/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: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: derivational complexity wrt. signature {0,activate,cons,f,n__0,n__f,n__s,p,s} + 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) = [4] p(activate) = [1] x1 + [4] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [0] p(n__0) = [0] p(n__f) = [1] x1 + [0] p(n__s) = [1] x1 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: 0() = [4] > [0] = n__0() activate(X) = [1] X + [4] > [1] X + [0] = X Following rules are (at-least) weakly oriented: activate(n__0()) = [4] >= [4] = 0() activate(n__f(X)) = [1] X + [4] >= [1] X + [4] = f(activate(X)) activate(n__s(X)) = [1] X + [4] >= [1] X + [4] = s(activate(X)) f(X) = [1] X + [0] >= [1] X + [0] = n__f(X) f(0()) = [4] >= [4] = cons(0(),n__f(n__s(n__0()))) f(s(0())) = [4] >= [4] = f(p(s(0()))) p(s(X)) = [1] X + [0] >= [1] X + [0] = X s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Weak TRS: 0() -> n__0() activate(X) -> X - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: derivational complexity wrt. signature {0,activate,cons,f,n__0,n__f,n__s,p,s} + 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 + [2] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [4] p(n__0) = [0] p(n__f) = [1] x1 + [4] p(n__s) = [1] x1 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: activate(n__0()) = [2] > [0] = 0() Following rules are (at-least) weakly oriented: 0() = [0] >= [0] = n__0() activate(X) = [1] X + [2] >= [1] X + [0] = X activate(n__f(X)) = [1] X + [6] >= [1] X + [6] = f(activate(X)) activate(n__s(X)) = [1] X + [2] >= [1] X + [2] = s(activate(X)) f(X) = [1] X + [4] >= [1] X + [4] = n__f(X) f(0()) = [4] >= [4] = cons(0(),n__f(n__s(n__0()))) f(s(0())) = [4] >= [4] = f(p(s(0()))) p(s(X)) = [1] X + [0] >= [1] X + [0] = X s(X) = [1] X + [0] >= [1] X + [0] = n__s(X) * Step 3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: derivational complexity wrt. signature {0,activate,cons,f,n__0,n__f,n__s,p,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(activate) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [0] p(n__0) = [0] p(n__f) = [1] x1 + [13] p(n__s) = [1] x1 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [3] Following rules are strictly oriented: activate(n__f(X)) = [1] X + [13] > [1] X + [0] = f(activate(X)) p(s(X)) = [1] X + [3] > [1] X + [0] = X s(X) = [1] X + [3] > [1] X + [0] = n__s(X) Following rules are (at-least) weakly oriented: 0() = [0] >= [0] = n__0() activate(X) = [1] X + [0] >= [1] X + [0] = X activate(n__0()) = [0] >= [0] = 0() activate(n__s(X)) = [1] X + [0] >= [1] X + [3] = s(activate(X)) f(X) = [1] X + [0] >= [1] X + [13] = n__f(X) f(0()) = [0] >= [13] = cons(0(),n__f(n__s(n__0()))) f(s(0())) = [3] >= [3] = f(p(s(0()))) 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: activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: derivational complexity wrt. signature {0,activate,cons,f,n__0,n__f,n__s,p,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] [0] p(activate) = [1 1] x1 + [0] [0 1] [3] p(cons) = [1 0] x1 + [1 0] x2 + [2] [0 0] [0 0] [2] p(f) = [1 0] x1 + [4] [0 1] [2] p(n__0) = [0] [0] p(n__f) = [1 0] x1 + [2] [0 1] [2] p(n__s) = [1 0] x1 + [0] [0 1] [4] p(p) = [1 0] x1 + [0] [0 1] [0] p(s) = [1 0] x1 + [0] [0 1] [4] Following rules are strictly oriented: activate(n__s(X)) = [1 1] X + [4] [0 1] [7] > [1 1] X + [0] [0 1] [7] = s(activate(X)) f(X) = [1 0] X + [4] [0 1] [2] > [1 0] X + [2] [0 1] [2] = n__f(X) Following rules are (at-least) weakly oriented: 0() = [0] [0] >= [0] [0] = n__0() activate(X) = [1 1] X + [0] [0 1] [3] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [0] [3] >= [0] [0] = 0() activate(n__f(X)) = [1 1] X + [4] [0 