Tool CaT
stdout:
YES(?,O(n^1))
Problem:
a(a(x1)) -> b(b(b(x1)))
b(x1) -> c(c(d(x1)))
c(x1) -> d(d(d(x1)))
b(c(x1)) -> c(b(x1))
b(c(d(x1))) -> a(x1)
Proof:
Bounds Processor:
bound: 2
enrichment: match
automaton:
final states: {4,3,2}
transitions:
d1(5) -> 6*
d1(17) -> 18*
d1(16) -> 17*
c1(7) -> 8*
c1(6) -> 7*
d2(20) -> 21*
d2(27) -> 28*
d2(29) -> 30*
d2(19) -> 20*
d2(21) -> 22*
d2(28) -> 29*
a0(1) -> 2*
b0(1) -> 3*
c0(1) -> 4*
d0(1) -> 1*
1 -> 5*
6 -> 27,16
7 -> 19*
8 -> 3*
18 -> 4*
22 -> 8*
30 -> 7*
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)))
, b(x1) -> c(c(d(x1)))
, c(x1) -> d(d(d(x1)))
, b(c(x1)) -> c(b(x1))
, b(c(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)))
, b(x1) -> c(c(d(x1)))
, c(x1) -> d(d(d(x1)))
, b(c(x1)) -> c(b(x1))
, b(c(d(x1))) -> a(x1)}
Proof Output:
The problem is match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 1
, b_0(2) -> 1
, c_0(2) -> 1
, c_1(3) -> 1
, c_1(4) -> 3
, d_0(2) -> 2
, d_1(2) -> 4
, d_1(4) -> 5
, d_1(5) -> 1
, d_2(3) -> 7
, d_2(4) -> 9
, d_2(6) -> 1
, d_2(7) -> 6
, d_2(8) -> 3
, d_2(9) -> 8}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)))
, b(x1) -> c(c(d(x1)))
, c(x1) -> d(d(d(x1)))
, b(c(x1)) -> c(b(x1))
, b(c(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)))
, b(x1) -> c(c(d(x1)))
, c(x1) -> d(d(d(x1)))
, b(c(x1)) -> c(b(x1))
, b(c(d(x1))) -> a(x1)}
Proof Output:
The problem is match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 1
, b_0(2) -> 1
, c_0(2) -> 1
, c_1(3) -> 1
, c_1(4) -> 3
, d_0(2) -> 2
, d_1(2) -> 4
, d_1(4) -> 5
, d_1(5) -> 1
, d_2(3) -> 7
, d_2(4) -> 9
, d_2(6) -> 1
, d_2(7) -> 6
, d_2(8) -> 3
, d_2(9) -> 8}