Tool CaT
stdout:
YES(?,O(n^1))
Problem:
g(h(x1)) -> g(f(s(x1)))
f(s(s(s(x1)))) -> h(f(s(h(x1))))
f(h(x1)) -> h(f(s(h(x1))))
h(x1) -> x1
f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
b(a(x1)) -> a(b(x1))
a(a(a(x1))) -> b(a(a(b(x1))))
b(b(b(b(x1)))) -> a(x1)
Proof:
Bounds Processor:
bound: 1
enrichment: match
automaton:
final states: {6,5,4,3,2}
transitions:
h1(15) -> 16*
h1(12) -> 13*
f1(14) -> 15*
s1(13) -> 14*
g0(1) -> 2*
h0(1) -> 4*
f0(1) -> 3*
s0(1) -> 1*
b0(1) -> 5*
a0(1) -> 6*
1 -> 4,12
12 -> 13*
15 -> 16,3
16 -> 15,3
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:
{ g(h(x1)) -> g(f(s(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, b(a(x1)) -> a(b(x1))
, a(a(a(x1))) -> b(a(a(b(x1))))
, b(b(b(b(x1)))) -> 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:
{ g(h(x1)) -> g(f(s(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, b(a(x1)) -> a(b(x1))
, a(a(a(x1))) -> b(a(a(b(x1))))
, b(b(b(b(x1)))) -> a(x1)}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ g_0(2) -> 1
, h_0(2) -> 1
, h_1(2) -> 5
, h_1(3) -> 1
, h_1(3) -> 3
, f_0(2) -> 1
, f_1(4) -> 1
, f_1(4) -> 3
, s_0(2) -> 1
, s_0(2) -> 2
, s_0(2) -> 5
, s_1(5) -> 4
, b_0(2) -> 1
, a_0(2) -> 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:
{ g(h(x1)) -> g(f(s(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, b(a(x1)) -> a(b(x1))
, a(a(a(x1))) -> b(a(a(b(x1))))
, b(b(b(b(x1)))) -> 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:
{ g(h(x1)) -> g(f(s(x1)))
, f(s(s(s(x1)))) -> h(f(s(h(x1))))
, f(h(x1)) -> h(f(s(h(x1))))
, h(x1) -> x1
, f(f(s(s(x1)))) -> s(s(s(f(f(x1)))))
, b(a(x1)) -> a(b(x1))
, a(a(a(x1))) -> b(a(a(b(x1))))
, b(b(b(b(x1)))) -> a(x1)}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ g_0(2) -> 1
, h_0(2) -> 1
, h_1(2) -> 5
, h_1(3) -> 1
, h_1(3) -> 3
, f_0(2) -> 1
, f_1(4) -> 1
, f_1(4) -> 3
, s_0(2) -> 1
, s_0(2) -> 2
, s_0(2) -> 5
, s_1(5) -> 4
, b_0(2) -> 1
, a_0(2) -> 1}