28
, 1_2(34) -> 33
, 1_2(37) -> 36
, 1_2(38) -> 17
, 1_2(40) -> 39
, 1_2(41) -> 40
, 1_2(43) -> 42
, 1_2(44) -> 40
, 1_2(46) -> 45
, 1_2(47) -> 40
, 1_2(49) -> 48
, 1_2(52) -> 51
, 1_2(53) -> 14
, 1_2(55) -> 54
, 1_2(56) -> 55
, 1_2(58) -> 57
, 1_2(59) -> 55
, 1_2(61) -> 60
, 1_2(62) -> 55
, 1_2(64) -> 63
, 1_2(65) -> 55
, 1_2(67) -> 66
, 1_2(70) -> 69
, 1_2(76) -> 75
, 1_2(79) -> 78
, 1_2(85) -> 84
, 1_2(86) -> 55
, 1_2(88) -> 87
, 1_2(89) -> 55
, 1_2(91) -> 90
, 1_2(92) -> 55
, 2_0(2) -> 3
, 2_0(3) -> 3
, 2_1(2) -> 10
, 2_1(3) -> 10
, 2_1(8) -> 7
, 2_1(14) -> 13
, 2_1(17) -> 16
, 2_1(20) -> 19
, 2_1(24) -> 23
, 2_2(23) -> 37
, 2_2(26) -> 52
, 2_2(27) -> 26
, 2_2(29) -> 31
, 2_2(32) -> 31
, 2_2(35) -> 34
, 2_2(38) -> 70
, 2_2(39) -> 38
, 2_2(41) -> 43
, 2_2(44) -> 43
, 2_2(47) -> 46
, 2_2(50) -> 49
, 2_2(53) -> 79
, 2_2(54) -> 53
, 2_2(59) -> 58
, 2_2(62) -> 61
, 2_2(65) -> 64
, 2_2(68) -> 67
, 2_2(77) -> 76
, 2_2(86) -> 85
, 2_2(89) -> 88
, 2_2(92) -> 91
, 0^#_0(2) -> 4
, 0^#_0(3) -> 4
, 0^#_1(6) -> 5
, 0^#_1(12) -> 11
, 0^#_1(15) -> 21
, 0^#_1(18) -> 22
, 0^#_2(30) -> 80
, 0^#_2(33) -> 71
, 0^#_2(42) -> 81
, 0^#_2(45) -> 95
, 0^#_2(48) -> 72
, 0^#_2(57) -> 82
, 0^#_2(60) -> 93
, 0^#_2(63) -> 96
, 0^#_2(66) -> 73
, 0^#_2(75) -> 74
, 0^#_2(84) -> 83
, 0^#_2(87) -> 94
, 0^#_2(90) -> 97
, c_0_1(5) -> 4
, c_0_2(71) -> 5
, c_0_2(72) -> 22
, c_0_2(73) -> 21
, c_0_2(74) -> 11
, c_1_1(22) -> 4
, c_1_2(80) -> 5
, c_1_2(95) -> 22
, c_1_2(96) -> 21
, c_1_2(97) -> 11
, c_2_1(21) -> 4
, c_2_2(80) -> 5
, c_2_2(81) -> 22
, c_2_2(93) -> 21
, c_2_2(94) -> 11
, c_3_1(11) -> 4
, c_3_2(80) -> 5
, c_3_2(81) -> 22
, c_3_2(82) -> 21
, c_3_2(83) -> 11}