Tool CaT
stdout:
YES(?,O(n^1))
Problem:
0(1(0(x1))) -> 0(2(1(0(x1))))
0(1(1(x1))) -> 0(3(1(4(1(x1)))))
0(1(1(x1))) -> 0(5(1(3(2(5(1(x1)))))))
0(1(1(x1))) -> 3(5(5(1(0(2(1(x1)))))))
0(1(4(x1))) -> 0(5(1(4(5(x1)))))
4(0(1(x1))) -> 3(5(1(4(1(2(0(x1)))))))
0(0(1(1(x1)))) -> 2(0(5(1(4(1(0(x1)))))))
0(0(3(4(x1)))) -> 0(3(2(1(4(5(5(0(x1))))))))
0(1(1(0(x1)))) -> 2(1(0(3(1(0(x1))))))
0(1(1(4(x1)))) -> 3(1(4(1(0(3(5(x1)))))))
0(1(3(0(x1)))) -> 0(0(3(2(1(x1)))))
0(1(3(4(x1)))) -> 3(1(4(2(5(0(5(x1)))))))
0(1(4(0(x1)))) -> 0(5(0(4(1(3(1(x1)))))))
0(3(4(1(x1)))) -> 4(3(0(3(1(x1)))))
2(0(1(0(x1)))) -> 2(0(5(1(0(x1)))))
2(0(1(1(x1)))) -> 2(0(2(1(3(1(x1))))))
2(3(1(1(x1)))) -> 2(3(1(4(1(x1)))))
2(3(1(1(x1)))) -> 3(1(3(2(1(x1)))))
2(4(1(1(x1)))) -> 2(4(2(1(4(1(x1))))))
3(0(1(0(x1)))) -> 3(1(2(2(0(0(x1))))))
3(3(1(1(x1)))) -> 5(3(5(1(3(2(2(1(x1))))))))
3(4(0(1(x1)))) -> 0(3(1(4(1(x1)))))
3(4(1(1(x1)))) -> 3(4(1(3(1(x1)))))
3(4(5(4(x1)))) -> 5(1(4(1(4(5(3(x1)))))))
4(0(0(1(x1)))) -> 4(0(0(3(1(x1)))))
4(3(0(1(x1)))) -> 3(5(0(4(2(1(x1))))))
4(3(0(1(x1)))) -> 3(1(4(2(5(5(2(0(x1))))))))
4(3(1(1(x1)))) -> 4(3(1(3(1(x1)))))
0(0(0(1(1(x1))))) -> 0(0(0(3(5(1(1(x1)))))))
0(0(0(3(4(x1))))) -> 0(0(3(1(0(4(5(x1)))))))
0(0(1(3(0(x1))))) -> 0(3(1(0(5(0(5(x1)))))))
0(0(2(0(1(x1))))) -> 0(0(0(3(2(1(x1))))))
0(1(1(3(4(x1))))) -> 0(5(4(1(3(1(x1))))))
0(1(1(3(4(x1))))) -> 4(1(0(3(5(5(2(1(x1))))))))
0(1(2(5(4(x1))))) -> 3(2(1(0(4(5(x1))))))
0(5(5(0(1(x1))))) -> 0(5(5(5(1(0(2(x1)))))))
2(0(1(4(1(x1))))) -> 5(5(1(0(4(2(1(x1)))))))
2(2(3(1(1(x1))))) -> 3(2(2(1(4(1(x1))))))
2(3(4(1(1(x1))))) -> 2(1(3(5(4(1(x1))))))
2(4(4(0(1(x1))))) -> 4(5(2(1(4(5(5(0(x1))))))))
3(0(4(0(1(x1))))) -> 4(0(3(1(2(2(0(x1)))))))
3(2(3(1(1(x1))))) -> 3(5(1(3(2(1(x1))))))
3(3(0(1(0(x1))))) -> 3(2(5(2(1(3(0(0(x1))))))))
3(3(4(0(1(x1))))) -> 3(5(0(3(1(2(4(1(x1))))))))
3(4(2(0(1(x1))))) -> 0(3(1(4(2(1(x1))))))
3(4(3(0(1(x1))))) -> 0(3(1(3(5(4(x1))))))
4(0(1(1(4(x1))))) -> 4(0(1(3(1(4(5(5(x1))))))))
4(3(2(0(1(x1))))) -> 3(1(2(2(4(0(x1))))))
4(3(4(1(1(x1))))) -> 3(1(4(1(4(2(x1))))))
0(0(2(1(1(1(x1)))))) -> 5(1(0(4(1(0(2(1(x1))))))))
0(1(1(2(5(4(x1)))))) -> 0(2(3(1(4(5(1(x1)))))))
0(1(5(1(4(0(x1)))))) -> 0(5(1(4(2(1(0(5(x1))))))))
0(2(5(3(1(1(x1)))))) -> 3(0(2(1(3(5(1(x1)))))))
0(3(0(5(4(1(x1)))))) -> 3(2(1(0(4(5(0(x1)))))))
0(5(5(3(3(4(x1)))))) -> 3(5(5(0(4(5(5(3(x1))))))))
2(0(4(4(1(1(x1)))))) -> 4(0(0(2(1(4(5(1(x1))))))))
2(2(3(1(0(1(x1)))))) -> 0(2(5(2(1(3(1(x1)))))))
2(3(0(1(1(1(x1)))))) -> 3(1(4(1(0(2(5(1(x1))))))))
