/export/starexec/sandbox2/solver/bin/starexec_run_tct_dc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^1)) * Step 1: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(c()) -> active(c()) mark(f(X)) -> active(f(X)) mark(g(X)) -> active(g(X)) - Signature: {active/1,f/1,g/1,mark/1} / {c/0} - Obligation: derivational complexity wrt. signature {active,c,f,g,mark} + 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(active) = [1] x1 + [0] p(c) = [12] p(f) = [1] x1 + [5] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [5] Following rules are strictly oriented: f(mark(X)) = [1] X + [10] > [1] X + [5] = f(X) g(mark(X)) = [1] X + [5] > [1] X + [0] = g(X) mark(c()) = [17] > [12] = active(c()) mark(f(X)) = [1] X + [10] > [1] X + [5] = active(f(X)) mark(g(X)) = [1] X + [5] > [1] X + [0] = active(g(X)) Following rules are (at-least) weakly oriented: active(c()) = [12] >= [22] = mark(f(g(c()))) active(f(g(X))) = [1] X + [5] >= [1] X + [5] = mark(g(X)) f(active(X)) = [1] X + [5] >= [1] X + [5] = f(X) g(active(X)) = [1] X + [0] >= [1] X + [0] = g(X) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) f(active(X)) -> f(X) g(active(X)) -> g(X) - Weak TRS: f(mark(X)) -> f(X) g(mark(X)) -> g(X) mark(c()) -> active(c()) mark(f(X)) -> active(f(X)) mark(g(X)) -> active(g(X)) - Signature: {active/1,f/1,g/1,mark/1} / {c/0} - Obligation: derivational complexity wrt. signature {active,c,f,g,mark} + 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(active) = [1] x1 + [2] p(c) = [14] p(f) = [1] x1 + [12] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [2] Following rules are strictly oriented: active(f(g(X))) = [1] X + [14] > [1] X + [2] = mark(g(X)) f(active(X)) = [1] X + [14] > [1] X + [12] = f(X) g(active(X)) = [1] X + [2] > [1] X + [0] = g(X) Following rules are (at-least) weakly oriented: active(c()) = [16] >= [28] = mark(f(g(c()))) f(mark(X)) = [1] X + [14] >= [1] X + [12] = f(X) g(mark(X)) = [1] X + [2] >= [1] X + [0] = g(X) mark(c()) = [16] >= [16] = active(c()) mark(f(X)) = [1] X + [14] >= [1] X + [14] = active(f(X)) mark(g(X)) = [1] X + [2] >= [1] X + [2] = active(g(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict TRS: active(c()) -> mark(f(g(c()))) - Weak TRS: active(f(g(X))) -> mark(g(X)) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(c()) -> active(c()) mark(f(X)) -> active(f(X)) mark(g(X)) -> active(g(X)) - Signature: {active/1,f/1,g/1,mark/1} / {c/0} - Obligation: derivational complexity wrt. signature {active,c,f,g,mark} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(active) = [1 4] x1 + [0] [0 0] [4] p(c) = [7] [2] p(f) = [1 2] x1 + [5] [0 0] [0] p(g) = [1 0] x1 + [1] [0 0] [0] p(mark) = [1 4] x1 + [0] [0 0] [4] Following rules are strictly oriented: active(c()) = [15] [4] > [13] [4] = mark(f(g(c()))) Following rules are (at-least) weakly oriented: active(f(g(X))) = [1 0] X + [6] [0 0] [4] >= [1 0] X + [1] [0 0] [4] = mark(g(X)) f(active(X)) = [1 4] X + [13] [0 0] [0] >= [1 2] X + [5] [0 0] [0] = f(X) f(mark(X)) = [1 4] X + [13] [0 0] [0] >= [1 2] X + [5] [0 0] [0] = f(X) g(active(X)) = [1 4] X + [1] [0 0] [0] >= [1 0] X + [1] [0 0] [0] = g(X) g(mark(X)) = [1 4] X + [1] [0 0] [0] >= [1 0] X + [1] [0 0] [0] = g(X) mark(c()) = [15] [4] >= [15] [4] = active(c()) mark(f(X)) = [1 2] X + [5] [0 0] [4] >= [1 2] X + [5] [0 0] [4] = active(f(X)) mark(g(X)) = [1 0] X + [1] [0 0] [4] >= [1 0] X + [1] [0 0] [4] = active(g(X)) * Step 4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(c()) -> mark(f(g(c()))) active(f(g(X))) -> mark(g(X)) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(c()) -> active(c()) mark(f(X)) -> active(f(X)) mark(g(X)) -> active(g(X)) - Signature: {active/1,f/1,g/1,mark/1} / {c/0} - Obligation: derivational complexity wrt. signature {active,c,f,g,mark} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^1))