MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { f(n__a(), X, X) -> f(activate(X), b(), n__b()) , activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE We add following weak dependency pairs: Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , activate^#(X) -> c_2() , activate^#(n__a()) -> c_3(a^#()) , activate^#(n__b()) -> c_4(b^#()) , a^#() -> c_7() , b^#() -> c_5() , b^#() -> c_6(a^#()) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , activate^#(X) -> c_2() , activate^#(n__a()) -> c_3(a^#()) , activate^#(n__b()) -> c_4(b^#()) , a^#() -> c_7() , b^#() -> c_5() , b^#() -> c_6(a^#()) } Strict Trs: { f(n__a(), X, X) -> f(activate(X), b(), n__b()) , activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE We replace rewrite rules by usable rules: Strict Usable Rules: { activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , activate^#(X) -> c_2() , activate^#(n__a()) -> c_3(a^#()) , activate^#(n__b()) -> c_4(b^#()) , a^#() -> c_7() , b^#() -> c_5() , b^#() -> c_6(a^#()) } Strict Trs: { activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE The weightgap principle applies (using the following constant growth matrix-interpretation) The following argument positions are usable: Uargs(f^#) = {1, 2}, Uargs(c_1) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_6) = {1} TcT has computed following constructor-restricted matrix interpretation. [n__a] = [0] [activate](x1) = [2] x1 + [2] [b] = [2] [n__b] = [1] [a] = [1] [f^#](x1, x2, x3) = [1] x1 + [1] x2 + [1] x3 + [1] [c_1](x1) = [1] x1 + [1] [activate^#](x1) = [2] x1 + [2] [c_2] = [1] [c_3](x1) = [1] x1 + [1] [a^#] = [1] [c_4](x1) = [1] x1 + [2] [b^#] = [1] [c_5] = [0] [c_6](x1) = [1] x1 + [2] [c_7] = [0] This order satisfies following ordering constraints: [activate(X)] = [2] X + [2] > [1] X + [0] = [X] [activate(n__a())] = [2] > [1] = [a()] [activate(n__b())] = [4] > [2] = [b()] [b()] = [2] > [1] = [n__b()] [b()] = [2] > [1] = [a()] [a()] = [1] > [0] = [n__a()] Further, it can be verified that all rules not oriented are covered by the weightgap condition. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , activate^#(n__a()) -> c_3(a^#()) , b^#() -> c_6(a^#()) } Weak DPs: { activate^#(X) -> c_2() , activate^#(n__b()) -> c_4(b^#()) , a^#() -> c_7() , b^#() -> c_5() } Weak Trs: { activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {2} by applications of Pre({2}) = {}. Here rules are labeled as follows: DPs: { 1: f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , 2: activate^#(n__a()) -> c_3(a^#()) , 3: b^#() -> c_6(a^#()) , 4: activate^#(X) -> c_2() , 5: activate^#(n__b()) -> c_4(b^#()) , 6: a^#() -> c_7() , 7: b^#() -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , b^#() -> c_6(a^#()) } Weak DPs: { activate^#(X) -> c_2() , activate^#(n__a()) -> c_3(a^#()) , activate^#(n__b()) -> c_4(b^#()) , a^#() -> c_7() , b^#() -> c_5() } Weak Trs: { activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { activate^#(X) -> c_2() , activate^#(n__a()) -> c_3(a^#()) , a^#() -> c_7() , b^#() -> c_5() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , b^#() -> c_6(a^#()) } Weak DPs: { activate^#(n__b()) -> c_4(b^#()) } Weak Trs: { activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { b^#() -> c_6(a^#()) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , b^#() -> c_2() } Weak DPs: { activate^#(n__b()) -> c_3(b^#()) } Weak Trs: { activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE Consider the dependency graph 1: f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) -->_1 f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) :1 2: b^#() -> c_2() 3: activate^#(n__b()) -> c_3(b^#()) -->_1 b^#() -> c_2() :2 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). { activate^#(n__b()) -> c_3(b^#()) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) , b^#() -> c_2() } Weak Trs: { activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE Consider the dependency graph 1: f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) -->_1 f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) :1 2: b^#() -> c_2() Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). { b^#() -> c_2() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { f^#(n__a(), X, X) -> c_1(f^#(activate(X), b(), n__b())) } Weak Trs: { activate(X) -> X , activate(n__a()) -> a() , activate(n__b()) -> b() , b() -> n__b() , b() -> a() , a() -> n__a() } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrices' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'matrix interpretation of dimension 4' failed due to the following reason: The input cannot be shown compatible 2) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 3) 'matrix interpretation of dimension 3' failed due to the following reason: The input cannot be shown compatible 4) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 5) 'matrix interpretation of dimension 2' failed due to the following reason: The input cannot be shown compatible 6) 'matrix interpretation of dimension 1' failed due to the following reason: The input cannot be shown compatible 2) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. Arrrr..