Tool CaT
stdout:
YES(?,O(n^1))
Problem:
a(a(x1)) -> b(b(b(x1)))
a(x1) -> d(c(d(x1)))
b(b(x1)) -> c(c(c(x1)))
c(c(x1)) -> d(d(d(x1)))
c(d(d(x1))) -> a(x1)
Proof:
Bounds Processor:
bound: 2
enrichment: match
automaton:
final states: {4,3,2}
transitions:
a1(13) -> 14*
d1(9) -> 10*
d1(11) -> 12*
c1(10) -> 11*
d2(21) -> 22*
d2(23) -> 24*
c2(22) -> 23*
a0(1) -> 2*
b0(1) -> 3*
d0(1) -> 1*
c0(1) -> 4*
1 -> 13,9
12 -> 2*
13 -> 21*
14 -> 23,11,4
24 -> 14,4
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) -> d(c(d(x1)))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(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) -> d(c(d(x1)))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(d(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
, a_1(2) -> 1
, a_1(2) -> 3
, a_1(2) -> 5
, b_0(2) -> 1
, d_0(2) -> 2
, d_1(2) -> 4
, d_1(3) -> 1
, d_2(2) -> 6
, d_2(5) -> 1
, d_2(5) -> 3
, d_2(5) -> 5
, c_0(2) -> 1
, c_1(4) -> 3
, c_2(6) -> 5}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) -> d(c(d(x1)))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(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) -> d(c(d(x1)))
, b(b(x1)) -> c(c(c(x1)))
, c(c(x1)) -> d(d(d(x1)))
, c(d(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
, a_1(2) -> 1
, a_1(2) -> 3
, a_1(2) -> 5
, b_0(2) -> 1
, d_0(2) -> 2
, d_1(2) -> 4
, d_1(3) -> 1
, d_2(2) -> 6
, d_2(5) -> 1
, d_2(5) -> 3
, d_2(5) -> 5
, c_0(2) -> 1
, c_1(4) -> 3
, c_2(6) -> 5}