/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: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(a()) -> a() mark(c(X)) -> c(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: derivational complexity wrt. signature {a,a__f,c,f,g,mark} + 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(a) = [2] p(a__f) = [1] x1 + [0] p(c) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [8] Following rules are strictly oriented: mark(a()) = [10] > [2] = a() mark(c(X)) = [1] X + [8] > [1] X + [0] = c(X) Following rules are (at-least) weakly oriented: a__f(X) = [1] X + [0] >= [1] X + [0] = f(X) a__f(f(a())) = [2] >= [2] = c(f(g(f(a())))) mark(f(X)) = [1] X + [8] >= [1] X + [8] = a__f(mark(X)) mark(g(X)) = [1] X + [8] >= [1] X + [8] = g(mark(X)) * Step 2: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(mark(X)) - Weak TRS: mark(a()) -> a() mark(c(X)) -> c(X) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: derivational complexity wrt. signature {a,a__f,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(a) = [0] p(a__f) = [1] x1 + [7] p(c) = [1] x1 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [0] p(mark) = [1] x1 + [0] Following rules are strictly oriented: a__f(X) = [1] X + [7] > [1] X + [0] = f(X) a__f(f(a())) = [7] > [0] = c(f(g(f(a())))) Following rules are (at-least) weakly oriented: mark(a()) = [0] >= [0] = a() mark(c(X)) = [1] X + [0] >= [1] X + [0] = c(X) mark(f(X)) = [1] X + [0] >= [1] X + [7] = a__f(mark(X)) mark(g(X)) = [1] X + [0] >= [1] X + [0] = 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: mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(mark(X)) - Weak TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(a()) -> a() mark(c(X)) -> c(X) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: derivational complexity wrt. signature {a,a__f,c,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(a__f) = [1 0] x1 + [7] [0 1] [4] p(c) = [1 0] x1 + [0] [0 0] [1] p(f) = [1 0] x1 + [0] [0 1] [4] p(g) = [1 1] x1 + [3] [0 1] [0] p(mark) = [1 2] x1 + [0] [0 1] [0] Following rules are strictly oriented: mark(f(X)) = [1 2] X + [8] [0 1] [4] > [1 2] X + [7] [0 1] [4] = a__f(mark(X)) Following rules are (at-least) weakly oriented: a__f(X) = [1 0] X + [7] [0 1] [4] >= [1 0] X + [0] [0 1] [4] = f(X) a__f(f(a())) = [8] [8] >= [8] [1] = c(f(g(f(a())))) mark(a()) = [1] [0] >= [1] [0] = a() mark(c(X)) = [1 0] X + [2] [0 0] [1] >= [1 0] X + [0] [0 0] [1] = c(X) mark(g(X)) = [1 3] X + [3] [0 1] [0] >= [1 3] X + [3] [0 1] [0] = g(mark(X)) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: mark(g(X)) -> g(mark(X)) - Weak TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(a()) -> a() mark(c(X)) -> c(X) mark(f(X)) -> a__f(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: derivational complexity wrt. signature {a,a__f,c,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) = [0] [0] p(a__f) = [1 0] x1 + [0] [0 1] [0] p(c) = [1 0] x1 + [0] [0 0] [0] p(f) = [1 0] x1 + [0] [0 1] [0] p(g) = [1 0] x1 + [0] [0 1] [1] p(mark) = [1 4] x1 + [4] [0 1] [4] Following rules are strictly oriented: mark(g(X)) = [1 4] X + [8] [0 1] [5] > [1 4] X + [4] [0 1] [5] = g(mark(X)) Following rules are (at-least) weakly oriented: a__f(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = f(X) a__f(f(a())) = [0] [0] >= [0] [0] = c(f(g(f(a())))) mark(a()) = [4] [4] >= [0] [0] = a() mark(c(X)) = [1 0] X + [4] [0 0] [4] >= [1 0] X + [0] [0 0] [0] = c(X) mark(f(X)) = [1 4] X + [4] [0 1] [4] >= [1 4] X + [4] [0 1] [4] = a__f(mark(X)) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__f(X) -> f(X) a__f(f(a())) -> c(f(g(f(a())))) mark(a()) -> a() mark(c(X)) -> c(X) mark(f(X)) -> a__f(mark(X)) mark(g(X)) -> g(mark(X)) - Signature: {a__f/1,mark/1} / {a/0,c/1,f/1,g/1} - Obligation: derivational complexity wrt. signature {a,a__f,c,f,g,mark} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))