2(3(2(3(0(1(x1)))))) -> 2(3(5(1(3(1(2(0(x1))))))))
2(3(4(1(0(1(x1)))))) -> 3(3(2(1(4(5(1(0(x1))))))))
3(0(1(4(1(1(x1)))))) -> 3(1(2(2(1(4(1(0(x1))))))))
3(0(5(2(5(4(x1)))))) -> 4(3(5(2(1(0(5(x1)))))))
3(3(0(2(5(0(x1)))))) -> 3(2(5(0(0(3(1(x1)))))))
3(3(1(4(0(0(x1)))))) -> 4(0(3(1(3(5(0(x1)))))))
3(4(1(2(5(0(x1)))))) -> 4(4(5(0(3(2(1(x1)))))))
3(4(2(2(0(1(x1)))))) -> 3(2(2(3(1(4(5(0(x1))))))))
3(4(5(3(0(1(x1)))))) -> 5(1(0(3(2(5(3(4(x1))))))))
4(0(1(3(0(1(x1)))))) -> 0(3(1(4(1(0(3(x1)))))))
4(0(3(4(1(1(x1)))))) -> 0(1(4(3(1(4(5(4(x1))))))))
4(2(0(5(1(1(x1)))))) -> 3(5(0(2(1(4(2(1(x1))))))))
4(3(0(1(0(4(x1)))))) -> 2(0(4(5(1(3(0(4(x1))))))))
0(1(2(3(4(0(1(x1))))))) -> 2(0(4(5(1(3(1(0(x1))))))))
2(3(4(5(4(1(1(x1))))))) -> 2(1(3(4(3(5(4(1(x1))))))))
2(4(0(2(1(0(1(x1))))))) -> 0(2(2(1(4(5(0(1(x1))))))))
2(4(3(4(2(1(1(x1))))))) -> 2(5(1(2(4(4(3(1(x1))))))))
2(4(5(0(3(0(1(x1))))))) -> 2(0(0(4(3(2(5(1(x1))))))))
3(0(0(4(0(1(1(x1))))))) -> 3(2(1(0(0(0(4(1(x1))))))))
3(2(4(5(1(0(1(x1))))))) -> 1(3(5(2(1(4(1(0(x1))))))))
4(0(3(2(0(0(1(x1))))))) -> 0(0(5(1(4(2(3(0(x1))))))))
4(3(0(5(0(1(4(x1))))))) -> 5(0(4(1(0(3(1(4(x1))))))))
Proof:
Bounds Processor:
bound: 1
enrichment: match
automaton:
final states: {6,5,4,3}
transitions:
31(40) -> 41*
31(31) -> 32*
31(16) -> 17*
51(39) -> 40*
51(29) -> 30*
51(38) -> 39*
51(33) -> 34*
11(15) -> 16*
11(37) -> 38*
11(32) -> 33*
11(19) -> 20*
11(13) -> 14*
01(17) -> 18*
01(36) -> 37*
21(35) -> 36*
21(30) -> 31*
41(14) -> 15*
00(2) -> 3*
00(1) -> 3*
10(2) -> 1*
10(1) -> 1*
20(2) -> 5*
20(1) -> 5*
30(2) -> 6*
30(1) -> 6*
40(2) -> 4*
40(1) -> 4*
50(2) -> 2*
50(1) -> 2*
1 -> 19*
2 -> 13*
14 -> 35,29
18 -> 3*
20 -> 14*
34 -> 17*
41 -> 3*
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(1(0(x1))) -> 0(2(1(0(x1))))
, 0(1(1(x1))) -> 0(3(1(4(1(x1)))))
, 0(1(1(x1))) -> 0(5(1(3(2(5(1(x1)))))))
, 0(1(1(x1))) -> 3(5(5(1(0(2(1(x1)))))))
, 0(1(4(x1))) -> 0(5(1(4(5(x1)))))
, 4(0(1(x1))) -> 3(5(1(4(1(2(0(x1)))))))
, 0(0(1(1(x1)))) -> 2(0(5(1(4(1(0(x1)))))))
, 0(0(3(4(x1)))) -> 0(3(2(1(4(5(5(0(x1))))))))
, 0(1(1(0(x1)))) -> 2(1(0(3(1(0(x1))))))
, 0(1(1(4(x1)))) -> 3(1(4(1(0(3(5(x1)))))))
, 0(1(3(0(x1)))) -> 0(0(3(2(1(x1)))))
, 0(1(3(4(x1)))) -> 3(1(4(2(5(0(5(x1)))))))
, 0(1(4(0(x1)))) -> 0(5(0(4(1(3(1(x1)))))))
, 0(3(4(1(x1)))) -> 4(3(0(3(1(x1)))))
, 2(0(1(0(x1)))) -> 2(0(5(1(0(x1)))))
, 2(0(1(1(x1)))) -> 2(0(2(1(3(1(x1))))))
, 2(3(1(1(x1)))) -> 2(3(1(4(1(x1)))))
, 2(3(1(1(x1)))) -> 3(1(3(2(1(x1)))))
, 2(4(1(1(x1)))) -> 2(4(2(1(4(1(x1))))))
, 3(0(1(0(x1)))) -> 3(1(2(2(0(0(x1))))))
