/export/starexec/sandbox/solver/bin/starexec_run_tct_dc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^2)) * Step 1: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__h(X)) -> h(activate(X)) f(X) -> g(n__h(n__f(X))) f(X) -> n__f(X) h(X) -> n__h(X) - Signature: {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1} - Obligation: derivational complexity wrt. signature {activate,f,g,h,n__f,n__h} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(activate) = [1] x1 + [11] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(h) = [1] x1 + [0] p(n__f) = [1] x1 + [0] p(n__h) = [1] x1 + [0] Following rules are strictly oriented: activate(X) = [1] X + [11] > [1] X + [0] = X Following rules are (at-least) weakly oriented: activate(n__f(X)) = [1] X + [11] >= [1] X + [11] = f(activate(X)) activate(n__h(X)) = [1] X + [11] >= [1] X + [11] = h(activate(X)) f(X) = [1] X + [0] >= [1] X + [0] = g(n__h(n__f(X))) f(X) = [1] X + [0] >= [1] X + [0] = n__f(X) h(X) = [1] X + [0] >= [1] X + [0] = n__h(X) * Step 2: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n__f(X)) -> f(activate(X)) activate(n__h(X)) -> h(activate(X)) f(X) -> g(n__h(n__f(X))) f(X) -> n__f(X) h(X) -> n__h(X) - Weak TRS: activate(X) -> X - Signature: {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1} - Obligation: derivational complexity wrt. signature {activate,f,g,h,n__f,n__h} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(activate) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(h) = [1] x1 + [1] p(n__f) = [1] x1 + [3] p(n__h) = [1] x1 + [0] Following rules are strictly oriented: activate(n__f(X)) = [1] X + [3] > [1] X + [0] = f(activate(X)) h(X) = [1] X + [1] > [1] X + [0] = n__h(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__h(X)) = [1] X + [0] >= [1] X + [1] = h(activate(X)) f(X) = [1] X + [0] >= [1] X + [3] = g(n__h(n__f(X))) f(X) = [1] X + [0] >= [1] X + [3] = n__f(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n__h(X)) -> h(activate(X)) f(X) -> g(n__h(n__f(X))) f(X) -> n__f(X) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) h(X) -> n__h(X) - Signature: {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1} - Obligation: derivational complexity wrt. signature {activate,f,g,h,n__f,n__h} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(activate) = [1 1] x1 + [0] [0 1] [0] p(f) = [1 4] x1 + [1] [0 1] [1] p(g) = [1 0] x1 + [0] [0 0] [1] p(h) = [1 0] x1 + [3] [0 1] [4] p(n__f) = [1 4] x1 + [1] [0 1] [1] p(n__h) = [1 0] x1 + [0] [0 1] [4] Following rules are strictly oriented: activate(n__h(X)) = [1 1] X + [4] [0 1] [4] > [1 1] X + [3] [0 1] [4] = h(activate(X)) Following rules are (at-least) weakly oriented: activate(X) = [1 1] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__f(X)) = [1 5] X + [2] [0 1] [1] >= [1 5] X + [1] [0 1] [1] = f(activate(X)) f(X) = [1 4] X + [1] [0 1] [1] >= [1 4] X + [1] [0 0] [1] = g(n__h(n__f(X))) f(X) = [1 4] X + [1] [0 1] [1] >= [1 4] X + [1] [0 1] [1] = n__f(X) h(X) = [1 0] X + [3] [0 1] [4] >= [1 0] X + [0] [0 1] [4] = n__h(X) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(X) -> g(n__h(n__f(X))) f(X) -> n__f(X) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__h(X)) -> h(activate(X)) h(X) -> n__h(X) - Signature: {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1} - Obligation: derivational complexity wrt. signature {activate,f,g,h,n__f,n__h} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(activate) = [1 4] x1 + [0] [0 1] [0] p(f) = [1 0] x1 + [1] [0 1] [2] p(g) = [1 0] x1 + [1] [0 0] [2] p(h) = [1 0] x1 + [1] [0 1] [1] p(n__f) = [1 0] x1 + [0] [0 1] [2] p(n__h) = [1 0] x1 + [0] [0 1] [1] Following rules are strictly oriented: f(X) = [1 0] X + [1] [0 1] [2] > [1 0] X + [0] [0 1] [2] = n__f(X) Following rules are (at-least) weakly oriented: activate(X) = [1 4] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X activate(n__f(X)) = [1 4] X + [8] [0 1] [2] >= [1 4] X + [1] [0 1] [2] = f(activate(X)) activate(n__h(X)) = [1 4] X + [4] [0 1] [1] >= [1 4] X + [1] [0 1] [1] = h(activate(X)) f(X) = [1 0] X + [1] [0 1] [2] >= [1 0] X + [1] [0 0] [2] = g(n__h(n__f(X))) h(X) = [1 0] X + [1] [0 1] [1] >= [1 0] X + [0] [0 1] [1] = n__h(X) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(X) -> g(n__h(n__f(X))) - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__h(X)) -> h(activate(X)) f(X) -> n__f(X) h(X) -> n__h(X) - Signature: {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1} - Obligation: derivational complexity wrt. signature {activate,f,g,h,n__f,n__h} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation: Following symbols are considered usable: all TcT has computed the following interpretation: p(activate) = [1 1] x1 + [4] [0 1] [1] p(f) = [1 0] x1 + [7] [0 1] [6] p(g) = [1 0] x1 + [0] [0 0] [1] p(h) = [1 0] x1 + [3] [0 1] [4] p(n__f) = [1 0] x1 + [4] [0 1] [6] p(n__h) = [1 0] x1 + [0] [0 1] [4] Following rules are strictly oriented: f(X) = [1 0] X + [7] [0 1] [6] > [1 0] X + [4] [0 0] [1] = g(n__h(n__f(X))) Following rules are (at-least) weakly oriented: activate(X) = [1 1] X + [4] [0 1] [1] >= [1 0] X + [0] [0 1] [0] = X activate(n__f(X)) = [1 1] X + [14] [0 1] [7] >= [1 1] X + [11] [0 1] [7] = f(activate(X)) activate(n__h(X)) = [1 1] X + [8] [0 1] [5] >= [1 1] X + [7] [0 1] [5] = h(activate(X)) f(X) = [1 0] X + [7] [0 1] [6] >= [1 0] X + [4] [0 1] [6] = n__f(X) h(X) = [1 0] X + [3] [0 1] [4] >= [1 0] X + [0] [0 1] [4] = n__h(X) * Step 6: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X activate(n__f(X)) -> f(activate(X)) activate(n__h(X)) -> h(activate(X)) f(X) -> g(n__h(n__f(X))) f(X) -> n__f(X) h(X) -> n__h(X) - Signature: {activate/1,f/1,h/1} / {g/1,n__f/1,n__h/1} - Obligation: derivational complexity wrt. signature {activate,f,g,h,n__f,n__h} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))