Tool CaT
stdout:
YES(?,O(n^1))
Problem:
0(0(1(2(x1)))) -> 0(2(1(3(3(0(x1))))))
0(1(0(4(x1)))) -> 0(1(3(0(4(x1)))))
0(1(0(5(x1)))) -> 0(1(3(5(0(x1)))))
0(1(5(0(x1)))) -> 0(1(3(5(0(x1)))))
0(3(1(0(x1)))) -> 0(1(3(5(0(x1)))))
3(1(0(2(x1)))) -> 1(3(5(0(2(x1)))))
3(1(0(2(x1)))) -> 3(2(1(3(0(x1)))))
3(1(0(5(x1)))) -> 1(3(5(3(0(x1)))))
3(1(2(0(x1)))) -> 2(1(3(3(0(x1)))))
3(1(2(0(x1)))) -> 3(0(2(1(4(x1)))))
3(1(2(0(x1)))) -> 3(0(3(2(1(x1)))))
3(1(2(0(x1)))) -> 3(3(2(1(0(x1)))))
3(1(2(0(x1)))) -> 3(5(2(1(0(x1)))))
3(1(2(2(x1)))) -> 3(5(2(2(1(x1)))))
3(4(2(0(x1)))) -> 3(3(2(4(0(x1)))))
4(0(1(0(x1)))) -> 1(3(0(4(0(x1)))))
5(1(0(5(x1)))) -> 1(3(3(5(5(0(x1))))))
5(2(5(0(x1)))) -> 5(2(3(5(0(x1)))))
5(3(1(0(x1)))) -> 0(1(3(5(5(x1)))))
0(0(4(1(0(x1))))) -> 0(4(1(3(0(0(x1))))))
0(1(2(0(5(x1))))) -> 0(3(0(5(1(2(x1))))))
0(1(3(1(2(x1))))) -> 3(0(2(2(1(1(x1))))))
0(1(5(0(4(x1))))) -> 0(4(1(3(5(0(x1))))))
0(1(5(3(5(x1))))) -> 0(1(3(5(3(5(x1))))))
0(2(0(3(4(x1))))) -> 0(4(5(2(3(0(x1))))))
0(2(3(1(5(x1))))) -> 0(1(3(5(5(2(x1))))))
0(2(4(1(2(x1))))) -> 0(2(1(1(4(2(x1))))))
0(2(5(0(2(x1))))) -> 0(0(3(5(2(2(x1))))))
0(2(5(5(0(x1))))) -> 0(0(2(5(1(5(x1))))))
0(3(1(5(0(x1))))) -> 1(3(0(5(3(0(x1))))))
0(5(3(2(0(x1))))) -> 0(0(2(1(3(5(x1))))))
3(1(0(5(2(x1))))) -> 2(4(1(3(5(0(x1))))))
3(1(3(0(2(x1))))) -> 2(1(3(3(3(0(x1))))))
3(1(3(2(0(x1))))) -> 3(1(3(0(5(2(x1))))))
3(1(4(1(2(x1))))) -> 1(4(3(2(2(1(x1))))))
3(1(4(2(0(x1))))) -> 2(1(3(3(0(4(x1))))))
3(1(5(0(2(x1))))) -> 2(5(1(3(0(5(x1))))))
3(3(1(0(0(x1))))) -> 5(1(3(3(0(0(x1))))))
3(4(0(2(0(x1))))) -> 3(0(0(2(1(4(x1))))))
3(4(2(3(2(x1))))) -> 3(3(2(5(4(2(x1))))))
4(3(1(0(2(x1))))) -> 4(2(1(3(0(1(x1))))))
4(3(1(2(0(x1))))) -> 1(3(0(1(4(2(x1))))))
4(5(5(1(2(x1))))) -> 5(4(5(2(1(1(x1))))))
4(5(5(5(0(x1))))) -> 5(5(1(5(0(4(x1))))))
5(1(0(1(2(x1))))) -> 2(1(1(3(5(0(x1))))))