1] [5] >= [1 1] X + [4] [0 1] [5] = f(activate(X)) f(0()) = [4] [2] >= [4] [2] = cons(0(),n__f(n__s(n__0()))) f(s(0())) = [4] [6] >= [4] [6] = f(p(s(0()))) p(s(X)) = [1 0] X + [0] [0 1] [4] >= [1 0] X + [0] [0 1] [0] = X s(X) = [1 0] X + [0] [0 1] [4] >= [1 0] X + [0] [0 1] [4] = n__s(X) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: derivational complexity wrt. signature {0,activate,cons,f,n__0,n__f,n__s,p,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) = [4] [1] p(activate) = [1 4] x1 + [5] [0 1] [2] p(cons) = [1 0] x1 + [1 0] x2 + [3] [0 0] [0 0] [0] p(f) = [1 4] x1 + [0] [0 1] [2] p(n__0) = [0] [0] p(n__f) = [1 4] x1 + [0] [0 1] [2] p(n__s) = [1 0] x1 + [0] [0 1] [0] p(p) = [1 0] x1 + [0] [0 1] [0] p(s) = [1 0] x1 + [0] [0 1] [0] Following rules are strictly oriented: f(0()) = [8] [3] > [7] [0] = cons(0(),n__f(n__s(n__0()))) Following rules are (at-least) weakly oriented: 0() = [4] [1] >= [0] [0] = n__0() activate(X) = [1 4] X + [5] [0 1] [2] >= [1 0] X + [0] [0 1] [0] = X activate(n__0()) = [5] [2] >= [4] [1] = 0() activate(n__f(X)) = [1 8] X + [13] [0 1] [4] >= [1 8] X + [13] [0 1] [4] = f(activate(X)) activate(n__s(X)) = [1 4] X + [5] [0 1] [2] >= [1 4] X + [5] [0 1] [2] = s(activate(X)) f(X) = [1 4] X + [0] [0 1] [2] >= [1 4] X + [0] [0 1] [2] = n__f(X) f(s(0())) = [8] [3] >= [8] [3] = f(p(s(0()))) p(s(X)) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = X s(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = n__s(X) * Step 6: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: f(s(0())) -> f(p(s(0()))) - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: derivational complexity wrt. signature {0,activate,cons,f,n__0,n__f,n__s,p,s} + Applied Processor: MI {miKind = Automaton Nothing, miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton Nothing: Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [3] [0] [2] p(activate) = [1 1 1] [1] [0 1 0] x_1 + [0] [0 0 1] [0] p(cons) = [1 0 0] [1 0 0] [0] [0 0 0] x_1 + [0 0 0] x_2 + [0] [0 0 1] [0 0 0] [2] p(f) = [1 5 4] [2] [0 0 0] x_1 + [0] [0 1 1] [2] p(n__0) = [0] [0] [2] p(n__f) = [1 5 4] [0] [0 0 0] x_1 + [0] [0 1 1] [2] p(n__s) = [1 0 0] [0] [0 0 1] x_1 + [0] [0 1 0] [0] p(p) = [1 0 0] [1] [0 0 1] x_1 + [0] [0 1 0] [0] p(s) = [1 0 0] [0] [0 0 1] x_1 + [0] [0 1 0] [0] Following rules are strictly oriented: f(s(0())) = [15] [0] [4] > [14] [0] [4] = f(p(s(0()))) Following rules are (at-least) weakly oriented: 0() = [3] [0] [2] >= [0] [0] [2] = n__0() activate(X) = [1 1 1] [1] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X activate(n__0()) = [3] [0] [2] >= [3] [0] [2] = 0() activate(n__f(X)) = [1 6 5] [3] [0 0 0] X + [0] [0 1 1] [2] >= [1 6 5] [3] [0 0 0] X + [0] [0 1 1] [2] = f(activate(X)) activate(n__s(X)) = [1 1 1] [1] [0 0 1] X + [0] [0 1 0] [0] >= [1 1 1] [1] [0 0 1] X + [0] [0 1 0] [0] = s(activate(X)) f(X) = [1 5 4] [2] [0 0 0] X + [0] [0 1 1] [2] >= [1 5 4] [0] [0 0 0] X + [0] [0 1 1] [2] = n__f(X) f(0()) = [13] [0] [4] >= [13] [0] [4] = cons(0(),n__f(n__s(n__0()))) p(s(X)) = [1 0 0] [1] [0 1 0] X + [0] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = X s(X) = [1 0 0] [0] [0 0 1] X + [0] [0 1 0] [0] >= [1 0 0] [0] [0 0 1] X + [0] [0 1 0] [0] = n__s(X) * Step 7: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0() -> n__0() activate(X) -> X activate(n__0()) -> 0() activate(n__f(X)) -> f(activate(X)) activate(n__s(X)) -> s(activate(X)) f(X) -> n__f(X) f(0()) -> cons(0(),n__f(n__s(n__0()))) f(s(0())) -> f(p(s(0()))) p(s(X)) -> X s(X) -> n__s(X) - Signature: {0/0,activate/1,f/1,p/1,s/1} / {cons/2,n__0/0,n__f/1,n__s/1} - Obligation: derivational complexity wrt. signature {0,activate,cons,f,n__0,n__f,n__s,p,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))