Tool CaT
stdout:
YES(?,O(n^1))
Problem:
a(a(x1)) -> b(b(b(x1)))
a(x1) -> c(d(x1))
b(b(x1)) -> c(c(c(x1)))
c(c(x1)) -> d(d(d(x1)))
e(d(x1)) -> a(b(c(d(e(x1)))))
b(x1) -> d(d(x1))
e(c(x1)) -> b(a(a(e(x1))))
c(d(d(x1))) -> a(x1)
Proof:
Bounds Processor:
bound: 4
enrichment: match
automaton:
final states: {5,4,3,2}
transitions:
a1(16) -> 17*
a1(28) -> 29*
d1(9) -> 10*
d1(26) -> 27*
d1(13) -> 14*
b1(15) -> 16*
c1(10) -> 11*
c1(14) -> 15*
e1(12) -> 13*
d2(45) -> 46*
d2(35) -> 36*
d2(44) -> 45*
d2(38) -> 39*
a0(1) -> 2*
c2(39) -> 40*
c2(36) -> 37*
b0(1) -> 3*
a3(57) -> 58*
a3(52) -> 53*
c0(1) -> 4*
c4(55) -> 56*
d0(1) -> 1*
d4(62) -> 63*
d4(54) -> 55*
e0(1) -> 5*
1 -> 28,12,9
10 -> 26*
11 -> 2*
15 -> 44*
16 -> 35*
17 -> 13,5
27 -> 3*
28 -> 38*
29 -> 40,11,2,4
37 -> 17,5
40 -> 29,4
44 -> 57*
45 -> 52*
46 -> 16*
52 -> 54*
53 -> 37*
56 -> 58,53,37,17
57 -> 62*
58 -> 56,53
63 -> 55*
problem:
QedTool IRC1
stdout:
YES(?,O(n^1))
Tool IRC2
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Proof Output:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:
Details:
--------
'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
'Bounds with minimal-enrichment and initial automaton 'match''
--------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: innermost runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Proof Output:
The problem is match-bounded by 4.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 1
, a_1(2) -> 1
, a_1(2) -> 7
, a_1(4) -> 1
, a_1(4) -> 7
, a_3(5) -> 1
, a_3(5) -> 7
, a_3(8) -> 1
, a_3(8) -> 7
, b_0(2) -> 1
, b_1(5) -> 4
, c_0(2) -> 1
, c_1(3) -> 1
, c_1(6) -> 5
, c_2(9) -> 1
, c_2(9) -> 7
, c_4(10) -> 1
, c_4(10) -> 7
, d_0(2) -> 2
, d_1(2) -> 3
, d_1(3) -> 1
, d_1(7) -> 6
, d_2(2) -> 9
, d_2(4) -> 9
, d_2(5) -> 8
, d_2(8) -> 4
, d_4(5) -> 10
, d_4(8) -> 10
, e_0(2) -> 1
, e_1(2) -> 7}Tool RC1
stdout:
YES(?,O(n^1))
Tool RC2
stdout:
YES(?,O(n^1))
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Proof Output:
'Bounds with minimal-enrichment and initial automaton 'match'' proved the best result:
Details:
--------
'Bounds with minimal-enrichment and initial automaton 'match'' succeeded with the following output:
'Bounds with minimal-enrichment and initial automaton 'match''
--------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ a(a(x1)) -> b(b(b(x1)))
, a(x1) -> c(d(x1))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, e(d(x1)) -> a(b(c(d(e(x1)))))
, b(x1) -> d(d(x1))
, e(c(x1)) -> b(a(a(e(x1))))
, c(d(d(x1))) -> a(x1)}
Proof Output:
The problem is match-bounded by 4.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 1
, a_1(2) -> 1
, a_1(2) -> 7
, a_1(4) -> 1
, a_1(4) -> 7
, a_3(5) -> 1
, a_3(5) -> 7
, a_3(8) -> 1
, a_3(8) -> 7
, b_0(2) -> 1
, b_1(5) -> 4
, c_0(2) -> 1
, c_1(3) -> 1
, c_1(6) -> 5
, c_2(9) -> 1
, c_2(9) -> 7
, c_4(10) -> 1
, c_4(10) -> 7
, d_0(2) -> 2
, d_1(2) -> 3
, d_1(3) -> 1
, d_1(7) -> 6
, d_2(2) -> 9
, d_2(4) -> 9
, d_2(5) -> 8
, d_2(8) -> 4
, d_4(5) -> 10
, d_4(8) -> 10
, e_0(2) -> 1
, e_1(2) -> 7}