/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: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) norm(g(x,y)) -> s(norm(x)) norm(nil()) -> 0() rem(g(x,y),0()) -> g(x,y) rem(g(x,y),s(z)) -> rem(x,z) rem(nil(),y) -> nil() - Signature: {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1} - Obligation: derivational complexity wrt. signature {0,f,g,nil,norm,rem,s} + 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(0) = [4] p(f) = [1] x1 + [1] x2 + [6] p(g) = [1] x1 + [1] x2 + [0] p(nil) = [0] p(norm) = [1] x1 + [0] p(rem) = [1] x1 + [1] x2 + [13] p(s) = [1] x1 + [3] Following rules are strictly oriented: f(x,nil()) = [1] x + [6] > [1] x + [0] = g(nil(),x) rem(g(x,y),0()) = [1] x + [1] y + [17] > [1] x + [1] y + [0] = g(x,y) rem(g(x,y),s(z)) = [1] x + [1] y + [1] z + [16] > [1] x + [1] z + [13] = rem(x,z) rem(nil(),y) = [1] y + [13] > [0] = nil() Following rules are (at-least) weakly oriented: f(x,g(y,z)) = [1] x + [1] y + [1] z + [6] >= [1] x + [1] y + [1] z + [6] = g(f(x,y),z) norm(g(x,y)) = [1] x + [1] y + [0] >= [1] x + [3] = s(norm(x)) norm(nil()) = [0] >= [4] = 0() 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: f(x,g(y,z)) -> g(f(x,y),z) norm(g(x,y)) -> s(norm(x)) norm(nil()) -> 0() - Weak TRS: f(x,nil()) -> g(nil(),x) rem(g(x,y),0()) -> g(x,y) rem(g(x,y),s(z)) -> rem(x,z) rem(nil(),y) -> nil() - Signature: {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1} - Obligation: derivational complexity wrt. signature {0,f,g,nil,norm,rem,s} + 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(0) = [0] p(f) = [1] x1 + [1] x2 + [13] p(g) = [1] x1 + [1] x2 + [13] p(nil) = [1] p(norm) = [1] x1 + [12] p(rem) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [2] Following rules are strictly oriented: norm(g(x,y)) = [1] x + [1] y + [25] > [1] x + [14] = s(norm(x)) norm(nil()) = [13] > [0] = 0() Following rules are (at-least) weakly oriented: f(x,g(y,z)) = [1] x + [1] y + [1] z + [26] >= [1] x + [1] y + [1] z + [26] = g(f(x,y),z) f(x,nil()) = [1] x + [14] >= [1] x + [14] = g(nil(),x) rem(g(x,y),0()) = [1] x + [1] y + [13] >= [1] x + [1] y + [13] = g(x,y) rem(g(x,y),s(z)) = [1] x + [1] y + [1] z + [15] >= [1] x + [1] z + [0] = rem(x,z) rem(nil(),y) = [1] y + [1] >= [1] = nil() 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: f(x,g(y,z)) -> g(f(x,y),z) - Weak TRS: f(x,nil()) -> g(nil(),x) norm(g(x,y)) -> s(norm(x)) norm(nil()) -> 0() rem(g(x,y),0()) -> g(x,y) rem(g(x,y),s(z)) -> rem(x,z) rem(nil(),y) -> nil() - Signature: {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1} - Obligation: derivational complexity wrt. signature {0,f,g,nil,norm,rem,s} + 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(0) = [0] [1] p(f) = [1 4] x1 + [1 2] x2 + [1] [0 0] [0 1] [4] p(g) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [1] p(nil) = [4] [3] p(norm) = [1 0] x1 + [5] [0 1] [0] p(rem) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [6] p(s) = [1 0] x1 + [0] [0 0] [1] Following rules are strictly oriented: f(x,g(y,z)) = [1 4] x + [1 2] y + [1 0] z + [3] [0 0] [0 1] [0 0] [5] > [1 4] x + [1 2] y + [1 0] z + [1] [0 0] [0 1] [0 0] [5] = g(f(x,y),z) Following rules are (at-least) weakly oriented: f(x,nil()) = [1 4] x + [11] [0 0] [7] >= [1 0] x + [4] [0 0] [4] = g(nil(),x) norm(g(x,y)) = [1 0] x + [1 0] y + [5] [0 1] [0 0] [1] >= [1 0] x + [5] [0 0] [1] = s(norm(x)) norm(nil()) = [9] [3] >= [0] [1] = 0() rem(g(x,y),0()) = [1 0] x + [1 0] y + [0] [0 1] [0 0] [7] >= [1 0] x + [1 0] y + [0] [0 1] [0 0] [1] = g(x,y) rem(g(x,y),s(z)) = [1 0] x + [1 0] y + [1 0] z + [0] [0 1] [0 0] [0 0] [7] >= [1 0] x + [1 0] z + [0] [0 1] [0 0] [6] = rem(x,z) rem(nil(),y) = [1 0] y + [4] [0 0] [9] >= [4] [3] = nil() * Step 4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(x,g(y,z)) -> g(f(x,y),z) f(x,nil()) -> g(nil(),x) norm(g(x,y)) -> s(norm(x)) norm(nil()) -> 0() rem(g(x,y),0()) -> g(x,y) rem(g(x,y),s(z)) -> rem(x,z) rem(nil(),y) -> nil() - Signature: {f/2,norm/1,rem/2} / {0/0,g/2,nil/0,s/1} - Obligation: derivational complexity wrt. signature {0,f,g,nil,norm,rem,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))