/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(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,U11,U12,activate,plus,s,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(0) = [0] p(U11) = [1] x1 + [1] x2 + [1] x3 + [0] p(U12) = [1] x1 + [1] x2 + [1] x3 + [0] p(activate) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [7] p(s) = [1] x1 + [0] p(tt) = [0] Following rules are strictly oriented: plus(N,0()) = [1] N + [7] > [1] N + [0] = N plus(N,s(M)) = [1] M + [1] N + [7] > [1] M + [1] N + [0] = U11(tt(),M,N) Following rules are (at-least) weakly oriented: U11(tt(),M,N) = [1] M + [1] N + [0] >= [1] M + [1] N + [0] = U12(tt(),activate(M),activate(N)) U12(tt(),M,N) = [1] M + [1] N + [0] >= [1] M + [1] N + [7] = s(plus(activate(N),activate(M))) 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: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X - Weak TRS: plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,U11,U12,activate,plus,s,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(0) = [0] p(U11) = [1] x1 + [1] x2 + [1] x3 + [0] p(U12) = [1] x1 + [1] x2 + [1] x3 + [9] p(activate) = [1] x1 + [11] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [0] p(tt) = [0] Following rules are strictly oriented: activate(X) = [1] X + [11] > [1] X + [0] = X Following rules are (at-least) weakly oriented: U11(tt(),M,N) = [1] M + [1] N + [0] >= [1] M + [1] N + [31] = U12(tt(),activate(M),activate(N)) U12(tt(),M,N) = [1] M + [1] N + [9] >= [1] M + [1] N + [22] = s(plus(activate(N),activate(M))) plus(N,0()) = [1] N + [0] >= [1] N + [0] = N plus(N,s(M)) = [1] M + [1] N + [0] >= [1] M + [1] N + [0] = U11(tt(),M,N) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) - Weak TRS: activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,U11,U12,activate,plus,s,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(0) = [9] p(U11) = [1] x1 + [1] x2 + [1] x3 + [0] p(U12) = [1] x1 + [1] x2 + [1] x3 + [8] p(activate) = [1] x1 + [0] p(plus) = [1] x1 + [1] x2 + [14] p(s) = [1] x1 + [8] p(tt) = [15] Following rules are strictly oriented: U12(tt(),M,N) = [1] M + [1] N + [23] > [1] M + [1] N + [22] = s(plus(activate(N),activate(M))) Following rules are (at-least) weakly oriented: U11(tt(),M,N) = [1] M + [1] N + [15] >= [1] M + [1] N + [23] = U12(tt(),activate(M),activate(N)) activate(X) = [1] X + [0] >= [1] X + [0] = X plus(N,0()) = [1] N + [23] >= [1] N + [0] = N plus(N,s(M)) = [1] M + [1] N + [22] >= [1] M + [1] N + [15] = U11(tt(),M,N) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) - Weak TRS: U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,U11,U12,activate,plus,s,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(0) = [0] [2] p(U11) = [1 0] x1 + [1 2] x2 + [1 0] x3 + [5] [0 1] [0 1] [0 1] [1] p(U12) = [1 1] x1 + [1 2] x2 + [1 0] x3 + [2] [0 1] [0 1] [0 1] [1] p(activate) = [1 0] x1 + [0] [0 1] [0] p(plus) = [1 0] x1 + [1 2] x2 + [7] [0 1] [0 1] [1] p(s) = [1 0] x1 + [1] [0 1] [1] p(tt) = [5] [1] Following rules are strictly oriented: U11(tt(),M,N) = [1 2] M + [1 0] N + [10] [0 1] [0 1] [2] > [1 2] M + [1 0] N + [8] [0 1] [0 1] [2] = U12(tt(),activate(M),activate(N)) Following rules are (at-least) weakly oriented: U12(tt(),M,N) = [1 2] M + [1 0] N + [8] [0 1] [0 1] [2] >= [1 2] M + [1 0] N + [8] [0 1] [0 1] [2] = s(plus(activate(N),activate(M))) activate(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X plus(N,0()) = [1 0] N + [11] [0 1] [3] >= [1 0] N + [0] [0 1] [0] = N plus(N,s(M)) = [1 2] M + [1 0] N + [10] [0 1] [0 1] [2] >= [1 2] M + [1 0] N + [10] [0 1] [0 1] [2] = U11(tt(),M,N) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: U11(tt(),M,N) -> U12(tt(),activate(M),activate(N)) U12(tt(),M,N) -> s(plus(activate(N),activate(M))) activate(X) -> X plus(N,0()) -> N plus(N,s(M)) -> U11(tt(),M,N) - Signature: {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,U11,U12,activate,plus,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))