5(1(1(0(4(x1))))) -> 5(1(1(4(3(0(x1))))))
5(1(5(2(0(x1))))) -> 2(1(3(5(0(5(x1))))))
5(4(1(0(5(x1))))) -> 4(1(3(5(5(0(x1))))))
Proof:
Bounds Processor:
bound: 1
enrichment: match
automaton:
final states: {6,5,4,3}
transitions:
31(17) -> 18*
51(16) -> 17*
21(15) -> 16*
21(14) -> 15*
11(19) -> 20*
11(13) -> 14*
00(2) -> 3*
00(1) -> 3*
10(2) -> 1*
10(1) -> 1*
20(2) -> 2*
20(1) -> 2*
30(2) -> 4*
30(1) -> 4*
40(2) -> 5*
40(1) -> 5*
50(2) -> 6*
50(1) -> 6*
1 -> 19*
2 -> 13*
18 -> 4*
20 -> 14*
problem:
QedTool IRC1
stdout:
MAYBE
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:
{ 0(0(1(2(x1)))) -> 0(2(1(3(3(0(x1))))))
, 0(1(0(4(x1)))) -> 0(1(3(0(4(x1)))))
, 0(1(0(5(x1)))) -> 0(1(3(5(0(x1)))))
, 0(1(5(0(x1)))) -> 0(1(3(5(0(x1)))))
, 0(3(1(0(x1)))) -> 0(1(3(5(0(x1)))))
, 3(1(0(2(x1)))) -> 1(3(5(0(2(x1)))))
, 3(1(0(2(x1)))) -> 3(2(1(3(0(x1)))))
, 3(1(0(5(x1)))) -> 1(3(5(3(0(x1)))))
, 3(1(2(0(x1)))) -> 2(1(3(3(0(x1)))))
, 3(1(2(0(x1)))) -> 3(0(2(1(4(x1)))))
, 3(1(2(0(x1)))) -> 3(0(3(2(1(x1)))))
, 3(1(2(0(x1)))) -> 3(3(2(1(0(x1)))))
, 3(1(2(0(x1)))) -> 3(5(2(1(0(x1)))))
, 3(1(2(2(x1)))) -> 3(5(2(2(1(x1)))))
, 3(4(2(0(x1)))) -> 3(3(2(4(0(x1)))))
, 4(0(1(0(x1)))) -> 1(3(0(4(0(x1)))))
, 5(1(0(5(x1)))) -> 1(3(3(5(5(0(x1))))))
, 5(2(5(0(x1)))) -> 5(2(3(5(0(x1)))))
, 5(3(1(0(x1)))) -> 0(1(3(5(5(x1)))))
, 0(0(4(1(0(x1))))) -> 0(4(1(3(0(0(x1))))))
, 0(1(2(0(5(x1))))) -> 0(3(0(5(1(2(x1))))))
, 0(1(3(1(2(x1))))) -> 3(0(2(2(1(1(x1))))))
, 0(1(5(0(4(x1))))) -> 0(4(1(3(5(0(x1))))))
, 0(1(5(3(5(x1))))) -> 0(1(3(5(3(5(x1))))))
, 0(2(0(3(4(x1))))) -> 0(4(5(2(3(0(x1))))))
, 0(2(3(1(5(x1))))) -> 0(1(3(5(5(2(x1))))))
, 0(2(4(1(2(x1))))) -> 0(2(1(1(4(2(x1))))))
, 0(2(5(0(2(x1))))) -> 0(0(3(5(2(2(x1))))))
, 0(2(5(5(0(x1))))) -> 0(0(2(5(1(5(x1))))))
, 0(3(1(5(0(x1))))) -> 1(3(0(5(3(0(x1))))))
, 0(5(3(2(0(x1))))) -> 0(0(2(1(3(5(x1))))))
