MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { ack(0(), y) -> s(y) , ack(s(x), y) -> f(x, x) , ack(s(x), 0()) -> ack(x, s(0())) , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) , f(x, y) -> ack(x, y) , f(x, s(y)) -> f(y, x) , f(s(x), y) -> f(x, s(x)) } Obligation: innermost runtime complexity Answer: MAYBE We add following dependency tuples: Strict DPs: { ack^#(0(), y) -> c_1() , ack^#(s(x), y) -> c_2(f^#(x, x)) , ack^#(s(x), 0()) -> c_3(ack^#(x, s(0()))) , ack^#(s(x), s(y)) -> c_4(ack^#(x, ack(s(x), y)), ack^#(s(x), y)) , f^#(x, y) -> c_5(ack^#(x, y)) , f^#(x, s(y)) -> c_6(f^#(y, x)) , f^#(s(x), y) -> c_7(f^#(x, s(x))) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { ack^#(0(), y) -> c_1() , ack^#(s(x), y) -> c_2(f^#(x, x)) , ack^#(s(x), 0()) -> c_3(ack^#(x, s(0()))) , ack^#(s(x), s(y)) -> c_4(ack^#(x, ack(s(x), y)), ack^#(s(x), y)) , f^#(x, y) -> c_5(ack^#(x, y)) , f^#(x, s(y)) -> c_6(f^#(y, x)) , f^#(s(x), y) -> c_7(f^#(x, s(x))) } Weak Trs: { ack(0(), y) -> s(y) , ack(s(x), y) -> f(x, x) , ack(s(x), 0()) -> ack(x, s(0())) , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) , f(x, y) -> ack(x, y) , f(x, s(y)) -> f(y, x) , f(s(x), y) -> f(x, s(x)) } Obligation: innermost runtime complexity Answer: MAYBE We estimate the number of application of {1} by applications of Pre({1}) = {3,4,5}. Here rules are labeled as follows: DPs: { 1: ack^#(0(), y) -> c_1() , 2: ack^#(s(x), y) -> c_2(f^#(x, x)) , 3: ack^#(s(x), 0()) -> c_3(ack^#(x, s(0()))) , 4: ack^#(s(x), s(y)) -> c_4(ack^#(x, ack(s(x), y)), ack^#(s(x), y)) , 5: f^#(x, y) -> c_5(ack^#(x, y)) , 6: f^#(x, s(y)) -> c_6(f^#(y, x)) , 7: f^#(s(x), y) -> c_7(f^#(x, s(x))) } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { ack^#(s(x), y) -> c_2(f^#(x, x)) , ack^#(s(x), 0()) -> c_3(ack^#(x, s(0()))) , ack^#(s(x), s(y)) -> c_4(ack^#(x, ack(s(x), y)), ack^#(s(x), y)) , f^#(x, y) -> c_5(ack^#(x, y)) , f^#(x, s(y)) -> c_6(f^#(y, x)) , f^#(s(x), y) -> c_7(f^#(x, s(x))) } Weak DPs: { ack^#(0(), y) -> c_1() } Weak Trs: { ack(0(), y) -> s(y) , ack(s(x), y) -> f(x, x) , ack(s(x), 0()) -> ack(x, s(0())) , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) , f(x, y) -> ack(x, y) , f(x, s(y)) -> f(y, x) , f(s(x), y) -> f(x, s(x)) } 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. { ack^#(0(), y) -> c_1() } We are left with following problem, upon which TcT provides the certificate MAYBE. Strict DPs: { ack^#(s(x), y) -> c_2(f^#(x, x)) , ack^#(s(x), 0()) -> c_3(ack^#(x, s(0()))) , ack^#(s(x), s(y)) -> c_4(ack^#(x, ack(s(x), y)), ack^#(s(x), y)) , f^#(x, y) -> c_5(ack^#(x, y)) , f^#(x, s(y)) -> c_6(f^#(y, x)) , f^#(s(x), y) -> c_7(f^#(x, s(x))) } Weak Trs: { ack(0(), y) -> s(y) , ack(s(x), y) -> f(x, x) , ack(s(x), 0()) -> ack(x, s(0())) , ack(s(x), s(y)) -> ack(x, ack(s(x), y)) , f(x, y) -> ack(x, y) , f(x, s(y)) -> f(y, x) , f(s(x), y) -> f(x, s(x)) } 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..