Tool CaT
stdout:
YES(?,O(n^1))
Problem:
a(d(x1)) -> d(b(x1))
a(x1) -> b(b(b(x1)))
d(x1) -> x1
a(x1) -> x1
b(d(b(x1))) -> a(d(x1))
b(c(x1)) -> c(d(d(x1)))
a(c(x1)) -> b(b(c(d(x1))))
Proof:
Bounds Processor:
bound: 2
enrichment: match
automaton:
final states: {4,3,2}
transitions:
b1(5) -> 6*
b1(7) -> 8*
b1(6) -> 7*
c1(19) -> 20*
c1(21) -> 22*
d1(17) -> 18*
d1(18) -> 19*
c2(27) -> 28*
c2(39) -> 40*
d2(25) -> 26*
d2(37) -> 38*
d2(26) -> 27*
d2(43) -> 44*
d2(38) -> 39*
a0(1) -> 2*
d0(1) -> 3*
b0(1) -> 4*
c0(1) -> 1*
1 -> 17,2,3,5
8 -> 2*
17 -> 18,21
18 -> 19,21
19 -> 43*
20 -> 6,4
21 -> 25*
22 -> 6*
25 -> 26*
26 -> 27*
27 -> 37*
28 -> 7*
37 -> 38*
38 -> 39*
40 -> 8*
43 -> 44*
44 -> 25*
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(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, d(x1) -> x1
, a(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(c(x1)) -> b(b(c(d(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(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, d(x1) -> x1
, a(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(c(x1)) -> b(b(c(d(x1))))}
Proof Output:
The problem is match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_0(2) -> 1
, d_0(2) -> 1
, d_1(2) -> 5
, d_1(2) -> 6
, d_1(2) -> 7
, d_1(2) -> 8
, d_1(2) -> 9
, d_1(2) -> 10
, d_1(6) -> 5
, d_1(6) -> 7
, d_1(6) -> 8
, d_1(6) -> 9
, d_1(6) -> 10
, d_2(5) -> 7
, d_2(5) -> 8
, d_2(5) -> 9
, d_2(5) -> 10
, d_2(6) -> 7
, d_2(6) -> 8
, d_2(6) -> 9
, d_2(6) -> 10
, d_2(7) -> 9
, d_2(7) -> 10
, d_2(8) -> 7
, d_2(8) -> 9
, d_2(8) -> 10
, d_2(10) -> 9
, b_0(2) -> 1
, b_1(2) -> 4
, b_1(3) -> 1
, b_1(4) -> 3
, c_0(2) -> 1
, c_0(2) -> 2
, c_0(2) -> 5
, c_0(2) -> 6
, c_0(2) -> 7
, c_0(2) -> 8
, c_0(2) -> 9
, c_0(2) -> 10
, c_1(5) -> 1
, c_1(5) -> 4
, c_1(6) -> 4
, c_2(7) -> 3
, c_2(9) -> 1}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(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, d(x1) -> x1
, a(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(c(x1)) -> b(b(c(d(x1))))}
Proof Output:
'Bounds with perSymbol-enrichment and initial automaton 'match'' proved the best result:
Details:
--------
'Bounds with perSymbol-enrichment and initial automaton 'match'' succeeded with the following output:
'Bounds with perSymbol-enrichment and initial automaton 'match''
----------------------------------------------------------------
Answer: YES(?,O(n^1))
Input Problem: runtime-complexity with respect to
Rules:
{ a(d(x1)) -> d(b(x1))
, a(x1) -> b(b(b(x1)))
, d(x1) -> x1
, a(x1) -> x1
, b(d(b(x1))) -> a(d(x1))
, b(c(x1)) -> c(d(d(x1)))
, a(c(x1)) -> b(b(c(d(x1))))}
Proof Output:
The problem is match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ a_0(4) -> 1
, d_0(4) -> 2
, d_1(4) -> 7
, d_1(4) -> 8
, d_1(4) -> 9
, d_1(4) -> 10
, d_1(4) -> 11
, d_1(4) -> 12
, d_1(8) -> 7
, d_1(8) -> 9
, d_1(8) -> 10
, d_1(8) -> 11
, d_1(8) -> 12
, d_2(7) -> 9
, d_2(7) -> 10
, d_2(7) -> 11
, d_2(7) -> 12
, d_2(8) -> 9
, d_2(8) -> 10
, d_2(8) -> 11
, d_2(8) -> 12
, d_2(9) -> 11
, d_2(9) -> 12
, d_2(10) -> 9
, d_2(10) -> 11
, d_2(10) -> 12
, d_2(12) -> 11
, b_0(4) -> 3
, b_1(4) -> 6
, b_1(5) -> 1
, b_1(6) -> 5
, c_0(4) -> 1
, c_0(4) -> 2
, c_0(4) -> 4
, c_0(4) -> 7
, c_0(4) -> 8
, c_0(4) -> 9
, c_0(4) -> 10
, c_0(4) -> 11
, c_0(4) -> 12
, c_1(7) -> 3
, c_1(7) -> 6
, c_1(8) -> 6
, c_2(9) -> 5
, c_2(11) -> 1}