Tool CaT
stdout:
YES(?,O(n^1))
Problem:
b(x,y) -> c(a(c(y),a(0(),x)))
a(y,x) -> y
a(y,c(b(a(0(),x),0()))) -> b(a(c(b(0(),y)),x),0())
Proof:
Bounds Processor:
bound: 1
enrichment: match
automaton:
final states: {4,3}
transitions:
c1(12) -> 3*
c1(2) -> 11*
c1(1) -> 11*
a1(9,2) -> 10*
a1(11,10) -> 12*
a1(9,1) -> 10*
01() -> 9*
b0(1,2) -> 3*
b0(2,1) -> 3*
b0(1,1) -> 3*
b0(2,2) -> 3*
c0(2) -> 1*
c0(1) -> 1*
a0(1,2) -> 4*
a0(2,1) -> 4*
a0(1,1) -> 4*
a0(2,2) -> 4*
00() -> 2*
1 -> 4*
2 -> 4*
9 -> 10*
11 -> 12*
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(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 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:
{ b(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ b_0(2, 2) -> 1
, c_0(2) -> 1
, c_0(2) -> 2
, c_1(2) -> 3
, c_1(2) -> 4
, c_1(3) -> 1
, a_0(2, 2) -> 1
, a_1(4, 5) -> 3
, a_1(6, 2) -> 5
, 0_0() -> 1
, 0_0() -> 2
, 0_1() -> 5
, 0_1() -> 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:
{ b(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 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:
{ b(x, y) -> c(a(c(y), a(0(), x)))
, a(y, x) -> y
, a(y, c(b(a(0(), x), 0()))) -> b(a(c(b(0(), y)), x), 0())}
Proof Output:
The problem is match-bounded by 1.
The enriched problem is compatible with the following automaton:
{ b_0(2, 2) -> 1
, c_0(2) -> 1
, c_0(2) -> 2
, c_1(2) -> 3
, c_1(2) -> 4
, c_1(3) -> 1
, a_0(2, 2) -> 1
, a_1(4, 5) -> 3
, a_1(6, 2) -> 5
, 0_0() -> 1
, 0_0() -> 2
, 0_1() -> 5
, 0_1() -> 6}