/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^2)) * Step 1: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(X)) -> mark(g(h(f(X)))) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) h(active(X)) -> h(X) h(mark(X)) -> h(X) mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(X)) mark(h(X)) -> active(h(mark(X))) - Signature: {active/1,f/1,g/1,h/1,mark/1} / {} - Obligation: derivational complexity wrt. signature {active,f,g,h,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(f) = [1] x1 + [15] p(g) = [1] x1 + [0] p(h) = [1] x1 + [0] p(mark) = [1] x1 + [1] Following rules are strictly oriented: active(f(X)) = [1] X + [17] > [1] X + [16] = mark(g(h(f(X)))) f(active(X)) = [1] X + [17] > [1] X + [15] = f(X) f(mark(X)) = [1] X + [16] > [1] X + [15] = f(X) g(active(X)) = [1] X + [2] > [1] X + [0] = g(X) g(mark(X)) = [1] X + [1] > [1] X + [0] = g(X) h(active(X)) = [1] X + [2] > [1] X + [0] = h(X) h(mark(X)) = [1] X + [1] > [1] X + [0] = h(X) Following rules are (at-least) weakly oriented: mark(f(X)) = [1] X + [16] >= [1] X + [18] = active(f(mark(X))) mark(g(X)) = [1] X + [1] >= [1] X + [2] = active(g(X)) mark(h(X)) = [1] X + [1] >= [1] X + [3] = active(h(mark(X))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(X)) mark(h(X)) -> active(h(mark(X))) - Weak TRS: active(f(X)) -> mark(g(h(f(X)))) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) h(active(X)) -> h(X) h(mark(X)) -> h(X) - Signature: {active/1,f/1,g/1,h/1,mark/1} / {} - Obligation: derivational complexity wrt. signature {active,f,g,h,mark} + 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(active) = [1 0] x1 + [0] [0 1] [0] p(f) = [1 4] x1 + [4] [0 1] [0] p(g) = [1 0] x1 + [0] [0 0] [0] p(h) = [1 0] x1 + [0] [0 1] [1] p(mark) = [1 4] x1 + [0] [0 1] [0] Following rules are strictly oriented: mark(h(X)) = [1 4] X + [4] [0 1] [1] > [1 4] X + [0] [0 1] [1] = active(h(mark(X))) Following rules are (at-least) weakly oriented: active(f(X)) = [1 4] X + [4] [0 1] [0] >= [1 4] X + [4] [0 0] [0] = mark(g(h(f(X)))) f(active(X)) = [1 4] X + [4] [0 1] [0] >= [1 4] X + [4] [0 1] [0] = f(X) f(mark(X)) = [1 8] X + [4] [0 1] [0] >= [1 4] X + [4] [0 1] [0] = f(X) g(active(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = g(X) g(mark(X)) = [1 4] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = g(X) h(active(X)) = [1 0] X + [0] [0 1] [1] >= [1 0] X + [0] [0 1] [1] = h(X) h(mark(X)) = [1 4] X + [0] [0 1] [1] >= [1 0] X + [0] [0 1] [1] = h(X) mark(f(X)) = [1 8] X + [4] [0 1] [0] >= [1 8] X + [4] [0 1] [0] = active(f(mark(X))) mark(g(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = active(g(X)) * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(X)) - Weak TRS: active(f(X)) -> mark(g(h(f(X)))) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) h(active(X)) -> h(X) h(mark(X)) -> h(X) mark(h(X)) -> active(h(mark(X))) - Signature: {active/1,f/1,g/1,h/1,mark/1} / {} - Obligation: derivational complexity wrt. signature {active,f,g,h,mark} + 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(active) = [1 0] x1 + [0] [0 1] [0] p(f) = [1 4] x1 + [0] [0 1] [1] p(g) = [1 0] x1 + [0] [0 0] [0] p(h) = [1 0] x1 + [0] [0 1] [3] p(mark) = [1 4] x1 + [0] [0 1] [0] Following rules are strictly oriented: mark(f(X)) = [1 