/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: a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(tail(X)) -> a__tail(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0} - Obligation: derivational complexity wrt. signature {0,a__tail,a__zeros,cons,mark,tail,zeros} + 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(a__tail) = [1] x1 + [3] p(a__zeros) = [13] p(cons) = [1] x1 + [1] x2 + [4] p(mark) = [1] x1 + [4] p(tail) = [1] x1 + [3] p(zeros) = [9] Following rules are strictly oriented: a__tail(cons(X,XS)) = [1] X + [1] XS + [7] > [1] XS + [4] = mark(XS) a__zeros() = [13] > [9] = zeros() mark(0()) = [4] > [0] = 0() Following rules are (at-least) weakly oriented: a__tail(X) = [1] X + [3] >= [1] X + [3] = tail(X) a__zeros() = [13] >= [13] = cons(0(),zeros()) mark(cons(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = cons(mark(X1),X2) mark(tail(X)) = [1] X + [7] >= [1] X + [7] = a__tail(mark(X)) mark(zeros()) = [13] >= [13] = a__zeros() * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__tail(X) -> tail(X) a__zeros() -> cons(0(),zeros()) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(tail(X)) -> a__tail(mark(X)) mark(zeros()) -> a__zeros() - Weak TRS: a__tail(cons(X,XS)) -> mark(XS) a__zeros() -> zeros() mark(0()) -> 0() - Signature: {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0} - Obligation: derivational complexity wrt. signature {0,a__tail,a__zeros,cons,mark,tail,zeros} + 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(a__tail) = [1] x1 + [2] p(a__zeros) = [0] p(cons) = [1] x1 + [1] x2 + [0] p(mark) = [1] x1 + [1] p(tail) = [1] x1 + [2] p(zeros) = [0] Following rules are strictly oriented: mark(zeros()) = [1] > [0] = a__zeros() Following rules are (at-least) weakly oriented: a__tail(X) = [1] X + [2] >= [1] X + [2] = tail(X) a__tail(cons(X,XS)) = [1] X + [1] XS + [2] >= [1] XS + [1] = mark(XS) a__zeros() = [0] >= [0] = cons(0(),zeros()) a__zeros() = [0] >= [0] = zeros() mark(0()) = [1] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [1] >= [1] X1 + [1] X2 + [1] = cons(mark(X1),X2) mark(tail(X)) = [1] X + [3] >= [1] X + [3] = a__tail(mark(X)) * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__tail(X) -> tail(X) a__zeros() -> cons(0(),zeros()) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(tail(X)) -> a__tail(mark(X)) - Weak TRS: a__tail(cons(X,XS)) -> mark(XS) a__zeros() -> zeros() mark(0()) -> 0() mark(zeros()) -> a__zeros() - Signature: {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0} - Obligation: derivational complexity wrt. signature {0,a__tail,a__zeros,cons,mark,tail,zeros} + 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(a__tail) = [1] x1 + [2] p(a__zeros) = [2] p(cons) = [1] x1 + [1] x2 + [0] p(mark) = [1] x1 + [2] p(tail) = [1] x1 + [2] p(zeros) = [0] Following rules are strictly oriented: a__zeros() = [2] > [0] = cons(0(),zeros()) Following rules are (at-least) weakly oriented: a__tail(X) = [1] X + [2] >= [1] X + [2] = tail(X) a__tail(cons(X,XS)) = [1] X + [1] XS + [2] >= [1] XS + [2] = mark(XS) a__zeros() = [2] >= [0] = zeros() mark(0()) = [2] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [2] >= [1] X1 + [1] X2 + [2] = cons(mark(X1),X2) mark(tail(X)) = [1] X + [4] >= [1] X + [4] = a__tail(mark(X)) mark(zeros()) = [2] >= [2] = a__zeros() * Step 4: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__tail(X) -> tail(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(tail(X)) -> a__tail(mark(X)) - Weak TRS: a__tail(cons(X,XS)) -> mark(XS) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(zeros()) -> a__zeros() - Signature: {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0} - Obligation: derivational complexity wrt. signature {0,a__tail,a__zeros,cons,mark,tail,zeros} + 