Tool CaT
stdout:
YES(?,O(n^1))
Problem:
f(x1) -> n(c(n(a(x1))))
c(f(x1)) -> f(n(a(c(x1))))
n(a(x1)) -> c(x1)
c(c(x1)) -> c(x1)
n(s(x1)) -> f(s(s(x1)))
n(f(x1)) -> f(n(x1))
Proof:
Bounds Processor:
bound: 3
enrichment: match
automaton:
final states: {5,4,3}
transitions:
f1(30) -> 31*
s1(29) -> 30*
s1(38) -> 39*
s1(28) -> 29*
c1(20) -> 21*
c1(26) -> 27*
c1(8) -> 9*
n1(7) -> 8*
n1(9) -> 10*
a1(6) -> 7*
a1(18) -> 19*
c2(52) -> 53*
c2(42) -> 43*
c2(54) -> 55*
f0(2) -> 3*
f0(1) -> 3*
n2(41) -> 42*
n2(43) -> 44*
n0(2) -> 5*
n0(1) -> 5*
a2(40) -> 41*
c0(2) -> 4*
c0(1) -> 4*
c3(60) -> 61*
a0(2) -> 1*
a0(1) -> 1*
s0(2) -> 2*
s0(1) -> 2*
1 -> 38,26,6
2 -> 28,20,18
6 -> 52*
10 -> 3*
18 -> 54*
19 -> 7*
21 -> 5*
27 -> 5*
30 -> 40*
31 -> 5*
39 -> 29*
40 -> 60*
44 -> 31*
53 -> 9,8
55 -> 9,8
61 -> 43,42
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(x1) -> n(c(n(a(x1))))
, c(f(x1)) -> f(n(a(c(x1))))
, n(a(x1)) -> c(x1)
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(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:
{ f(x1) -> n(c(n(a(x1))))
, c(f(x1)) -> f(n(a(c(x1))))
, n(a(x1)) -> c(x1)
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))}
Proof Output:
The problem is match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 1
, f_1(6) -> 1
, n_0(2) -> 1
, n_1(3) -> 1
, n_1(5) -> 4
, n_2(8) -> 1
, n_2(10) -> 9
, c_0(2) -> 1
, c_1(2) -> 1
, c_1(4) -> 3
, c_2(2) -> 3
, c_2(2) -> 4
, c_2(9) -> 8
, c_3(6) -> 8
, c_3(6) -> 9
, a_0(2) -> 2
, a_1(2) -> 5
, a_2(6) -> 10
, s_0(2) -> 2
, s_1(2) -> 7
, s_1(7) -> 6}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(x1) -> n(c(n(a(x1))))
, c(f(x1)) -> f(n(a(c(x1))))
, n(a(x1)) -> c(x1)
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(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:
{ f(x1) -> n(c(n(a(x1))))
, c(f(x1)) -> f(n(a(c(x1))))
, n(a(x1)) -> c(x1)
, c(c(x1)) -> c(x1)
, n(s(x1)) -> f(s(s(x1)))
, n(f(x1)) -> f(n(x1))}
Proof Output:
The problem is match-bounded by 3.
The enriched problem is compatible with the following automaton:
{ f_0(2) -> 1
, f_1(6) -> 1
, n_0(2) -> 1
, n_1(3) -> 1
, n_1(5) -> 4
, n_2(8) -> 1
, n_2(10) -> 9
, c_0(2) -> 1
, c_1(2) -> 1
, c_1(4) -> 3
, c_2(2) -> 3
, c_2(2) -> 4
, c_2(9) -> 8
, c_3(6) -> 8
, c_3(6) -> 9
, a_0(2) -> 2
, a_1(2) -> 5
, a_2(6) -> 10
, s_0(2) -> 2
, s_1(2) -> 7
, s_1(7) -> 6}