/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: U11(tt()) -> U12(tt()) U12(tt()) -> tt() __(X,nil()) -> X __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil(),X) -> X activate(X) -> X isNePal(__(I,__(P,I))) -> U11(tt()) - Signature: {U11/1,U12/1,__/2,activate/1,isNePal/1} / {nil/0,tt/0} - Obligation: derivational complexity wrt. signature {U11,U12,__,activate,isNePal,nil,tt} + 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(U11) = [1] x1 + [0] p(U12) = [1] x1 + [0] p(__) = [1] x1 + [1] x2 + [0] p(activate) = [1] x1 + [0] p(isNePal) = [1] x1 + [1] p(nil) = [0] p(tt) = [0] Following rules are strictly oriented: isNePal(__(I,__(P,I))) = [2] I + [1] P + [1] > [0] = U11(tt()) Following rules are (at-least) weakly oriented: U11(tt()) = [0] >= [0] = U12(tt()) U12(tt()) = [0] >= [0] = tt() __(X,nil()) = [1] X + [0] >= [1] X + [0] = X __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [0] >= [1] X + [1] Y + [1] Z + [0] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [0] >= [1] X + [0] = X activate(X) = [1] X + [0] >= [1] X + [0] = 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: U11(tt()) -> U12(tt()) U12(tt()) -> tt() __(X,nil()) -> X __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil(),X) -> X activate(X) -> X - Weak TRS: isNePal(__(I,__(P,I))) -> U11(tt()) - Signature: {U11/1,U12/1,__/2,activate/1,isNePal/1} / {nil/0,tt/0} - Obligation: derivational complexity wrt. signature {U11,U12,__,activate,isNePal,nil,tt} + 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(U11) = [1] x1 + [13] p(U12) = [1] x1 + [12] p(__) = [1] x1 + [1] x2 + [9] p(activate) = [1] x1 + [0] p(isNePal) = [1] x1 + [5] p(nil) = [5] p(tt) = [10] Following rules are strictly oriented: U11(tt()) = [23] > [22] = U12(tt()) U12(tt()) = [22] > [10] = tt() __(X,nil()) = [1] X + [14] > [1] X + [0] = X __(nil(),X) = [1] X + [14] > [1] X + [0] = X Following rules are (at-least) weakly oriented: __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [18] >= [1] X + [1] Y + [1] Z + [18] = __(X,__(Y,Z)) activate(X) = [1] X + [0] >= [1] X + [0] = X isNePal(__(I,__(P,I))) = [2] I + [1] P + [23] >= [23] = U11(tt()) * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: __(__(X,Y),Z) -> __(X,__(Y,Z)) activate(X) -> X - Weak TRS: U11(tt()) -> U12(tt()) U12(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X isNePal(__(I,__(P,I))) -> U11(tt()) - Signature: {U11/1,U12/1,__/2,activate/1,isNePal/1} / {nil/0,tt/0} - Obligation: derivational complexity wrt. signature {U11,U12,__,activate,isNePal,nil,tt} + 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(U11) = [1] x1 + [6] p(U12) = [1] x1 + [6] p(__) = [1] x1 + [1] x2 + [8] p(activate) = [1] x1 + [9] p(isNePal) = [1] x1 + [4] p(nil) = [6] p(tt) = [4] Following rules are strictly oriented: activate(X) = [1] X + [9] > [1] X + [0] = X Following rules are (at-least) weakly oriented: U11(tt()) = [10] >= [10] = U12(tt()) U12(tt()) = [10] >= [4] = tt() __(X,nil()) = [1] X + [14] >= [1] X + [0] = X __(__(X,Y),Z) = [1] X + [1] Y + [1] Z + [16] >= [1] X + [1] Y + [1] Z + [16] = __(X,__(Y,Z)) __(nil(),X) = [1] X + [14] >= [1] X + [0] = X isNePal(__(I,__(P,I))) = [2] I + [1] P + [20] >= [10] = U11(tt()) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: __(__(X,Y),Z) -> __(X,__(Y,Z)) - Weak TRS: U11(tt()) -> U12(tt()) U12(tt()) -> tt() __(X,nil()) -> X __(nil(),X) -> X activate(X) -> X isNePal(__(I,__(P,I))) -> U11(tt()) - Signature: {U11/1,U12/1,__/2,activate/1,isNePal/1} / {nil/0,tt/0} - Obligation: derivational complexity wrt. signature {U11,U12,__,activate,isNePal,nil,tt} + 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(U11) = [1 1] x1 + [6] [0 1] [2] p(U12) = [1 0] x1 + [3] [0 1] [0] p(__) = [1 1] x1 + [1 0] x2 + [0] [0 1] [0 1] [1] p(activate) = [1 2] x1 + [0] [0 1] [0] p(isNePal) = [1 2] x1 + [4] [0 1] [1] p(nil) = [4] [2] p(tt) = [1] [0] Following rules are strictly oriented: __(__(X,Y),Z) = [1 2] X + [1 1] Y + [1 0] Z + [1] [0 1] [0 1] [0 1] [2] > [1 1] X + [1 1] Y + [1 0] Z + [0] [0 1] [0 1] [0 1] [2] = __(X,__(Y,Z)) Following rules are (at-least) weakly oriented: U11(tt()) = [7] [2] >= [4] [0] = U12(tt()) U12(tt()) = [4] [0] >= [1] [0] = tt() __(X,nil()) = [1 1] X + [4] [0 1] [3] >= [1 0] X + [0] [0 1] [0] = X __(nil(),X) = [1 0] X + [6] [0 1] [3] >= [1 0] X + [0] [0 1] [0] = X activate(X) = [1 2] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X isNePal(__(I,__(P,I))) = [2 5] I + [1 3] P + [8] [0 2] [0 1] [3] >= [7] [2] = U11(tt()) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: U11(tt()) -> U12(tt()) U12(tt()) -> tt() __(X,nil()) -> X __(__(X,Y),Z) -> __(X,__(Y,Z)) __(nil(),X) -> X activate(X) -> X isNePal(__(I,__(P,I))) -> U11(tt()) - Signature: {U11/1,U12/1,__/2,activate/1,isNePal/1} / {nil/0,tt/0} - Obligation: derivational complexity wrt. signature {U11,U12,__,activate,isNePal,nil,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))