/export/starexec/sandbox/solver/bin/starexec_run_tct_dci /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,y),z)) -> f(.(x,.(y,z))) f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost derivational complexity wrt. signature {.,f,g,nil} + 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(.) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [0] p(g) = [1] x1 + [9] p(nil) = [0] Following rules are strictly oriented: g(nil()) = [9] > [0] = nil() Following rules are (at-least) weakly oriented: f(.(.(x,y),z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = f(.(x,.(y,z))) f(.(nil(),y)) = [1] y + [0] >= [1] y + [0] = .(nil(),f(y)) f(nil()) = [0] >= [0] = nil() g(.(x,.(y,z))) = [1] x + [1] y + [1] z + [9] >= [1] x + [1] y + [1] z + [9] = g(.(.(x,y),z)) g(.(x,nil())) = [1] x + [9] >= [1] x + [9] = .(g(x),nil()) 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,y),z)) -> f(.(x,.(y,z))) f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) - Weak TRS: g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost derivational complexity wrt. signature {.,f,g,nil} + 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(.) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [8] p(g) = [1] x1 + [1] p(nil) = [0] Following rules are strictly oriented: f(nil()) = [8] > [0] = nil() Following rules are (at-least) weakly oriented: f(.(.(x,y),z)) = [1] x + [1] y + [1] z + [8] >= [1] x + [1] y + [1] z + [8] = f(.(x,.(y,z))) f(.(nil(),y)) = [1] y + [8] >= [1] y + [8] = .(nil(),f(y)) g(.(x,.(y,z))) = [1] x + [1] y + [1] z + [1] >= [1] x + [1] y + [1] z + [1] = g(.(.(x,y),z)) g(.(x,nil())) = [1] x + [1] >= [1] x + [1] = .(g(x),nil()) g(nil()) = [1] >= [0] = 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,y),z)) -> f(.(x,.(y,z))) f(.(nil(),y)) -> .(nil(),f(y)) g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) - Weak TRS: f(nil()) -> nil() g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost derivational complexity wrt. signature {.,f,g,nil} + 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(.) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [1] p(f) = [1 2] x1 + [6] [0 1] [0] p(g) = [1 0] x1 + [0] [0 1] [0] p(nil) = [0] [0] Following rules are strictly oriented: f(.(nil(),y)) = [1 2] y + [8] [0 1] [1] > [1 2] y + [6] [0 1] [1] = .(nil(),f(y)) Following rules are (at-least) weakly oriented: f(.(.(x,y),z)) = [1 2] x + [1 2] y + [1 2] z + [10] [0 1] [0 1] [0 1] [2] >= [1 2] x + [1 2] y + [1 2] z + [10] [0 1] [0 1] [0 1] [2] = f(.(x,.(y,z))) f(nil()) = [6] [0] >= [0] [0] = nil() g(.(x,.(y,z))) = [1 0] x + [1 0] y + [1 0] z + [0] [0 1] [0 1] [0 1] [2] >= [1 0] x + [1 0] y + [1 0] z + [0] [0 1] [0 1] [0 1] [2] = g(.(.(x,y),z)) g(.(x,nil())) = [1 0] x + [0] [0 1] [1] >= [1 0] x + [0] [0 1] [1] = .(g(x),nil()) g(nil()) = [0] [0] >= [0] [0] = nil() * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(.(.(x,y),z)) -> f(.(x,.(y,z))) g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) - Weak TRS: f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost derivational complexity wrt. signature {.,f,g,nil} + 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(.) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 1] [4] p(f) = [1 0] x1 + [0] [0 1] [2] p(g) = [1 1] x1 + [0] [0 1] [1] p(nil) = [7] [4] Following rules are strictly oriented: g(.(x,nil())) = [1 1] x + [15] [0 1] [9] > [1 1] x + [7] [0 1] [9] = .(g(x),nil()) Following rules are (at-least) weakly oriented: f(.(.(x,y),z)) = [1 0] x + [1 0] y + [1 0] z + [0] [0 1] [0 1] [0 1] [10] >= [1 0] x + [1 0] y + [1 0] z + [0] [0 1] [0 1] [0 1] [10] = f(.(x,.(y,z))) f(.(nil(),y)) = [1 0] y + [7] [0 1] [10] >= [1 0] y + [7] [0 1] [10] = .