/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(f(a()))) -> mark(f(g(f(a())))) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(f(X)) -> active(f(X)) mark(g(X)) -> active(g(mark(X))) - Signature: {active/1,f/1,g/1,mark/1} / {a/0} - Obligation: derivational complexity wrt. signature {a,active,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(a) = [2] p(active) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [6] p(mark) = [1] x1 + [1] Following rules are strictly oriented: f(mark(X)) = [1] X + [1] > [1] X + [0] = f(X) g(mark(X)) = [1] X + [7] > [1] X + [6] = g(X) mark(a()) = [3] > [2] = active(a()) mark(f(X)) = [1] X + [1] > [1] X + [0] = active(f(X)) Following rules are (at-least) weakly oriented: active(f(f(a()))) = [2] >= [9] = mark(f(g(f(a())))) f(active(X)) = [1] X + [0] >= [1] X + [0] = f(X) g(active(X)) = [1] X + [6] >= [1] X + [6] = g(X) mark(g(X)) = [1] X + [7] >= [1] X + [7] = active(g(mark(X))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: active(f(f(a()))) -> mark(f(g(f(a())))) f(active(X)) -> f(X) g(active(X)) -> g(X) mark(g(X)) -> active(g(mark(X))) - Weak TRS: f(mark(X)) -> f(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(f(X)) -> active(f(X)) - Signature: {active/1,f/1,g/1,mark/1} / {a/0} - Obligation: derivational complexity wrt. signature {a,active,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(a) = [1] p(active) = [1] x1 + [3] p(f) = [1] x1 + [1] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [3] Following rules are strictly oriented: f(active(X)) = [1] X + [4] > [1] X + [1] = f(X) g(active(X)) = [1] X + [3] > [1] X + [0] = g(X) Following rules are (at-least) weakly oriented: active(f(f(a()))) = [6] >= [6] = mark(f(g(f(a())))) f(mark(X)) = [1] X + [4] >= [1] X + [1] = f(X) g(mark(X)) = [1] X + [3] >= [1] X + [0] = g(X) mark(a()) = [4] >= [4] = active(a()) mark(f(X)) = [1] X + [4] >= [1] X + [4] = active(f(X)) mark(g(X)) = [1] X + [3] >= [1] X + [6] = active(g(mark(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: active(f(f(a()))) -> mark(f(g(f(a())))) mark(g(X)) -> active(g(mark(X))) - Weak TRS: f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(f(X)) -> active(f(X)) - Signature: {active/1,f/1,g/1,mark/1} / {a/0} - Obligation: derivational complexity wrt. signature {a,active,f,g,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(a) = [6] [0] p(active) = [1 0] x1 + [0] [0 1] [0] p(f) = [1 4] x1 + [1] [0 0] [1] p(g) = [1 2] x1 + [0] [0 0] [0] p(mark) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: active(f(f(a()))) = [12] [1] > [10] [1] = mark(f(g(f(a())))) Following rules are (at-least) weakly oriented: f(active(X)) = [1 4] X + [1] [0 0] [1] >= [1 4] X + [1] [0 0] [1] = f(X) f(mark(X)) = [1 4] X + [1] [0 0] [1] >= [1 4] X + [1] [0 0] [1] = f(X) g(active(X)) = [1 2] X + [0] [0 0] [0] >= [1 2] X + [0] [0 0] [0] = g(X) g(mark(X)) = [1 2] X + [0] [0 0] [0] >= [1 2] X + [0] [0 0] [0] = g(X) mark(a()) = [6] [0] >= [6] [0] = active(a()) mark(f(X)) = [1 4] X + [1] [0 0] [1] >= [1 4] X + [1] [0 0] [1] = active(f(X)) mark(g(X)) = [1 2] X + [0] [0 0] [0] >= [1 2] X + [0] [0 0] [0] = active(g(mark(X))) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(g(X)) -> active(g(mark(X))) - Weak TRS: active(f(f(a()))) -> mark(f(g(f(a())))) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(f(X)) -> active(f(X)) - Signature: {active/1,f/1,g/1,mark/1} / {a/0} - Obligation: derivational complexity wrt. signature {a,active,f,g,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(a) = [1] [0] p(active) = [1 0] x1 + [0] [0 1] [0] p(f) = [1 0] x1 + [4] [0 0] [0] p(g) = [1 0] x1 + [0] [0 1] [1] p(mark) = [1 2] x1 + [0] [0 1] [0] Following rules are strictly oriented: mark(g(X)) = [1 2] X + [2] [0 1] [1] > [1 2] X + [0] [0 1] [1] = active(g(mark(X))) Following rules are (at-least) weakly oriented: active(f(f(a()))) = [9] [0] >= [9] [0] = mark(f(g(f(a())))) f(active(X)) = [1 0] X + [4] [0 0] [0] >= [1 0] X + [4] [0 0] [0] = f(X) f(mark(X)) = [1 2] X + [4] [0 0] [0] >= [1 0] X + [4] [0 0] [0] = f(X) g(active(X)) = [1 0] X + [0] [0 1] [1] >= [1 0] X + [0] [0 1] [1] = g(X) g(mark(X)) = [1 2] X + [0] [0 1] [1] >= [1 0] X + [0] [0 1] [1] = g(X) mark(a()) = [1] [0] >= [1] [0] = active(a()) mark(f(X)) = [1 0] X + [4] [0 0] [0] >= [1 0] X + [4] [0 0] [0] = active(f(X)) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: active(f(f(a()))) -> mark(f(g(f(a())))) f(active(X)) -> f(X) f(mark(X)) -> f(X) g(active(X)) -> g(X) g(mark(X)) -> g(X) mark(a()) -> active(a()) mark(f(X)) -> active(f(X)) mark(g(X)) -> active(g(mark(X))) - Signature: {active/1,f/1,g/1,mark/1} / {a/0} - Obligation: derivational complexity wrt. signature {a,active,f,g,mark} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))