, 3(3(1(1(x1)))) -> 5(3(5(1(3(2(2(1(x1))))))))
, 3(4(0(1(x1)))) -> 0(3(1(4(1(x1)))))
, 3(4(1(1(x1)))) -> 3(4(1(3(1(x1)))))
, 3(4(5(4(x1)))) -> 5(1(4(1(4(5(3(x1)))))))
, 4(0(0(1(x1)))) -> 4(0(0(3(1(x1)))))
, 4(3(0(1(x1)))) -> 3(5(0(4(2(1(x1))))))
, 4(3(0(1(x1)))) -> 3(1(4(2(5(5(2(0(x1))))))))
, 4(3(1(1(x1)))) -> 4(3(1(3(1(x1)))))
, 0(0(0(1(1(x1))))) -> 0(0(0(3(5(1(1(x1)))))))
, 0(0(0(3(4(x1))))) -> 0(0(3(1(0(4(5(x1)))))))
, 0(0(1(3(0(x1))))) -> 0(3(1(0(5(0(5(x1)))))))
, 0(0(2(0(1(x1))))) -> 0(0(0(3(2(1(x1))))))
, 0(1(1(3(4(x1))))) -> 0(5(4(1(3(1(x1))))))
, 0(1(1(3(4(x1))))) -> 4(1(0(3(5(5(2(1(x1))))))))
, 0(1(2(5(4(x1))))) -> 3(2(1(0(4(5(x1))))))
, 0(5(5(0(1(x1))))) -> 0(5(5(5(1(0(2(x1)))))))
, 2(0(1(4(1(x1))))) -> 5(5(1(0(4(2(1(x1)))))))
, 2(2(3(1(1(x1))))) -> 3(2(2(1(4(1(x1))))))
, 2(3(4(1(1(x1))))) -> 2(1(3(5(4(1(x1))))))
, 2(4(4(0(1(x1))))) -> 4(5(2(1(4(5(5(0(x1))))))))
, 3(0(4(0(1(x1))))) -> 4(0(3(1(2(2(0(x1)))))))
, 3(2(3(1(1(x1))))) -> 3(5(1(3(2(1(x1))))))
, 3(3(0(1(0(x1))))) -> 3(2(5(2(1(3(0(0(x1))))))))
, 3(3(4(0(1(x1))))) -> 3(5(0(3(1(2(4(1(x1))))))))
, 3(4(2(0(1(x1))))) -> 0(3(1(4(2(1(x1))))))
, 3(4(3(0(1(x1))))) -> 0(3(1(3(5(4(x1))))))
, 4(0(1(1(4(x1))))) -> 4(0(1(3(1(4(5(5(x1))))))))
, 4(3(2(0(1(x1))))) -> 3(1(2(2(4(0(x1))))))
, 4(3(4(1(1(x1))))) -> 3(1(4(1(4(2(x1))))))
, 0(0(2(1(1(1(x1)))))) -> 5(1(0(4(1(0(2(1(x1))))))))
, 0(1(1(2(5(4(x1)))))) -> 0(2(3(1(4(5(1(x1)))))))
, 0(1(5(1(4(0(x1)))))) -> 0(5(1(4(2(1(0(5(x1))))))))
, 0(2(5(3(1(1(x1)))))) -> 3(0(2(1(3(5(1(x1)))))))
, 0(3(0(5(4(1(x1)))))) -> 3(2(1(0(4(5(0(x1)))))))
, 0(5(5(3(3(4(x1)))))) -> 3(5(5(0(4(5(5(3(x1))))))))
, 2(0(4(4(1(1(x1)))))) -> 4(0(0(2(1(4(5(1(x1))))))))
, 2(2(3(1(0(1(x1)))))) -> 0(2(5(2(1(3(1(x1)))))))
, 2(3(0(1(1(1(x1)))))) -> 3(1(4(1(0(2(5(1(x1))))))))
, 2(3(2(3(0(1(x1)))))) -> 2(3(5(1(3(1(2(0(x1))))))))
, 2(3(4(1(0(1(x1)))))) -> 3(3(2(1(4(5(1(0(x1))))))))
, 3(0(1(4(1(1(x1)))))) -> 3(1(2(2(1(4(1(0(x1))))))))
, 3(0(5(2(5(4(x1)))))) -> 4(3(5(2(1(0(5(x1)))))))
, 3(3(0(2(5(0(x1)))))) -> 3(2(5(0(0(3(1(x1)))))))
, 3(3(1(4(0(0(x1)))))) -> 4(0(3(1(3(5(0(x1)))))))
, 3(4(1(2(5(0(x1)))))) -> 4(4(5(0(3(2(1(x1)))))))
, 3(4(2(2(0(1(x1)))))) -> 3(2(2(3(1(4(5(0(x1))))))))
, 3(4(5(3(0(1(x1)))))) -> 5(1(0(3(2(5(3(4(x1))))))))
, 4(0(1(3(0(1(x1)))))) -> 0(3(1(4(1(0(3(x1)))))))
, 4(0(3(4(1(1(x1)))))) -> 0(1(4(3(1(4(5(4(x1))))))))
, 4(2(0(5(1(1(x1)))))) -> 3(5(0(2(1(4(2(1(x1))))))))
, 4(3(0(1(0(4(x1)))))) -> 2(0(4(5(1(3(0(4(x1))))))))
, 0(1(2(3(4(0(1(x1))))))) -> 2(0(4(5(1(3(1(0(x1))))))))