, 3(1(0(5(2(x1))))) -> 2(4(1(3(5(0(x1))))))
, 3(1(3(0(2(x1))))) -> 2(1(3(3(3(0(x1))))))
, 3(1(3(2(0(x1))))) -> 3(1(3(0(5(2(x1))))))
, 3(1(4(1(2(x1))))) -> 1(4(3(2(2(1(x1))))))
, 3(1(4(2(0(x1))))) -> 2(1(3(3(0(4(x1))))))
, 3(1(5(0(2(x1))))) -> 2(5(1(3(0(5(x1))))))
, 3(3(1(0(0(x1))))) -> 5(1(3(3(0(0(x1))))))
, 3(4(0(2(0(x1))))) -> 3(0(0(2(1(4(x1))))))
, 3(4(2(3(2(x1))))) -> 3(3(2(5(4(2(x1))))))
, 4(3(1(0(2(x1))))) -> 4(2(1(3(0(1(x1))))))
, 4(3(1(2(0(x1))))) -> 1(3(0(1(4(2(x1))))))
, 4(5(5(1(2(x1))))) -> 5(4(5(2(1(1(x1))))))
, 4(5(5(5(0(x1))))) -> 5(5(1(5(0(4(x1))))))
, 5(1(0(1(2(x1))))) -> 2(1(1(3(5(0(x1))))))
, 5(1(1(0(4(x1))))) -> 5(1(1(4(3(0(x1))))))
, 5(1(5(2(0(x1))))) -> 2(1(3(5(0(5(x1))))))
, 5(4(1(0(5(x1))))) -> 4(1(3(5(5(0(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:
{ 0(0(1(2(x1)))) -> 0(2(1(3(3(0(x1))))))
, 0(1(0(4(x1)))) -> 0(1(3(0(4(x1)))))
, 0(1(0(5(x1)))) -> 0(1(3(5(0(x1)))))
, 0(1(5(0(x1)))) -> 0(1(3(5(0(x1)))))
, 0(3(1(0(x1)))) -> 0(1(3(5(0(x1)))))
, 3(1(0(2(x1)))) -> 1(3(5(0(2(x1)))))
, 3(1(0(2(x1)))) -> 3(2(1(3(0(x1)))))
, 3(1(0(5(x1)))) -> 1(3(5(3(0(x1)))))
, 3(1(2(0(x1)))) -> 2(1(3(3(0(x1)))))
, 3(1(2(0(x1)))) -> 3(0(2(1(4(x1)))))
, 3(1(2(0(x1)))) -> 3(0(3(2(1(x1)))))
, 3(1(2(0(x1)))) -> 3(3(2(1(0(x1)))))
, 3(1(2(0(x1)))) -> 3(5(2(1(0(x1)))))
, 3(1(2(2(x1)))) -> 3(5(2(2(1(x1)))))
, 3(4(2(0(x1)))) -> 3(3(2(4(0(x1)))))
, 4(0(1(0(x1)))) -> 1(3(0(4(0(x1)))))
, 5(1(0(5(x1)))) -> 1(3(3(5(5(0(x1))))))
, 5(2(5(0(x1)))) -> 5(2(3(5(0(x1)))))
, 5(3(1(0(x1)))) -> 0(1(3(5(5(x1)))))
, 0(0(4(1(0(x1))))) -> 0(4(1(3(0(0(x1))))))
, 0(1(2(0(5(x1))))) -> 0(3(0(5(1(2(x1))))))
, 0(1(3(1(2(x1))))) -> 3(0(2(2(1(1(x1))))))
, 0(1(5(0(4(x1))))) -> 0(4(1(3(5(0(x1))))))
, 0(1(5(3(5(x1))))) -> 0(1(3(5(3(5(x1))))))
, 0(2(0(3(4(x1))))) -> 0(4(5(2(3(0(x1))))))
, 0(2(3(1(5(x1))))) -> 0(1(3(5(5(2(x1))))))
