/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: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) - Signature: {activate/1,and/2,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,activate,and,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(activate) = [1] x1 + [1] p(and) = [1] x1 + [1] x2 + [1] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [0] p(tt) = [0] Following rules are strictly oriented: activate(X) = [1] X + [1] > [1] X + [0] = X Following rules are (at-least) weakly oriented: and(tt(),X) = [1] X + [1] >= [1] X + [1] = activate(X) plus(N,0()) = [1] N + [0] >= [1] N + [0] = N plus(N,s(M)) = [1] M + [1] N + [0] >= [1] M + [1] N + [0] = s(plus(N,M)) 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: and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) - Weak TRS: activate(X) -> X - Signature: {activate/1,and/2,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,activate,and,plus,s,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(0) = [0] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [8] p(plus) = [1] x1 + [1] x2 + [0] p(s) = [1] x1 + [0] p(tt) = [0] Following rules are strictly oriented: and(tt(),X) = [1] X + [8] > [1] X + [0] = activate(X) Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X plus(N,0()) = [1] N + [0] >= [1] N + [0] = N plus(N,s(M)) = [1] M + [1] N + [0] >= [1] M + [1] N + [0] = s(plus(N,M)) * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) - Signature: {activate/1,and/2,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,activate,and,plus,s,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(0) = [13] p(activate) = [1] x1 + [0] p(and) = [1] x1 + [1] x2 + [3] p(plus) = [1] x1 + [1] x2 + [8] p(s) = [1] x1 + [9] p(tt) = [5] Following rules are strictly oriented: plus(N,0()) = [1] N + [21] > [1] N + [0] = N Following rules are (at-least) weakly oriented: activate(X) = [1] X + [0] >= [1] X + [0] = X and(tt(),X) = [1] X + [8] >= [1] X + [0] = activate(X) plus(N,s(M)) = [1] M + [1] N + [17] >= [1] M + [1] N + [17] = s(plus(N,M)) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: plus(N,s(M)) -> s(plus(N,M)) - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N - Signature: {activate/1,and/2,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,activate,and,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) = [2] [4] p(activate) = [1 0] x1 + [8] [0 1] [2] p(and) = [1 7] x1 + [1 0] x2 + [5] [0 0] [0 1] [10] p(plus) = [1 0] x1 + [1 2] x2 + [0] [0 1] [0 1] [8] p(s) = [1 0] x1 + [2] [0 1] [8] p(tt) = [0] [3] Following rules are strictly oriented: plus(N,s(M)) = [1 2] M + [1 0] N + [18] [0 1] [0 1] [16] > [1 2] M + [1 0] N + [2] [0 1] [0 1] [16] = s(plus(N,M)) Following rules are (at-least) weakly oriented: activate(X) = [1 0] X + [8] [0 1] [2] >= [1 0] X + [0] [0 1] [0] = X and(tt(),X) = [1 0] X + [26] [0 1] [10] >= [1 0] X + [8] [0 1] [2] = activate(X) plus(N,0()) = [1 0] N + [10] [0 1] [12] >= [1 0] N + [0] [0 1] [0] = N * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: activate(X) -> X and(tt(),X) -> activate(X) plus(N,0()) -> N plus(N,s(M)) -> s(plus(N,M)) - Signature: {activate/1,and/2,plus/2} / {0/0,s/1,tt/0} - Obligation: derivational complexity wrt. signature {0,activate,and,plus,s,tt} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))