, 2(3(4(5(4(1(1(x1))))))) -> 2(1(3(4(3(5(4(1(x1))))))))
, 2(4(0(2(1(0(1(x1))))))) -> 0(2(2(1(4(5(0(1(x1))))))))
, 2(4(3(4(2(1(1(x1))))))) -> 2(5(1(2(4(4(3(1(x1))))))))
, 2(4(5(0(3(0(1(x1))))))) -> 2(0(0(4(3(2(5(1(x1))))))))
, 3(0(0(4(0(1(1(x1))))))) -> 3(2(1(0(0(0(4(1(x1))))))))
, 3(2(4(5(1(0(1(x1))))))) -> 1(3(5(2(1(4(1(0(x1))))))))
, 4(0(3(2(0(0(1(x1))))))) -> 0(0(5(1(4(2(3(0(x1))))))))
, 4(3(0(5(0(1(4(x1))))))) -> 5(0(4(1(0(3(1(4(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(1(0(x1))) -> 0(2(1(0(x1))))
, 0(1(1(x1))) -> 0(3(1(4(1(x1)))))
, 0(1(1(x1))) -> 0(5(1(3(2(5(1(x1)))))))
, 0(1(1(x1))) -> 3(5(5(1(0(2(1(x1)))))))
, 0(1(4(x1))) -> 0(5(1(4(5(x1)))))
, 4(0(1(x1))) -> 3(5(1(4(1(2(0(x1)))))))
, 0(0(1(1(x1)))) -> 2(0(5(1(4(1(0(x1)))))))
, 0(0(3(4(x1)))) -> 0(3(2(1(4(5(5(0(x1))))))))
, 0(1(1(0(x1)))) -> 2(1(0(3(1(0(x1))))))
, 0(1(1(4(x1)))) -> 3(1(4(1(0(3(5(x1)))))))
, 0(1(3(0(x1)))) -> 0(0(3(2(1(x1)))))
, 0(1(3(4(x1)))) -> 3(1(4(2(5(0(5(x1)))))))
, 0(1(4(0(x1)))) -> 0(5(0(4(1(3(1(x1)))))))
, 0(3(4(1(x1)))) -> 4(3(0(3(1(x1)))))
, 2(0(1(0(x1)))) -> 2(0(5(1(0(x1)))))
, 2(0(1(1(x1)))) -> 2(0(2(1(3(1(x1))))))
, 2(3(1(1(x1)))) -> 2(3(1(4(1(x1)))))
, 2(3(1(1(x1)))) -> 3(1(3(2(1(x1)))))
, 2(4(1(1(x1)))) -> 2(4(2(1(4(1(x1))))))
, 3(0(1(0(x1)))) -> 3(1(2(2(0(0(x1))))))
, 3(3(1(1(x1)))) -> 5(3(5(1(3(2(2(1(x1))))))))
, 3(4(0(1(x1)))) -> 0(3(1(4(1(x1)))))
, 3(4(1(1(x1)))) -> 3(4(1(3(1(x1)))))
, 3(4(5(4(x1)))) -> 5(1(4(1(4(5(3(x1)))))))
, 4(0(0(1(x1)))) -> 4(0(0(3(1(x1)))))
, 4(3(0(1(x1)))) -> 3(5(0(4(2(1(x1))))))
, 4(3(0(1(x1)))) -> 3(1(4(2(5(5(2(0(x1))))))))
, 4(3(1(1(x1)))) -> 4(3(1(3(1(x1)))))
, 0(0(0(1(1(x1))))) -> 0(0(0(3(5(1(1(x1)))))))
, 0(0(0(3(4(x1))))) -> 0(0(3(1(0(4(5(x1)))))))
, 0(0(1(3(0(x1))))) -> 0(3(1(0(5(0(5(x1)))))))
, 0(0(2(0(1(x1))))) -> 0(0(0(3(2(1(x1))))))
, 0(1(1(3(4(x1))))) -> 0(5(4(1(3(1(x1))))))
, 0(1(1(3(4(x1))))) -> 4(1(0(3(5(5(2(1(x1))))))))
, 0(1(2(5(4(x1))))) -> 3(2(1(0(4(5(x1))))))
, 0(5(5(0(1(x1))))) -> 0(5(5(5(1(0(2(x1)))))))
, 2(0(1(4(1(x1))))) -> 5(5(1(0(4(2(1(x1)))))))
, 2(2(3(1(1(x1))))) -> 3(2(2(1(4(1(x1))))))
, 2(3(4(1(1(x1))))) -> 2(1(3(5(4(1(x1))))))
, 2(4(4(0(1(x1))))) -> 4(5(2(1(4(5(5(0(x1))))))))
, 3(0(4(0(1(x1))))) -> 4(0(3(1(2(2(0(x1)))))))
, 3(2(3(1(1(x1))))) -> 3(5(1(3(2(1(x1))))))
, 3(3(0(1(0(x1))))) -> 3(2(5(2(1(3(0(0(x1))))))))
, 3(3(4(0(1(x1))))) -> 3(5(0(3(1(2(4(1(x1))))))))
, 3(4(2(0(1(x1))))) -> 0(3(1(4(2(1(x1))))))
, 3(4(3(0(1(x1))))) -> 0(3(1(3(5(4(x1))))))
, 4(0(1(1(4(x1))))) -> 4(0(1(3(1(4(5(5(x1))))))))