, 0(2(4(1(2(x1))))) -> 0(2(1(1(4(2(x1))))))
, 0(2(5(0(2(x1))))) -> 0(0(3(5(2(2(x1))))))
, 0(2(5(5(0(x1))))) -> 0(0(2(5(1(5(x1))))))
, 0(3(1(5(0(x1))))) -> 1(3(0(5(3(0(x1))))))
, 0(5(3(2(0(x1))))) -> 0(0(2(1(3(5(x1))))))
, 3(1(0(5(2(x1))))) -> 2(4(1(3(5(0(x1))))))
, 3(1(3(0(2(x1))))) -> 2(1(3(3(3(0(x1))))))
, 3(1(3(2(0(x1))))) -> 3(1(3(0(5(2(x1))))))
, 3(1(4(1(2(x1))))) -> 1(4(3(2(2(1(x1))))))
, 3(1(4(2(0(x1))))) -> 2(1(3(3(0(4(x1))))))
, 3(1(5(0(2(x1))))) -> 2(5(1(3(0(5(x1))))))
, 3(3(1(0(0(x1))))) -> 5(1(3(3(0(0(x1))))))
, 3(4(0(2(0(x1))))) -> 3(0(0(2(1(4(x1))))))
, 3(4(2(3(2(x1))))) -> 3(3(2(5(4(2(x1))))))
, 4(3(1(0(2(x1))))) -> 4(2(1(3(0(1(x1))))))
, 4(3(1(2(0(x1))))) -> 1(3(0(1(4(2(x1))))))
, 4(5(5(1(2(x1))))) -> 5(4(5(2(1(1(x1))))))
, 4(5(5(5(0(x1))))) -> 5(5(1(5(0(4(x1))))))
, 5(1(0(1(2(x1))))) -> 2(1(1(3(5(0(x1))))))
, 5(1(1(0(4(x1))))) -> 5(1(1(4(3(0(x1))))))
, 5(1(5(2(0(x1))))) -> 2(1(3(5(0(5(x1))))))
, 5(4(1(0(5(x1))))) -> 4(1(3(5(5(0(x1))))))}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 0_0(2) -> 1
, 1_0(2) -> 2
, 1_1(2) -> 6
, 2_0(2) -> 2
, 2_1(5) -> 4
, 2_1(6) -> 5
, 3_0(2) -> 1
, 3_1(3) -> 1
, 4_0(2) -> 1
, 5_0(2) -> 1
, 5_1(4) -> 3}Tool RC1
stdout:
MAYBE
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:
{ 0(0(1(2(x1)))) -> 0(2(1(3(3(0(x1))))))
, 0(1(0(4(x1)))) -> 0(1(3(0(4(x1)))))
, 0(1(0(5(x1)))) -> 0(1(3(5(0(x1)))))
, 0(1(5(0(x1)))) -> 0(1(3(5(0(x1)))))
, 0(3(1(0(x1)))) -> 0(1(3(5(0(x1)))))
, 3(1(0(2(x1)))) -> 1(3(5(0(2(x1)))))
, 3(1(0(2(x1)))) -> 3(2(1(3(0(x1)))))
, 3(1(0(5(x1)))) -> 1(3(5(3(0(x1)))))
, 3(1(2(0(x1)))) -> 2(1(3(3(0(x1)))))
, 3(1(2(0(x1)))) -> 3(0(2(1(4(x1)))))
, 3(1(2(0(x1)))) -> 3(0(3(2(1(x1)))))
, 3(1(2(0(x1)))) -> 3(3(2(1(0(x1)))))
, 3(1(2(0(x1)))) -> 3(5(2(1(0(x1)))))
, 3(1(2(2(x1)))) -> 3(5(2(2(1(x1)))))
, 3(4(2(0(x1)))) -> 3(3(2(4(0(x1)))))
, 4(0(1(0(x1)))) -> 1(3(0(4(0(x1)))))
