Tool CaT
stdout:
YES(?,O(n^1))
Problem:
b(b(x1)) -> c(d(x1))
c(c(x1)) -> d(d(d(x1)))
c(x1) -> g(x1)
d(d(x1)) -> c(f(x1))
d(d(d(x1))) -> g(c(x1))
f(x1) -> a(g(x1))
g(x1) -> d(a(b(x1)))
g(g(x1)) -> b(c(x1))
Proof:
Bounds Processor:
bound: 2
enrichment: match
automaton:
final states: {6,5,4,3,2}
transitions:
d1(17) -> 18*
a1(16) -> 17*
a1(13) -> 14*
b1(15) -> 16*
g1(7) -> 8*
d2(29) -> 30*
a2(28) -> 29*
b2(27) -> 28*
b0(1) -> 2*
c0(1) -> 3*
d0(1) -> 4*
g0(1) -> 6*
f0(1) -> 5*
a0(1) -> 1*
1 -> 15,7
7 -> 27*
8 -> 13,3
14 -> 5*
18 -> 6*
30 -> 8,13
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:
{ b(b(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(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:
{ b(b(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))}
Proof Output:
The problem is match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_0(2) -> 1
, b_1(2) -> 4
, b_2(2) -> 6
, c_0(2) -> 1
, d_0(2) -> 1
, d_1(3) -> 1
, d_2(5) -> 1
, g_0(2) -> 1
, g_1(2) -> 1
, f_0(2) -> 1
, a_0(2) -> 2
, a_1(1) -> 1
, a_1(4) -> 3
, a_2(6) -> 5}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:
{ b(b(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(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:
{ b(b(x1)) -> c(d(x1))
, c(c(x1)) -> d(d(d(x1)))
, c(x1) -> g(x1)
, d(d(x1)) -> c(f(x1))
, d(d(d(x1))) -> g(c(x1))
, f(x1) -> a(g(x1))
, g(x1) -> d(a(b(x1)))
, g(g(x1)) -> b(c(x1))}
Proof Output:
The problem is match-bounded by 2.
The enriched problem is compatible with the following automaton:
{ b_0(2) -> 1
, b_1(2) -> 4
, b_2(2) -> 6
, c_0(2) -> 1
, d_0(2) -> 1
, d_1(3) -> 1
, d_2(5) -> 1
, g_0(2) -> 1
, g_1(2) -> 1
, f_0(2) -> 1
, a_0(2) -> 2
, a_1(1) -> 1
, a_1(4) -> 3
, a_2(6) -> 5}