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(a__tail) = [1] x1 + [1] p(a__zeros) = [9] p(cons) = [1] x1 + [1] x2 + [9] p(mark) = [1] x1 + [10] p(tail) = [1] x1 + [4] p(zeros) = [0] Following rules are strictly oriented: mark(tail(X)) = [1] X + [14] > [1] X + [11] = a__tail(mark(X)) Following rules are (at-least) weakly oriented: a__tail(X) = [1] X + [1] >= [1] X + [4] = tail(X) a__tail(cons(X,XS)) = [1] X + [1] XS + [10] >= [1] XS + [10] = mark(XS) a__zeros() = [9] >= [9] = cons(0(),zeros()) a__zeros() = [9] >= [0] = zeros() mark(0()) = [10] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [19] >= [1] X1 + [1] X2 + [19] = cons(mark(X1),X2) mark(zeros()) = [10] >= [9] = a__zeros() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__tail(X) -> tail(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) - Weak TRS: a__tail(cons(X,XS)) -> mark(XS) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(tail(X)) -> a__tail(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0} - Obligation: derivational complexity wrt. signature {0,a__tail,a__zeros,cons,mark,tail,zeros} + 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(a__tail) = [1 3] x1 + [0] [0 1] [3] p(a__zeros) = [4] [2] p(cons) = [1 0] x1 + [1 4] x2 + [0] [0 1] [0 1] [1] p(mark) = [1 4] x1 + [0] [0 1] [4] p(tail) = [1 3] x1 + [0] [0 1] [3] p(zeros) = [0] [1] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 4] X1 + [1 8] X2 + [4] [0 1] [0 1] [5] > [1 4] X1 + [1 4] X2 + [0] [0 1] [0 1] [5] = cons(mark(X1),X2) Following rules are (at-least) weakly oriented: a__tail(X) = [1 3] X + [0] [0 1] [3] >= [1 3] X + [0] [0 1] [3] = tail(X) a__tail(cons(X,XS)) = [1 3] X + [1 7] XS + [3] [0 1] [0 1] [4] >= [1 4] XS + [0] [0 1] [4] = mark(XS) a__zeros() = [4] [2] >= [4] [2] = cons(0(),zeros()) a__zeros() = [4] [2] >= [0] [1] = zeros() mark(0()) = [0] [4] >= [0] [0] = 0() mark(tail(X)) = [1 7] X + [12] [0 1] [7] >= [1 7] X + [12] [0 1] [7] = a__tail(mark(X)) mark(zeros()) = [4] [5] >= [4] [2] = a__zeros() * Step 6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a__tail(X) -> tail(X) - Weak TRS: a__tail(cons(X,XS)) -> mark(XS) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(tail(X)) -> a__tail(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0} - Obligation: derivational complexity wrt. signature {0,a__tail,a__zeros,cons,mark,tail,zeros} + 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(a__tail) = [1 6] x1 + [1] [0 1] [1] p(a__zeros) = [0] [0] p(cons) = [1 4] x1 + [1 0] x2 + [0] [0 1] [0 1] [0] p(mark) = [1 2] x1 + [0] [0 1] [0] p(tail) = [1 6] x1 + [0] [0 1] [1] p(zeros) = [0] [0] Following rules are strictly oriented: a__tail(X) = [1 6] X + [1] [0 1] [1] > [1 6] X + [0] [0 1] [1] = tail(X) Following rules are (at-least) weakly oriented: a__tail(cons(X,XS)) = [1 10] X + [1 6] XS + [1] [0 1] [0 1] [1] >= [1 2] XS + [0] [0 1] [0] = mark(XS) a__zeros() = [0] [0] >= [0] [0] = cons(0(),zeros()) a__zeros() = [0] [0] >= [0] [0] = zeros() mark(0()) = [0] [0] >= [0] [0] = 0() mark(cons(X1,X2)) = [1 6] X1 + [1 2] X2 + [0] [0 1] [0 1] [0] >= [1 6] X1 + [1 0] X2 + [0] [0 1] [0 1] [0] = cons(mark(X1),X2) mark(tail(X)) = [1 8] X + [2] [0 1] [1] >= [1 8] X + [1] [0 1] [1] = a__tail(mark(X)) mark(zeros()) = [0] [0] >= [0] [0] = a__zeros() * Step 7: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__tail(X) -> tail(X) a__tail(cons(X,XS)) -> mark(XS) a__zeros() -> cons(0(),zeros()) a__zeros() -> zeros() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(tail(X)) -> a__tail(mark(X)) mark(zeros()) -> a__zeros() - Signature: {a__tail/1,a__zeros/0,mark/1} / {0/0,cons/2,tail/1,zeros/0} - Obligation: derivational complexity wrt. signature {0,a__tail,a__zeros,cons,mark,tail,zeros} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))