Tool CaT
stdout:
MAYBE
Problem:
b(y,z) -> f(c(c(y,z,z),a(),a()))
b(b(z,y),a()) -> z
c(f(z),f(c(a(),x,a())),y) -> c(f(b(x,z)),c(z,y,a()),a())
Proof:
OpenTool IRC1
stdout:
MAYBE
Tool IRC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z
, c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: b^#(y, z) -> c_0(c^#(c(y, z, z), a(), a()))
, 2: b^#(b(z, y), a()) -> c_1()
, 3: c^#(f(z), f(c(a(), x, a())), y) ->
c_2(c^#(f(b(x, z)), c(z, y, a()), a()))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ NA ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())
, b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b^#(y, z) -> c_0(c^#(c(y, z, z), a(), a()))
, c(f(z), f(c(a(), x, a())), y) -> c(f(b(x, z)), c(z, y, a()), a())
, b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(b) = {}, Uargs(f) = {}, Uargs(c) = {}, Uargs(b^#) = {},
Uargs(c_0) = {}, Uargs(c^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
b(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
c(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
a() = [0]
[0]
b^#(x1, x2) = [0 0] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1() = [0]
[0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {b^#(b(z, y), a()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(b) = {}, Uargs(b^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
b(x1, x2) = [0 0] x1 + [0 0] x2 + [2]
[0 0] [0 0] [2]
a() = [2]
[0]
b^#(x1, x2) = [2 2] x1 + [2 0] x2 + [3]
[0 0] [0 0] [3]
c_1() = [0]
[1]
* Path {3}: NA
------------
The usable rules for this path are:
{ b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z
, c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: b^#(y, z) -> c_0(c^#(c(y, z, z), a(), a()))
, 2: b^#(b(z, y), a()) -> c_1()
, 3: c^#(f(z), f(c(a(), x, a())), y) ->
c_2(c^#(f(b(x, z)), c(z, y, a()), a()))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ NA ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())
, b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: innermost runtime-complexity with respect to
Rules:
{ b^#(y, z) -> c_0(c^#(c(y, z, z), a(), a()))
, c(f(z), f(c(a(), x, a())), y) -> c(f(b(x, z)), c(z, y, a()), a())
, b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(b) = {}, Uargs(f) = {}, Uargs(c) = {}, Uargs(b^#) = {},
Uargs(c_0) = {}, Uargs(c^#) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
b(x1, x2) = [0] x1 + [0] x2 + [0]
f(x1) = [0] x1 + [0]
c(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
b^#(x1, x2) = [0] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1() = [0]
c_2(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: innermost DP runtime-complexity with respect to
Strict Rules: {b^#(b(z, y), a()) -> c_1()}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(b) = {}, Uargs(b^#) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
b(x1, x2) = [0] x1 + [0] x2 + [2]
a() = [2]
b^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_1() = [0]
* Path {3}: NA
------------
The usable rules for this path are:
{ b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z
, c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
Tool RC1
stdout:
MAYBE
Tool RC2
stdout:
MAYBE
'Fastest (timeout of 60.0 seconds)'
-----------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z
, c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())}
Proof Output:
None of the processors succeeded.
