Tool CaT
stdout:
YES(?,O(n^1))
Problem:
f(f(x)) -> f(c(f(x)))
f(f(x)) -> f(d(f(x)))
g(c(x)) -> x
g(d(x)) -> x
g(c(0())) -> g(d(1()))
g(c(1())) -> g(d(0()))
Proof:
Bounds Processor:
bound: 1
enrichment: match
automaton:
final states: {6,5}
transitions:
g1(11) -> 12*
d1(10) -> 11*
01() -> 13*
11() -> 10*
f0(2) -> 5*
f0(4) -> 5*
f0(1) -> 5*
f0(3) -> 5*
c0(2) -> 1*
c0(4) -> 1*
c0(1) -> 1*
c0(3) -> 1*
d0(2) -> 2*
d0(4) -> 2*
d0(1) -> 2*
d0(3) -> 2*
g0(2) -> 6*
g0(4) -> 6*
g0(1) -> 6*
g0(3) -> 6*
00() -> 3*
10() -> 4*
1 -> 6*
2 -> 6*
3 -> 6*
4 -> 6*
10 -> 12,6
12 -> 6*
13 -> 10*
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:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, g(c(x)) -> x
, g(d(x)) -> x
, g(c(0())) -> g(d(1()))
, g(c(1())) -> g(d(0()))}
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:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, g(c(x)) -> x
, g(d(x)) -> x
, g(c(0())) -> g(d(1()))
, g(c(1())) -> g(d(0()))}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 1
, c_0(2) -> 1
, c_0(2) -> 2
, d_0(2) -> 1
, d_0(2) -> 2
, d_1(4) -> 3
, g_0(2) -> 1
, g_1(3) -> 1
, 0_0() -> 1
, 0_0() -> 2
, 0_1() -> 1
, 0_1() -> 4
, 1_0() -> 1
, 1_0() -> 2
, 1_1() -> 1
, 1_1() -> 4}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:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, g(c(x)) -> x
, g(d(x)) -> x
, g(c(0())) -> g(d(1()))
, g(c(1())) -> g(d(0()))}
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:
{ f(f(x)) -> f(c(f(x)))
, f(f(x)) -> f(d(f(x)))
, g(c(x)) -> x
, g(d(x)) -> x
, g(c(0())) -> g(d(1()))
, g(c(1())) -> g(d(0()))}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 1
, c_0(2) -> 1
, c_0(2) -> 2
, d_0(2) -> 1
, d_0(2) -> 2
, d_1(4) -> 3
, g_0(2) -> 1
, g_1(3) -> 1
, 0_0() -> 1
, 0_0() -> 2
, 0_1() -> 1
, 0_1() -> 4
, 1_0() -> 1
, 1_0() -> 2
, 1_1() -> 1
, 1_1() -> 4}