, 4(3(2(0(1(x1))))) -> 3(1(2(2(4(0(x1))))))
, 4(3(4(1(1(x1))))) -> 3(1(4(1(4(2(x1))))))
, 0(0(2(1(1(1(x1)))))) -> 5(1(0(4(1(0(2(1(x1))))))))
, 0(1(1(2(5(4(x1)))))) -> 0(2(3(1(4(5(1(x1)))))))
, 0(1(5(1(4(0(x1)))))) -> 0(5(1(4(2(1(0(5(x1))))))))
, 0(2(5(3(1(1(x1)))))) -> 3(0(2(1(3(5(1(x1)))))))
, 0(3(0(5(4(1(x1)))))) -> 3(2(1(0(4(5(0(x1)))))))
, 0(5(5(3(3(4(x1)))))) -> 3(5(5(0(4(5(5(3(x1))))))))
, 2(0(4(4(1(1(x1)))))) -> 4(0(0(2(1(4(5(1(x1))))))))
, 2(2(3(1(0(1(x1)))))) -> 0(2(5(2(1(3(1(x1)))))))
, 2(3(0(1(1(1(x1)))))) -> 3(1(4(1(0(2(5(1(x1))))))))
, 2(3(2(3(0(1(x1)))))) -> 2(3(5(1(3(1(2(0(x1))))))))
, 2(3(4(1(0(1(x1)))))) -> 3(3(2(1(4(5(1(0(x1))))))))
, 3(0(1(4(1(1(x1)))))) -> 3(1(2(2(1(4(1(0(x1))))))))
, 3(0(5(2(5(4(x1)))))) -> 4(3(5(2(1(0(5(x1)))))))
, 3(3(0(2(5(0(x1)))))) -> 3(2(5(0(0(3(1(x1)))))))
, 3(3(1(4(0(0(x1)))))) -> 4(0(3(1(3(5(0(x1)))))))
, 3(4(1(2(5(0(x1)))))) -> 4(4(5(0(3(2(1(x1)))))))
, 3(4(2(2(0(1(x1)))))) -> 3(2(2(3(1(4(5(0(x1))))))))
, 3(4(5(3(0(1(x1)))))) -> 5(1(0(3(2(5(3(4(x1))))))))
, 4(0(1(3(0(1(x1)))))) -> 0(3(1(4(1(0(3(x1)))))))
, 4(0(3(4(1(1(x1)))))) -> 0(1(4(3(1(4(5(4(x1))))))))
, 4(2(0(5(1(1(x1)))))) -> 3(5(0(2(1(4(2(1(x1))))))))
, 4(3(0(1(0(4(x1)))))) -> 2(0(4(5(1(3(0(4(x1))))))))
, 0(1(2(3(4(0(1(x1))))))) -> 2(0(4(5(1(3(1(0(x1))))))))
, 2(3(4(5(4(1(1(x1))))))) -> 2(1(3(4(3(5(4(1(x1))))))))
, 2(4(0(2(1(0(1(x1))))))) -> 0(2(2(1(4(5(0(1(x1))))))))
, 2(4(3(4(2(1(1(x1))))))) -> 2(5(1(2(4(4(3(1(x1))))))))
, 2(4(5(0(3(0(1(x1))))))) -> 2(0(0(4(3(2(5(1(x1))))))))
, 3(0(0(4(0(1(1(x1))))))) -> 3(2(1(0(0(0(4(1(x1))))))))
, 3(2(4(5(1(0(1(x1))))))) -> 1(3(5(2(1(4(1(0(x1))))))))
, 4(0(3(2(0(0(1(x1))))))) -> 0(0(5(1(4(2(3(0(x1))))))))
, 4(3(0(5(0(1(4(x1))))))) -> 5(0(4(1(0(3(1(4(x1))))))))}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 0_0(2) -> 1
, 0_1(3) -> 1
, 0_1(15) -> 14
, 1_0(2) -> 2
, 1_1(2) -> 6
, 1_1(5) -> 4
, 1_1(8) -> 7
, 1_1(14) -> 13
, 2_0(2) -> 1
, 2_1(6) -> 15
, 2_1(10) -> 9
, 3_0(2) -> 1
, 3_1(4) -> 3
, 3_1(9) -> 8
, 3_1(11) -> 1
, 4_0(2) -> 1
, 4_1(6) -> 5
, 5_0(2) -> 2
, 5_1(6) -> 10
, 5_1(7) -> 3
, 5_1(12) -> 11
, 5_1(13) -> 12}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(1(0(x1))) -> 0(2(1(0(x1))))
, 0(1(1(x1))) -> 0(3(1(4(1(x1)))))
, 0(1(1(x1))) -> 0(5(1(3(2(5(1(x1)))))))
, 0(1(1(x1))) -> 3(5(5(1(0(2(1(x1)))))))
, 0(1(4(x1))) -> 0(5(1(4(5(x1)))))
, 4(0(1(x1))) -> 3(5(1(4(1(2(0(x1)))))))
, 0(0(1(1(x1)))) -> 2(0(5(1(4(1(0(x1)))))))
, 0(0(3(4(x1)))) -> 0(3(2(1(4(5(5(0(x1))))))))
, 0(1(1(0(x1)))) -> 2(1(0(3(1(0(x1))))))
