'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)))))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(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))))))))))))))))
, 0^#(1(2(1(x1)))) ->
c_4(0^#(1(2(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)))))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(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_4(0^#(1(2(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_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_4(0^#(1(2(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_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_4(0^#(1(2(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_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_4(0^#(1(2(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_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)))))))}
{0^#(1(2(1(x1)))) ->
c_4(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))}
==> {0^#(1(2(1(x1)))) ->
c_4(0^#(1(2(0(1(2(0(1(2(0(1(2(0(1(2(0(1(2(x1)))))))))))))))))))}
{0^#(1(2(1(x1)))) ->
c_4(0^#(1(2(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_4(0^#(1(2(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_4(0^#(1(2(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_4(0^#(1(2(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_4(0^#(1(2(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_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)))))))))))))))))))
, 0(1(2(1(x1)))) ->
1(2(1(1(0(1(2(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)))) ->
1(2(1(1(0(1(2(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_4(0^#(1(2(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_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)))) ->
1(2(1(1(0(1(2(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_4(0^#(1(2(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_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)))) ->
1(2(1(1(0(1(2(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_4(0^#(1(2(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_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(4) -> 21
, 0_1(7) -> 6
, 0_1(10) -> 12
, 0_1(13) -> 12
, 0_1(16) -> 15
, 0_1(19) -> 18
, 0_2(32) -> 31
, 0_2(35) -> 34
, 0_2(38) -> 37
, 0_2(44) -> 43
, 0_2(47) -> 46
, 0_2(50) -> 49
, 0_2(53) -> 52
, 0_2(59) -> 58
, 0_2(62) -> 61
, 0_2(65) -> 64
, 0_2(68) -> 67
, 0_2(71) -> 70
, 0_2(77) -> 76
, 0_2(80) -> 79
, 0_2(83) -> 82
, 0_2(86) -> 85
, 0_2(89) -> 88
, 0_2(92) -> 91
, 0_2(99) -> 120
, 0_2(102) -> 101
, 0_2(109) -> 76
, 0_2(112) -> 111
, 0_2(115) -> 114
, 0_2(118) -> 117
, 1_0(2) -> 2
, 1_1(5) -> 4
, 1_1(8) -> 7
, 1_1(11) -> 10
, 1_1(12) -> 27
, 1_1(14) -> 13
, 1_1(15) -> 27
, 1_1(17) -> 16
, 1_1(18) -> 27
, 1_1(20) -> 19
, 1_1(21) -> 27
, 1_1(25) -> 6
, 1_1(27) -> 26
, 1_2(28) -> 21
, 1_2(30) -> 29
, 1_2(31) -> 30
, 1_2(33) -> 32
, 1_2(34) -> 30
, 1_2(36) -> 35
, 1_2(39) -> 38
, 1_2(40) -> 18
, 1_2(42) -> 41
, 1_2(43) -> 42
, 1_2(45) -> 44
, 1_2(46) -> 42
, 1_2(48) -> 47
, 1_2(49) -> 42
, 1_2(51) -> 50
, 1_2(54) -> 53
, 1_2(55) -> 15
, 1_2(57) -> 56
, 1_2(58) -> 57
, 1_2(60) -> 59
, 1_2(61) -> 57
, 1_2(63) -> 62
, 1_2(64) -> 57
, 1_2(66) -> 65
, 1_2(67) -> 57
, 1_2(69) -> 68
, 1_2(72) -> 71
, 1_2(73) -> 12
, 1_2(75) -> 74
, 1_2(76) -> 75
, 1_2(78) -> 77
, 1_2(79) -> 75
, 1_2(81) -> 80
, 1_2(82) -> 75
, 1_2(84) -> 83
, 1_2(85) -> 75
, 1_2(87) -> 86
, 1_2(88) -> 75
, 1_2(90) -> 89
, 1_2(93) -> 92
, 1_2(100) -> 99
, 1_2(103) -> 102
, 1_2(110) -> 109
, 1_2(111) -> 75
, 1_2(113) -> 112
, 1_2(114) -> 75
, 1_2(116) -> 115
, 1_2(117) -> 75
, 1_2(119) -> 118
, 1_2(120) -> 75
, 2_0(2) -> 2
, 2_1(2) -> 8
, 2_1(6) -> 5
, 2_1(12) -> 11
, 2_1(15) -> 14
, 2_1(18) -> 17
, 2_1(21) -> 20
, 2_1(26) -> 25
, 2_2(25) -> 39
, 2_2(28) -> 54
, 2_2(29) -> 28
, 2_2(31) -> 33
, 2_2(34) -> 33
, 2_2(37) -> 36
, 2_2(40) -> 72
, 2_2(41) -> 40
, 2_2(43) -> 45
, 2_2(46) -> 45
, 2_2(49) -> 48
, 2_2(52) -> 51
, 2_2(55) -> 93
, 2_2(56) -> 55
, 2_2(58) -> 60
, 2_2(61) -> 60
, 2_2(64) -> 63
, 2_2(67) -> 66
, 2_2(70) -> 69
, 2_2(73) -> 103
, 2_2(74) -> 73
, 2_2(79) -> 78
, 2_2(82) -> 81
, 2_2(85) -> 84
, 2_2(88) -> 87
, 2_2(91) -> 90
, 2_2(101) -> 100
, 2_2(111) -> 110
, 2_2(114) -> 113
, 2_2(117) -> 116
, 2_2(120) -> 119
, 0^#_0(2) -> 1
, 0^#_1(4) -> 3
, 0^#_1(10) -> 9
, 0^#_1(13) -> 22
, 0^#_1(16) -> 23
, 0^#_1(19) -> 24
, 0^#_2(32) -> 104
, 0^#_2(35) -> 94
, 0^#_2(44) -> 105
, 0^#_2(47) -> 126
, 0^#_2(50) -> 95
, 0^#_2(59) -> 106
, 0^#_2(62) -> 123
, 0^#_2(65) -> 127
, 0^#_2(68) -> 96
, 0^#_2(77) -> 107
, 0^#_2(80) -> 121
, 0^#_2(83) -> 124
, 0^#_2(86) -> 128
, 0^#_2(89) -> 97
, 0^#_2(99) -> 98
, 0^#_2(109) -> 108
, 0^#_2(112) -> 122
, 0^#_2(115) -> 125
, 0^#_2(118) -> 129
, c_0_1(3) -> 1
, c_0_2(94) -> 3
, c_0_2(95) -> 24
, c_0_2(96) -> 23
, c_0_2(97) -> 22
, c_0_2(98) -> 9
, c_1_1(24) -> 1
, c_1_2(104) -> 3
, c_1_2(126) -> 24
, c_1_2(127) -> 23
, c_1_2(128) -> 22
, c_1_2(129) -> 9
, c_2_1(23) -> 1
, c_2_2(104) -> 3
, c_2_2(105) -> 24
, c_2_2(123) -> 23
, c_2_2(124) -> 22
, c_2_2(125) -> 9
, c_3_1(22) -> 1
, c_3_2(104) -> 3
, c_3_2(105) -> 24
, c_3_2(106) -> 23
, c_3_2(121) -> 22
, c_3_2(122) -> 9
, c_4_1(9) -> 1
, c_4_2(104) -> 3
, c_4_2(105) -> 24
, c_4_2(106) -> 23
, c_4_2(107) -> 22
, c_4_2(108) -> 9}