, 5(1(0(5(x1)))) -> 1(3(3(5(5(0(x1))))))
, 5(2(5(0(x1)))) -> 5(2(3(5(0(x1)))))
, 5(3(1(0(x1)))) -> 0(1(3(5(5(x1)))))
, 0(0(4(1(0(x1))))) -> 0(4(1(3(0(0(x1))))))
, 0(1(2(0(5(x1))))) -> 0(3(0(5(1(2(x1))))))
, 0(1(3(1(2(x1))))) -> 3(0(2(2(1(1(x1))))))
, 0(1(5(0(4(x1))))) -> 0(4(1(3(5(0(x1))))))
, 0(1(5(3(5(x1))))) -> 0(1(3(5(3(5(x1))))))
, 0(2(0(3(4(x1))))) -> 0(4(5(2(3(0(x1))))))
, 0(2(3(1(5(x1))))) -> 0(1(3(5(5(2(x1))))))
, 0(2(4(1(2(x1))))) -> 0(2(1(1(4(2(x1))))))
, 0(2(5(0(2(x1))))) -> 0(0(3(5(2(2(x1))))))
, 0(2(5(5(0(x1))))) -> 0(0(2(5(1(5(x1))))))
, 0(3(1(5(0(x1))))) -> 1(3(0(5(3(0(x1))))))
, 0(5(3(2(0(x1))))) -> 0(0(2(1(3(5(x1))))))
, 3(1(0(5(2(x1))))) -> 2(4(1(3(5(0(x1))))))
, 3(1(3(0(2(x1))))) -> 2(1(3(3(3(0(x1))))))
, 3(1(3(2(0(x1))))) -> 3(1(3(0(5(2(x1))))))
, 3(1(4(1(2(x1))))) -> 1(4(3(2(2(1(x1))))))
, 3(1(4(2(0(x1))))) -> 2(1(3(3(0(4(x1))))))
, 3(1(5(0(2(x1))))) -> 2(5(1(3(0(5(x1))))))
, 3(3(1(0(0(x1))))) -> 5(1(3(3(0(0(x1))))))
, 3(4(0(2(0(x1))))) -> 3(0(0(2(1(4(x1))))))
, 3(4(2(3(2(x1))))) -> 3(3(2(5(4(2(x1))))))
, 4(3(1(0(2(x1))))) -> 4(2(1(3(0(1(x1))))))
, 4(3(1(2(0(x1))))) -> 1(3(0(1(4(2(x1))))))
, 4(5(5(1(2(x1))))) -> 5(4(5(2(1(1(x1))))))
, 4(5(5(5(0(x1))))) -> 5(5(1(5(0(4(x1))))))
, 5(1(0(1(2(x1))))) -> 2(1(1(3(5(0(x1))))))
, 5(1(1(0(4(x1))))) -> 5(1(1(4(3(0(x1))))))
, 5(1(5(2(0(x1))))) -> 2(1(3(5(0(5(x1))))))
, 5(4(1(0(5(x1))))) -> 4(1(3(5(5(0(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:
{ 0(0(1(2(x1)))) -> 0(2(1(3(3(0(x1))))))
, 0(1(0(4(x1)))) -> 0(1(3(0(4(x1)))))
, 0(1(0(5(x1)))) -> 0(1(3(5(0(x1)))))
, 0(1(5(0(x1)))) -> 0(1(3(5(0(x1)))))
, 0(3(1(0(x1)))) -> 0(1(3(5(0(x1)))))
, 3(1(0(2(x1)))) -> 1(3(5(0(2(x1)))))
, 3(1(0(2(x1)))) -> 3(2(1(3(0(x1)))))
, 3(1(0(5(x1)))) -> 1(3(5(3(0(x1)))))
, 3(1(2(0(x1)))) -> 2(1(3(3(0(x1)))))
, 3(1(2(0(x1)))) -> 3(0(2(1(4(x1)))))