, 0(1(1(4(x1)))) -> 3(1(4(1(0(3(5(x1)))))))
, 0(1(3(0(x1)))) -> 0(0(3(2(1(x1)))))
, 0(1(3(4(x1)))) -> 3(1(4(2(5(0(5(x1)))))))
, 0(1(4(0(x1)))) -> 0(5(0(4(1(3(1(x1)))))))
, 0(3(4(1(x1)))) -> 4(3(0(3(1(x1)))))
, 2(0(1(0(x1)))) -> 2(0(5(1(0(x1)))))
, 2(0(1(1(x1)))) -> 2(0(2(1(3(1(x1))))))
, 2(3(1(1(x1)))) -> 2(3(1(4(1(x1)))))
, 2(3(1(1(x1)))) -> 3(1(3(2(1(x1)))))
, 2(4(1(1(x1)))) -> 2(4(2(1(4(1(x1))))))
, 3(0(1(0(x1)))) -> 3(1(2(2(0(0(x1))))))
, 3(3(1(1(x1)))) -> 5(3(5(1(3(2(2(1(x1))))))))
, 3(4(0(1(x1)))) -> 0(3(1(4(1(x1)))))
, 3(4(1(1(x1)))) -> 3(4(1(3(1(x1)))))
, 3(4(5(4(x1)))) -> 5(1(4(1(4(5(3(x1)))))))
, 4(0(0(1(x1)))) -> 4(0(0(3(1(x1)))))
, 4(3(0(1(x1)))) -> 3(5(0(4(2(1(x1))))))
, 4(3(0(1(x1)))) -> 3(1(4(2(5(5(2(0(x1))))))))
, 4(3(1(1(x1)))) -> 4(3(1(3(1(x1)))))
, 0(0(0(1(1(x1))))) -> 0(0(0(3(5(1(1(x1)))))))
, 0(0(0(3(4(x1))))) -> 0(0(3(1(0(4(5(x1)))))))
, 0(0(1(3(0(x1))))) -> 0(3(1(0(5(0(5(x1)))))))
, 0(0(2(0(1(x1))))) -> 0(0(0(3(2(1(x1))))))
, 0(1(1(3(4(x1))))) -> 0(5(4(1(3(1(x1))))))
, 0(1(1(3(4(x1))))) -> 4(1(0(3(5(5(2(1(x1))))))))
, 0(1(2(5(4(x1))))) -> 3(2(1(0(4(5(x1))))))
, 0(5(5(0(1(x1))))) -> 0(5(5(5(1(0(2(x1)))))))
, 2(0(1(4(1(x1))))) -> 5(5(1(0(4(2(1(x1)))))))
, 2(2(3(1(1(x1))))) -> 3(2(2(1(4(1(x1))))))
, 2(3(4(1(1(x1))))) -> 2(1(3(5(4(1(x1))))))
, 2(4(4(0(1(x1))))) -> 4(5(2(1(4(5(5(0(x1))))))))
, 3(0(4(0(1(x1))))) -> 4(0(3(1(2(2(0(x1)))))))
, 3(2(3(1(1(x1))))) -> 3(5(1(3(2(1(x1))))))
, 3(3(0(1(0(x1))))) -> 3(2(5(2(1(3(0(0(x1))))))))
, 3(3(4(0(1(x1))))) -> 3(5(0(3(1(2(4(1(x1))))))))
, 3(4(2(0(1(x1))))) -> 0(3(1(4(2(1(x1))))))
, 3(4(3(0(1(x1))))) -> 0(3(1(3(5(4(x1))))))
, 4(0(1(1(4(x1))))) -> 4(0(1(3(1(4(5(5(x1))))))))
, 4(3(2(0(1(x1))))) -> 3(1(2(2(4(0(x1))))))
, 4(3(4(1(1(x1))))) -> 3(1(4(1(4(2(x1))))))
, 0(0(2(1(1(1(x1)))))) -> 5(1(0(4(1(0(2(1(x1))))))))
, 0(1(1(2(5(4(x1)))))) -> 0(2(3(1(4(5(1(x1)))))))
, 0(1(5(1(4(0(x1)))))) -> 0(5(1(4(2(1(0(5(x1))))))))
, 0(2(5(3(1(1(x1)))))) -> 3(0(2(1(3(5(1(x1)))))))
, 0(3(0(5(4(1(x1)))))) -> 3(2(1(0(4(5(0(x1)))))))
, 0(5(5(3(3(4(x1)))))) -> 3(5(5(0(4(5(5(3(x1))))))))
, 2(0(4(4(1(1(x1)))))) -> 4(0(0(2(1(4(5(1(x1))))))))
, 2(2(3(1(0(1(x1)))))) -> 0(2(5(2(1(3(1(x1)))))))
, 2(3(0(1(1(1(x1)))))) -> 3(1(4(1(0(2(5(1(x1))))))))
, 2(3(2(3(0(1(x1)))))) -> 2(3(5(1(3(1(2(0(x1))))))))
, 2(3(4(1(0(1(x1)))))) -> 3(3(2(1(4(5(1(0(x1))))))))
, 3(0(1(4(1(1(x1)))))) -> 3(1(2(2(1(4(1(0(x1))))))))
, 3(0(5(2(5(4(x1)))))) -> 4(3(5(2(1(0(5(x1)))))))
, 3(3(0(2(5(0(x1)))))) -> 3(2(5(0(0(3(1(x1)))))))
, 3(3(1(4(0(0(x1)))))) -> 4(0(3(1(3(5(0(x1)))))))
