Tool CaT
stdout:
YES(?,O(n^1))
Problem:
a(d(x)) -> d(c(b(a(x))))
b(c(x)) -> c(d(a(b(x))))
a(c(x)) -> x
b(d(x)) -> x
Proof:
Bounds Processor:
bound: 1
enrichment: match
automaton:
final states: {4,3}
transitions:
c1(22) -> 23*
c1(7) -> 8*
d1(21) -> 22*
d1(8) -> 9*
a1(20) -> 21*
a1(5) -> 6*
a1(17) -> 18*
b1(19) -> 20*
b1(31) -> 32*
b1(6) -> 7*
a0(2) -> 3*
a0(1) -> 3*
d0(2) -> 1*
d0(1) -> 1*
c0(2) -> 2*
c0(1) -> 2*
b0(2) -> 4*
b0(1) -> 4*
1 -> 7,21,32,20,6,4,3,31,17
2 -> 7,21,32,20,6,4,3,19,5
8 -> 7*
9 -> 21,18,6,3
18 -> 6*
22 -> 21*
23 -> 7,20,4
32 -> 20*
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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 1
, a_1(2) -> 5
, a_1(8) -> 7
, d_0(2) -> 1
, d_0(2) -> 2
, d_0(2) -> 4
, d_0(2) -> 5
, d_0(2) -> 7
, d_0(2) -> 8
, d_1(3) -> 1
, d_1(3) -> 5
, d_1(3) -> 7
, d_1(7) -> 6
, d_1(7) -> 7
, c_0(2) -> 1
, c_0(2) -> 2
, c_0(2) -> 4
, c_0(2) -> 5
, c_0(2) -> 7
, c_0(2) -> 8
, c_1(4) -> 3
, c_1(4) -> 4
, c_1(6) -> 1
, c_1(6) -> 4
, c_1(6) -> 8
, b_0(2) -> 1
, b_1(2) -> 8
, b_1(5) -> 4}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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
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(d(x)) -> d(c(b(a(x))))
, b(c(x)) -> c(d(a(b(x))))
, a(c(x)) -> x
, b(d(x)) -> x}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 1
, a_1(2) -> 5
, a_1(8) -> 7
, d_0(2) -> 1
, d_0(2) -> 2
, d_0(2) -> 4
, d_0(2) -> 5
, d_0(2) -> 7
, d_0(2) -> 8
, d_1(3) -> 1
, d_1(3) -> 5
, d_1(3) -> 7
, d_1(7) -> 6
, d_1(7) -> 7
, c_0(2) -> 1
, c_0(2) -> 2
, c_0(2) -> 4
, c_0(2) -> 5
, c_0(2) -> 7
, c_0(2) -> 8
, c_1(4) -> 3
, c_1(4) -> 4
, c_1(6) -> 1
, c_1(6) -> 4
, c_1(6) -> 8
, b_0(2) -> 1
, b_1(2) -> 8
, b_1(5) -> 4}