Tool CaT
stdout:
YES(?,O(n^1))
Problem:
a(b(b(x1))) -> P(a(b(x1)))
a(P(x1)) -> P(a(x(x1)))
a(x(x1)) -> x(a(x1))
b(P(x1)) -> b(Q(x1))
Q(x(x1)) -> a(Q(x1))
Q(a(x1)) -> b(b(a(x1)))
Proof:
Bounds Processor:
bound: 2
enrichment: match
automaton:
final states: {5,4,3}
transitions:
a1(7) -> 8*
a1(26) -> 27*
a1(38) -> 39*
a1(18) -> 19*
Q1(36) -> 37*
Q1(28) -> 29*
b1(29) -> 30*
x1(19) -> 20*
x1(16) -> 17*
x1(6) -> 7*
P1(8) -> 9*
x2(46) -> 47*
a0(2) -> 3*
a0(1) -> 3*
a2(50) -> 51*
a2(45) -> 46*
b0(2) -> 4*
b0(1) -> 4*
P0(2) -> 1*
P0(1) -> 1*
x0(2) -> 2*
x0(1) -> 2*
Q0(2) -> 5*
Q0(1) -> 5*
1 -> 36,26,16
2 -> 28,18,6
6 -> 50*
9 -> 46,27,19,3
16 -> 45*
17 -> 7*
20 -> 51,46,19,3
27 -> 19*
29 -> 38*
30 -> 4*
37 -> 29*
39 -> 29,38,5
47 -> 8*
51 -> 46*
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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(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) -> 5
, a_1(4) -> 3
, a_1(7) -> 1
, a_1(7) -> 7
, a_2(2) -> 6
, b_0(2) -> 1
, b_1(7) -> 1
, P_0(2) -> 2
, P_1(3) -> 1
, P_1(3) -> 5
, P_1(3) -> 6
, x_0(2) -> 2
, x_1(2) -> 4
, x_1(5) -> 1
, x_1(5) -> 5
, x_1(5) -> 6
, x_2(6) -> 3
, Q_0(2) -> 1
, Q_1(2) -> 7}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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(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(b(b(x1))) -> P(a(b(x1)))
, a(P(x1)) -> P(a(x(x1)))
, a(x(x1)) -> x(a(x1))
, b(P(x1)) -> b(Q(x1))
, Q(x(x1)) -> a(Q(x1))
, Q(a(x1)) -> b(b(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) -> 5
, a_1(4) -> 3
, a_1(7) -> 1
, a_1(7) -> 7
, a_2(2) -> 6
, b_0(2) -> 1
, b_1(7) -> 1
, P_0(2) -> 2
, P_1(3) -> 1
, P_1(3) -> 5
, P_1(3) -> 6
, x_0(2) -> 2
, x_1(2) -> 4
, x_1(5) -> 1
, x_1(5) -> 5
, x_1(5) -> 6
, x_2(6) -> 3
, Q_0(2) -> 1
, Q_1(2) -> 7}