, 3(4(1(2(5(0(x1)))))) -> 4(4(5(0(3(2(1(x1)))))))
, 3(4(2(2(0(1(x1)))))) -> 3(2(2(3(1(4(5(0(x1))))))))
, 3(4(5(3(0(1(x1)))))) -> 5(1(0(3(2(5(3(4(x1))))))))
, 4(0(1(3(0(1(x1)))))) -> 0(3(1(4(1(0(3(x1)))))))
, 4(0(3(4(1(1(x1)))))) -> 0(1(4(3(1(4(5(4(x1))))))))
, 4(2(0(5(1(1(x1)))))) -> 3(5(0(2(1(4(2(1(x1))))))))
, 4(3(0(1(0(4(x1)))))) -> 2(0(4(5(1(3(0(4(x1))))))))
, 0(1(2(3(4(0(1(x1))))))) -> 2(0(4(5(1(3(1(0(x1))))))))
, 2(3(4(5(4(1(1(x1))))))) -> 2(1(3(4(3(5(4(1(x1))))))))
, 2(4(0(2(1(0(1(x1))))))) -> 0(2(2(1(4(5(0(1(x1))))))))
, 2(4(3(4(2(1(1(x1))))))) -> 2(5(1(2(4(4(3(1(x1))))))))
, 2(4(5(0(3(0(1(x1))))))) -> 2(0(0(4(3(2(5(1(x1))))))))
, 3(0(0(4(0(1(1(x1))))))) -> 3(2(1(0(0(0(4(1(x1))))))))
, 3(2(4(5(1(0(1(x1))))))) -> 1(3(5(2(1(4(1(0(x1))))))))
, 4(0(3(2(0(0(1(x1))))))) -> 0(0(5(1(4(2(3(0(x1))))))))
, 4(3(0(5(0(1(4(x1))))))) -> 5(0(4(1(0(3(1(4(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(1(0(x1))) -> 0(2(1(0(x1))))
, 0(1(1(x1))) -> 0(3(1(4(1(x1)))))
, 0(1(1(x1))) -> 0(5(1(3(2(5(1(x1)))))))
, 0(1(1(x1))) -> 3(5(5(1(0(2(1(x1)))))))
, 0(1(4(x1))) -> 0(5(1(4(5(x1)))))
, 4(0(1(x1))) -> 3(5(1(4(1(2(0(x1)))))))
, 0(0(1(1(x1)))) -> 2(0(5(1(4(1(0(x1)))))))
, 0(0(3(4(x1)))) -> 0(3(2(1(4(5(5(0(x1))))))))
, 0(1(1(0(x1)))) -> 2(1(0(3(1(0(x1))))))
, 0(1(1(4(x1)))) -> 3(1(4(1(0(3(5(x1)))))))
, 0(1(3(0(x1)))) -> 0(0(3(2(1(x1)))))
, 0(1(3(4(x1)))) -> 3(1(4(2(5(0(5(x1)))))))
, 0(1(4(0(x1)))) -> 0(5(0(4(1(3(1(x1)))))))
, 0(3(4(1(x1)))) -> 4(3(0(3(1(x1)))))
, 2(0(1(0(x1)))) -> 2(0(5(1(0(x1)))))
, 2(0(1(1(x1)))) -> 2(0(2(1(3(1(x1))))))
, 2(3(1(1(x1)))) -> 2(3(1(4(1(x1)))))
, 2(3(1(1(x1)))) -> 3(1(3(2(1(x1)))))
, 2(4(1(1(x1)))) -> 2(4(2(1(4(1(x1))))))
, 3(0(1(0(x1)))) -> 3(1(2(2(0(0(x1))))))
, 3(3(1(1(x1)))) -> 5(3(5(1(3(2(2(1(x1))))))))
, 3(4(0(1(x1)))) -> 0(3(1(4(1(x1)))))
, 3(4(1(1(x1)))) -> 3(4(1(3(1(x1)))))
, 3(4(5(4(x1)))) -> 5(1(4(1(4(5(3(x1)))))))
, 4(0(0(1(x1)))) -> 4(0(0(3(1(x1)))))
, 4(3(0(1(x1)))) -> 3(5(0(4(2(1(x1))))))
, 4(3(0(1(x1)))) -> 3(1(4(2(5(5(2(0(x1))))))))
, 4(3(1(1(x1)))) -> 4(3(1(3(1(x1)))))
, 0(0(0(1(1(x1))))) -> 0(0(0(3(5(1(1(x1)))))))
, 0(0(0(3(4(x1))))) -> 0(0(3(1(0(4(5(x1)))))))
, 0(0(1(3(0(x1))))) -> 0(3(1(0(5(0(5(x1)))))))
, 0(0(2(0(1(x1))))) -> 0(0(0(3(2(1(x1))))))
, 0(1(1(3(4(x1))))) -> 0(5(4(1(3(1(x1))))))
, 0(1(1(3(4(x1))))) -> 4(1(0(3(5(5(2(1(x1))))))))
, 0(1(2(5(4(x1))))) -> 3(2(1(0(4(5(x1))))))
, 0(5(5(0(1(x1))))) -> 0(5(5(5(1(0(2(x1)))))))
, 2(0(1(4(1(x1))))) -> 5(5(1(0(4(2(1(x1)))))))
, 2(2(3(1(1(x1))))) -> 3(2(2(1(4(1(x1))))))