, 3(1(2(0(x1)))) -> 3(0(3(2(1(x1)))))
, 3(1(2(0(x1)))) -> 3(3(2(1(0(x1)))))
, 3(1(2(0(x1)))) -> 3(5(2(1(0(x1)))))
, 3(1(2(2(x1)))) -> 3(5(2(2(1(x1)))))
, 3(4(2(0(x1)))) -> 3(3(2(4(0(x1)))))
, 4(0(1(0(x1)))) -> 1(3(0(4(0(x1)))))
, 5(1(0(5(x1)))) -> 1(3(3(5(5(0(x1))))))
, 5(2(5(0(x1)))) -> 5(2(3(5(0(x1)))))
, 5(3(1(0(x1)))) -> 0(1(3(5(5(x1)))))
, 0(0(4(1(0(x1))))) -> 0(4(1(3(0(0(x1))))))
, 0(1(2(0(5(x1))))) -> 0(3(0(5(1(2(x1))))))
, 0(1(3(1(2(x1))))) -> 3(0(2(2(1(1(x1))))))
, 0(1(5(0(4(x1))))) -> 0(4(1(3(5(0(x1))))))
, 0(1(5(3(5(x1))))) -> 0(1(3(5(3(5(x1))))))
, 0(2(0(3(4(x1))))) -> 0(4(5(2(3(0(x1))))))
, 0(2(3(1(5(x1))))) -> 0(1(3(5(5(2(x1))))))
, 0(2(4(1(2(x1))))) -> 0(2(1(1(4(2(x1))))))
, 0(2(5(0(2(x1))))) -> 0(0(3(5(2(2(x1))))))
, 0(2(5(5(0(x1))))) -> 0(0(2(5(1(5(x1))))))
, 0(3(1(5(0(x1))))) -> 1(3(0(5(3(0(x1))))))
, 0(5(3(2(0(x1))))) -> 0(0(2(1(3(5(x1))))))
, 3(1(0(5(2(x1))))) -> 2(4(1(3(5(0(x1))))))
, 3(1(3(0(2(x1))))) -> 2(1(3(3(3(0(x1))))))
, 3(1(3(2(0(x1))))) -> 3(1(3(0(5(2(x1))))))
, 3(1(4(1(2(x1))))) -> 1(4(3(2(2(1(x1))))))
, 3(1(4(2(0(x1))))) -> 2(1(3(3(0(4(x1))))))
, 3(1(5(0(2(x1))))) -> 2(5(1(3(0(5(x1))))))
, 3(3(1(0(0(x1))))) -> 5(1(3(3(0(0(x1))))))
, 3(4(0(2(0(x1))))) -> 3(0(0(2(1(4(x1))))))
, 3(4(2(3(2(x1))))) -> 3(3(2(5(4(2(x1))))))
, 4(3(1(0(2(x1))))) -> 4(2(1(3(0(1(x1))))))
, 4(3(1(2(0(x1))))) -> 1(3(0(1(4(2(x1))))))
, 4(5(5(1(2(x1))))) -> 5(4(5(2(1(1(x1))))))
, 4(5(5(5(0(x1))))) -> 5(5(1(5(0(4(x1))))))
, 5(1(0(1(2(x1))))) -> 2(1(1(3(5(0(x1))))))
, 5(1(1(0(4(x1))))) -> 5(1(1(4(3(0(x1))))))
, 5(1(5(2(0(x1))))) -> 2(1(3(5(0(5(x1))))))
, 5(4(1(0(5(x1))))) -> 4(1(3(5(5(0(x1))))))}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 0_0(2) -> 1
, 1_0(2) -> 2
, 1_1(2) -> 6
, 2_0(2) -> 2
, 2_1(5) -> 4
, 2_1(6) -> 5
, 3_0(2) -> 1
, 3_1(3) -> 1
, 4_0(2) -> 1
, 5_0(2) -> 1
, 5_1(4) -> 3}