Details of failed attempt(s):
-----------------------------
1) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: b^#(y, z) -> c_0(c^#(c(y, z, z), a(), a()))
, 2: b^#(b(z, y), a()) -> c_1(z)
, 3: c^#(f(z), f(c(a(), x, a())), y) ->
c_2(c^#(f(b(x, z)), c(z, y, a()), a()))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ NA ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())
, b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ b^#(y, z) -> c_0(c^#(c(y, z, z), a(), a()))
, c(f(z), f(c(a(), x, a())), y) -> c(f(b(x, z)), c(z, y, a()), a())
, b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(b) = {}, Uargs(f) = {}, Uargs(c) = {}, Uargs(b^#) = {},
Uargs(c_0) = {}, Uargs(c^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
b(x1, x2) = [0 0] x1 + [3 3] x2 + [0]
[0 0] [2 2] [0]
f(x1) = [0 0] x1 + [0]
[0 0] [0]
c(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
a() = [0]
[0]
b^#(x1, x2) = [1 3] x1 + [0 0] x2 + [0]
[0 0] [0 0] [0]
c_0(x1) = [0 0] x1 + [0]
[0 0] [0]
c^#(x1, x2, x3) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [0]
[0 0] [0 0] [0 0] [0]
c_1(x1) = [1 1] x1 + [0]
[0 0] [0]
c_2(x1) = [0 0] x1 + [0]
[0 0] [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 2'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {b^#(b(z, y), a()) -> c_1(z)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(b) = {}, Uargs(b^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
b(x1, x2) = [2 2] x1 + [0 0] x2 + [0]
[2 2] [0 0] [2]
a() = [2]
[2]
b^#(x1, x2) = [2 2] x1 + [2 2] x2 + [3]
[2 2] [2 0] [7]
c_1(x1) = [0 0] x1 + [0]
[0 0] [1]
* Path {3}: NA
------------
The usable rules for this path are:
{ b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z
, c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
2) 'wdg' failed due to the following reason:
Transformation Details:
-----------------------
We have computed the following set of weak (innermost) dependency pairs:
{ 1: b^#(y, z) -> c_0(c^#(c(y, z, z), a(), a()))
, 2: b^#(b(z, y), a()) -> c_1(z)
, 3: c^#(f(z), f(c(a(), x, a())), y) ->
c_2(c^#(f(b(x, z)), c(z, y, a()), a()))}
Following Dependency Graph (modulo SCCs) was computed. (Answers to
subproofs are indicated to the right.)
->{3} [ NA ]
->{2} [ YES(?,O(1)) ]
->{1} [ MAYBE ]
Sub-problems:
-------------
* Path {1}: MAYBE
---------------
The usable rules for this path are:
{ c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())
, b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: MAYBE
Input Problem: runtime-complexity with respect to
Rules:
{ b^#(y, z) -> c_0(c^#(c(y, z, z), a(), a()))
, c(f(z), f(c(a(), x, a())), y) -> c(f(b(x, z)), c(z, y, a()), a())
, b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z}
Proof Output:
The input cannot be shown compatible
* Path {2}: YES(?,O(1))
---------------------
The usable rules of this path are empty.
The weightgap principle applies, using the following adequate RMI:
The following argument positions are usable:
Uargs(b) = {}, Uargs(f) = {}, Uargs(c) = {}, Uargs(b^#) = {},
Uargs(c_0) = {}, Uargs(c^#) = {}, Uargs(c_1) = {}, Uargs(c_2) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
b(x1, x2) = [0] x1 + [2] x2 + [0]
f(x1) = [0] x1 + [0]
c(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
a() = [0]
b^#(x1, x2) = [3] x1 + [0] x2 + [0]
c_0(x1) = [0] x1 + [0]
c^#(x1, x2, x3) = [0] x1 + [0] x2 + [0] x3 + [0]
c_1(x1) = [0] x1 + [0]
c_2(x1) = [0] x1 + [0]
We apply the sub-processor on the resulting sub-problem:
'matrix-interpretation of dimension 1'
--------------------------------------
Answer: YES(?,O(1))
Input Problem: DP runtime-complexity with respect to
Strict Rules: {b^#(b(z, y), a()) -> c_1(z)}
Weak Rules: {}
Proof Output:
The following argument positions are usable:
Uargs(b) = {}, Uargs(b^#) = {}, Uargs(c_1) = {}
We have the following constructor-restricted matrix interpretation:
Interpretation Functions:
b(x1, x2) = [7] x1 + [0] x2 + [2]
a() = [2]
b^#(x1, x2) = [2] x1 + [2] x2 + [7]
c_1(x1) = [0] x1 + [0]
* Path {3}: NA
------------
The usable rules for this path are:
{ b(y, z) -> f(c(c(y, z, z), a(), a()))
, b(b(z, y), a()) -> z
, c(f(z), f(c(a(), x, a())), y) ->
c(f(b(x, z)), c(z, y, a()), a())}
The weight gap principle does not apply:
The input cannot be shown compatible
Complexity induced by the adequate RMI: MAYBE
We have not generated a proof for the resulting sub-problem.
3) 'matrix-interpretation of dimension 1' failed due to the following reason:
The input cannot be shown compatible
4) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.
5) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason:
match-boundness of the problem could not be verified.