(nil(),f(y)) f(nil()) = [7] [6] >= [7] [4] = nil() g(.(x,.(y,z))) = [1 1] x + [1 1] y + [1 1] z + [8] [0 1] [0 1] [0 1] [9] >= [1 1] x + [1 1] y + [1 1] z + [8] [0 1] [0 1] [0 1] [9] = g(.(.(x,y),z)) g(nil()) = [11] [5] >= [7] [4] = nil() * Step 5: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(.(.(x,y),z)) -> f(.(x,.(y,z))) g(.(x,.(y,z))) -> g(.(.(x,y),z)) - Weak TRS: f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(.(x,nil())) -> .(g(x),nil()) g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost derivational complexity wrt. signature {.,f,g,nil} + Applied Processor: MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton (Just 2): Following symbols are considered usable: all TcT has computed the following interpretation: p(.) = [1 0 0] [1 0 0] [0] [0 1 0] x_1 + [0 1 0] x_2 + [1] [0 0 1] [0 1 0] [0] p(f) = [1 0 0] [0] [0 1 0] x_1 + [3] [0 1 0] [1] p(g) = [1 0 2] [0] [0 1 0] x_1 + [0] [0 2 0] [0] p(nil) = [0] [1] [0] Following rules are strictly oriented: g(.(x,.(y,z))) = [1 0 2] [1 2 0] [1 2 0] [2] [0 1 0] x + [0 1 0] y + [0 1 0] z + [2] [0 2 0] [0 2 0] [0 2 0] [4] > [1 0 2] [1 2 0] [1 2 0] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [2] [0 2 0] [0 2 0] [0 2 0] [4] = g(.(.(x,y),z)) Following rules are (at-least) weakly oriented: f(.(.(x,y),z)) = [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [5] [0 1 0] [0 1 0] [0 1 0] [3] >= [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [5] [0 1 0] [0 1 0] [0 1 0] [3] = f(.(x,.(y,z))) f(.(nil(),y)) = [1 0 0] [0] [0 1 0] y + [5] [0 1 0] [3] >= [1 0 0] [0] [0 1 0] y + [5] [0 1 0] [3] = .(nil(),f(y)) f(nil()) = [0] [4] [2] >= [0] [1] [0] = nil() g(.(x,nil())) = [1 0 2] [2] [0 1 0] x + [2] [0 2 0] [4] >= [1 0 2] [0] [0 1 0] x + [2] [0 2 0] [1] = .(g(x),nil()) g(nil()) = [0] [1] [2] >= [0] [1] [0] = nil() * Step 6: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(.(.(x,y),z)) -> f(.(x,.(y,z))) - Weak TRS: f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost derivational complexity wrt. signature {.,f,g,nil} + Applied Processor: MI {miKind = Automaton (Just 2), miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton (Just 2): Following symbols are considered usable: all TcT has computed the following interpretation: p(.) = [1 1 0] [1 0 0] [0] [0 0 0] x_1 + [0 1 0] x_2 + [1] [0 0 1] [0 0 1] [0] p(f) = [1 0 0] [0] [0 0 1] x_1 + [0] [0 0 1] [0] p(g) = [1 1 0] [0] [0 0 0] x_1 + [1] [0 0 1] [0] p(nil) = [0] [0] [2] Following rules are strictly oriented: f(.(.(x,y),z)) = [1 1 0] [1 1 0] [1 0 0] [1] [0 0 1] x + [0 0 1] y + [0 0 1] z + [0] [0 0 1] [0 0 1] [0 0 1] [0] > [1 1 0] [1 1 0] [1 0 0] [0] [0 0 1] x + [0 0 1] y + [0 0 1] z + [0] [0 0 1] [0 0 1] [0 0 1] [0] = f(.(x,.(y,z))) Following rules are (at-least) weakly oriented: f(.(nil(),y)) = [1 0 0] [0] [0 0 1] y + [2] [0 0 1] [2] >= [1 0 0] [0] [0 0 1] y + [1] [0 0 1] [2] = .(nil(),f(y)) f(nil()) = [0] [2] [2] >= [0] [0] [2] = nil() g(.(x,.(y,z))) = [1 1 0] [1 1 0] [1 1 0] [2] [0 0 0] x + [0 0 0] y + [0 0 0] z + [1] [0 0 1] [0 0 1] [0 0 1] [0] >= [1 1 0] [1 1 0] [1 1 0] [2] [0 0 0] x + [0 0 0] y + [0 0 0] z + [1] [0 0 1] [0 0 1] [0 0 1] [0] = g(.(.(x,y),z)) g(.(x,nil())) = [1 1 0] [1] [0 0 0] x + [1] [0 0 1] [2] >= [1 1 0] [1] [0 0 0] x + [1] [0 0 1] [2] = .(g(x),nil()) g(nil()) = [0] [1] [2] >= [0] [0] [2] = nil() * Step 7: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: f(.(.(x,y),z)) -> f(.(x,.(y,z))) f(.(nil(),y)) -> .(nil(),f(y)) f(nil()) -> nil() g(.(x,.(y,z))) -> g(.(.(x,y),z)) g(.(x,nil())) -> .(g(x),nil()) g(nil()) -> nil() - Signature: {f/1,g/1} / {./2,nil/0} - Obligation: innermost derivational complexity wrt. signature {.,f,g,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))