, 2(3(4(1(1(x1))))) -> 2(1(3(5(4(1(x1))))))
, 2(4(4(0(1(x1))))) -> 4(5(2(1(4(5(5(0(x1))))))))
, 3(0(4(0(1(x1))))) -> 4(0(3(1(2(2(0(x1)))))))
, 3(2(3(1(1(x1))))) -> 3(5(1(3(2(1(x1))))))
, 3(3(0(1(0(x1))))) -> 3(2(5(2(1(3(0(0(x1))))))))
, 3(3(4(0(1(x1))))) -> 3(5(0(3(1(2(4(1(x1))))))))
, 3(4(2(0(1(x1))))) -> 0(3(1(4(2(1(x1))))))
, 3(4(3(0(1(x1))))) -> 0(3(1(3(5(4(x1))))))
, 4(0(1(1(4(x1))))) -> 4(0(1(3(1(4(5(5(x1))))))))
, 4(3(2(0(1(x1))))) -> 3(1(2(2(4(0(x1))))))
, 4(3(4(1(1(x1))))) -> 3(1(4(1(4(2(x1))))))
, 0(0(2(1(1(1(x1)))))) -> 5(1(0(4(1(0(2(1(x1))))))))
, 0(1(1(2(5(4(x1)))))) -> 0(2(3(1(4(5(1(x1)))))))
, 0(1(5(1(4(0(x1)))))) -> 0(5(1(4(2(1(0(5(x1))))))))
, 0(2(5(3(1(1(x1)))))) -> 3(0(2(1(3(5(1(x1)))))))
, 0(3(0(5(4(1(x1)))))) -> 3(2(1(0(4(5(0(x1)))))))
, 0(5(5(3(3(4(x1)))))) -> 3(5(5(0(4(5(5(3(x1))))))))
, 2(0(4(4(1(1(x1)))))) -> 4(0(0(2(1(4(5(1(x1))))))))
, 2(2(3(1(0(1(x1)))))) -> 0(2(5(2(1(3(1(x1)))))))
, 2(3(0(1(1(1(x1)))))) -> 3(1(4(1(0(2(5(1(x1))))))))
, 2(3(2(3(0(1(x1)))))) -> 2(3(5(1(3(1(2(0(x1))))))))
, 2(3(4(1(0(1(x1)))))) -> 3(3(2(1(4(5(1(0(x1))))))))
, 3(0(1(4(1(1(x1)))))) -> 3(1(2(2(1(4(1(0(x1))))))))
, 3(0(5(2(5(4(x1)))))) -> 4(3(5(2(1(0(5(x1)))))))
, 3(3(0(2(5(0(x1)))))) -> 3(2(5(0(0(3(1(x1)))))))
, 3(3(1(4(0(0(x1)))))) -> 4(0(3(1(3(5(0(x1)))))))
, 3(4(1(2(5(0(x1)))))) -> 4(4(5(0(3(2(1(x1)))))))
, 3(4(2(2(0(1(x1)))))) -> 3(2(2(3(1(4(5(0(x1))))))))
, 3(4(5(3(0(1(x1)))))) -> 5(1(0(3(2(5(3(4(x1))))))))
, 4(0(1(3(0(1(x1)))))) -> 0(3(1(4(1(0(3(x1)))))))
, 4(0(3(4(1(1(x1)))))) -> 0(1(4(3(1(4(5(4(x1))))))))
, 4(2(0(5(1(1(x1)))))) -> 3(5(0(2(1(4(2(1(x1))))))))
, 4(3(0(1(0(4(x1)))))) -> 2(0(4(5(1(3(0(4(x1))))))))
, 0(1(2(3(4(0(1(x1))))))) -> 2(0(4(5(1(3(1(0(x1))))))))
, 2(3(4(5(4(1(1(x1))))))) -> 2(1(3(4(3(5(4(1(x1))))))))
, 2(4(0(2(1(0(1(x1))))))) -> 0(2(2(1(4(5(0(1(x1))))))))
, 2(4(3(4(2(1(1(x1))))))) -> 2(5(1(2(4(4(3(1(x1))))))))
, 2(4(5(0(3(0(1(x1))))))) -> 2(0(0(4(3(2(5(1(x1))))))))
, 3(0(0(4(0(1(1(x1))))))) -> 3(2(1(0(0(0(4(1(x1))))))))
, 3(2(4(5(1(0(1(x1))))))) -> 1(3(5(2(1(4(1(0(x1))))))))
, 4(0(3(2(0(0(1(x1))))))) -> 0(0(5(1(4(2(3(0(x1))))))))
, 4(3(0(5(0(1(4(x1))))))) -> 5(0(4(1(0(3(1(4(x1))))))))}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ 0_0(2) -> 1
, 0_1(3) -> 1
, 0_1(15) -> 14
, 1_0(2) -> 2
, 1_1(2) -> 6
, 1_1(5) -> 4
, 1_1(8) -> 7
, 1_1(14) -> 13
, 2_0(2) -> 1
, 2_1(6) -> 15
, 2_1(10) -> 9
, 3_0(2) -> 1
, 3_1(4) -> 3
, 3_1(9) -> 8
, 3_1(11) -> 1
, 4_0(2) -> 1
, 4_1(6) -> 5
, 5_0(2) -> 2
, 5_1(6) -> 10
, 5_1(7) -> 3
, 5_1(12) -> 11
, 5_1(13) -> 12}