8] X + [4] [0 1] [1] > [1 8] X + [0] [0 1] [1] = active(f(mark(X))) Following rules are (at-least) weakly oriented: active(f(X)) = [1 4] X + [0] [0 1] [1] >= [1 4] X + [0] [0 0] [0] = mark(g(h(f(X)))) f(active(X)) = [1 4] X + [0] [0 1] [1] >= [1 4] X + [0] [0 1] [1] = f(X) f(mark(X)) = [1 8] X + [0] [0 1] [1] >= [1 4] X + [0] [0 1] [1] = f(X) g(active(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = g(X) g(mark(X)) = [1 4] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = g(X) h(active(X)) = [1 0] X + [0] [0 1] [3] >= [1 0] X + [0] [0 1] [3] = h(X) h(mark(X)) = [1 4] X + [0] [0 1] [3] >= [1 0] X + [0] [0 1] [3] = h(X) mark(g(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = active(g(X)) mark(h(X)) = [1 4] X + [12] [0 1] [3] >= [1 4] X + [0] [0 1] [3] = active(h(mark(X))) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(g(X)) -> active(g(X)) - Weak TRS: active(f(X)) -> mark(g(h(f(X)))) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) h(active(X)) -> h(X) h(mark(X)) -> h(X) mark(f(X)) -> active(f(mark(X))) mark(h(X)) -> active(h(mark(X))) - Signature: {active/1,f/1,g/1,h/1,mark/1} / {} - Obligation: derivational complexity wrt. signature {active,f,g,h,mark} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, 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 2 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 0 2] [0] [0 1 2] x1 + [0] [0 0 0] [0] p(f) = [1 0 0] [1] [0 1 2] x1 + [0] [0 0 0] [3] p(g) = [1 0 0] [0] [0 0 0] x1 + [2] [0 0 0] [0] p(h) = [1 0 0] [2] [0 1 2] x1 + [0] [0 0 0] [2] p(mark) = [1 2 2] [0] [0 1 2] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: mark(g(X)) = [1 0 0] [4] [0 0 0] X + [2] [0 0 0] [0] > [1 0 0] [0] [0 0 0] X + [2] [0 0 0] [0] = active(g(X)) Following rules are (at-least) weakly oriented: active(f(X)) = [1 0 0] [7] [0 1 2] X + [6] [0 0 0] [0] >= [1 0 0] [7] [0 0 0] X + [2] [0 0 0] [0] = mark(g(h(f(X)))) f(active(X)) = [1 0 2] [1] [0 1 2] X + [0] [0 0 0] [3] >= [1 0 0] [1] [0 1 2] X + [0] [0 0 0] [3] = f(X) f(mark(X)) = [1 2 2] [1] [0 1 2] X + [0] [0 0 0] [3] >= [1 0 0] [1] [0 1 2] X + [0] [0 0 0] [3] = f(X) g(active(X)) = [1 0 2] [0] [0 0 0] X + [2] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [2] [0 0 0] [0] = g(X) g(mark(X)) = [1 2 2] [0] [0 0 0] X + [2] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] X + [2] [0 0 0] [0] = g(X) h(active(X)) = [1 0 2] [2] [0 1 2] X + [0] [0 0 0] [2] >= [1 0 0] [2] [0 1 2] X + [0] [0 0 0] [2] = h(X) h(mark(X)) = [1 2 2] [2] [0 1 2] X + [0] [0 0 0] [2] >= [1 0 0] [2] [0 1 2] X + [0] [0 0 0] [2] = h(X) mark(f(X)) = [1 2 4] [7] [0 1 2] X + [6] [0 0 0] [0] >= [1 2 2] [7] [0 1 2] X + [6] [0 0 0] [0] = active(f(mark(X))) mark(h(X)) = [1 2 4] [6] [0 1 2] X + [4] [0 0 0] [0] >= [1 2 2] [6] [0 1 2] X + [4] [0 0 0] [0] = active(h(mark(X))) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(f(X)) -> mark(g(h(f(X)))) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) h(active(X)) -> h(X) h(mark(X)) -> h(X) mark(f(X)) -> active(f(mark(X))) mark(g(X)) -> active(g(X)) mark(h(X)) -> active(h(mark(X))) - Signature: {active/1,f/1,g/1,h/1,mark/1} / {} - Obligation: derivational complexity wrt. signature {active,f,g,h,mark} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))