/export/starexec/sandbox2/solver/bin/starexec_run_tct_dci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(?,O(n^3)) * Step 1: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) a__p(s(0())) -> 0() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost derivational complexity wrt. signature {0,a__f,a__p,cons,f,mark,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(a__f) = [1] x1 + [0] p(a__p) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [0] p(mark) = [1] x1 + [8] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: mark(0()) = [8] > [0] = 0() Following rules are (at-least) weakly oriented: a__f(X) = [1] X + [0] >= [1] X + [0] = f(X) a__f(0()) = [0] >= [0] = cons(0(),f(s(0()))) a__f(s(0())) = [0] >= [0] = a__f(a__p(s(0()))) a__p(X) = [1] X + [0] >= [1] X + [0] = p(X) a__p(s(0())) = [0] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [8] >= [1] X1 + [1] X2 + [8] = cons(mark(X1),X2) mark(f(X)) = [1] X + [8] >= [1] X + [8] = a__f(mark(X)) mark(p(X)) = [1] X + [8] >= [1] X + [8] = a__p(mark(X)) mark(s(X)) = [1] X + [8] >= [1] X + [8] = s(mark(X)) * Step 2: WeightGap. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) a__p(s(0())) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: mark(0()) -> 0() - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost derivational complexity wrt. signature {0,a__f,a__p,cons,f,mark,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(a__f) = [1] x1 + [10] p(a__p) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [0] p(mark) = [1] x1 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__f(X) = [1] X + [10] > [1] X + [0] = f(X) a__f(0()) = [10] > [0] = cons(0(),f(s(0()))) Following rules are (at-least) weakly oriented: a__f(s(0())) = [10] >= [10] = a__f(a__p(s(0()))) a__p(X) = [1] X + [0] >= [1] X + [0] = p(X) a__p(s(0())) = [0] >= [0] = 0() mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(mark(X1),X2) mark(f(X)) = [1] X + [0] >= [1] X + [10] = a__f(mark(X)) mark(p(X)) = [1] X + [0] >= [1] X + [0] = a__p(mark(X)) mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) a__p(s(0())) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) mark(0()) -> 0() - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost derivational complexity wrt. signature {0,a__f,a__p,cons,f,mark,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(a__f) = [1] x1 + [9] p(a__p) = [1] x1 + [5] p(cons) = [1] x1 + [1] x2 + [0] p(f) = [1] x1 + [9] p(mark) = [1] x1 + [0] p(p) = [1] x1 + [12] p(s) = [1] x1 + [0] Following rules are strictly oriented: a__p(s(0())) = [5] > [0] = 0() mark(p(X)) = [1] X + [12] > [1] X + [5] = a__p(mark(X)) Following rules are (at-least) weakly oriented: a__f(X) = [1] X + [9] >= [1] X + [9] = f(X) a__f(0()) = [9] >= [9] = cons(0(),f(s(0()))) a__f(s(0())) = [9] >= [14] = a__f(a__p(s(0()))) a__p(X) = [1] X + [5] >= [1] X + [12] = p(X) mark(0()) = [0] >= [0] = 0() mark(cons(X1,X2)) = [1] X1 + [1] X2 + [0] >= [1] X1 + [1] X2 + [0] = cons(mark(X1),X2) mark(f(X)) = [1] X + [9] >= [1] X + [9] = a__f(mark(X)) mark(s(X)) = [1] X + [0] >= [1] X + [0] = s(mark(X)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(s(X)) -> s(mark(X)) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__p(s(0())) -> 0() mark(0()) -> 0() mark(p(X)) -> a__p(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost derivational complexity wrt. signature {0,a__f,a__p,cons,f,mark,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(a__f) = [1 0] x1 + [0] [0 1] [0] p(a__p) = [1 0] x1 + [0] [0 1] [0] p(cons) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [0] p(f) = [1 0] x1 + [0] [0 1] [0] p(mark) = [1 1] x1 + [0] [0 1] [0] p(p) = [1 0] x1 + [0] [0 1] [0] p(s) = [1 0] x1 + [0] [0 1] [2] Following rules are strictly oriented: mark(s(X)) = [1 1] X + [2] [0 1] [2] > [1 1] X + [0] [0 1] [2] = s(mark(X)) Following rules are (at-least) weakly oriented: a__f(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = f(X) a__f(0()) = [0] [0] >= [0] [0] = cons(0(),f(s(0()))) a__f(s(0())) = [0] [2] >= [0] [2] = a__f(a__p(s(0()))) a__p(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = p(X) a__p(s(0())) = [0] [2] >= [0] [0] = 0() mark(0()) = [0] [0] >= [0] [0] = 0() mark(cons(X1,X2)) = [1 1] X1 + [1 0] X2 + [0] [0 1] [0 0] [0] >= [1 1] X1 + [1 0] X2 + [0] [0 1] [0 0] [0] = cons(mark(X1),X2) mark(f(X)) = [1 1] X + [0] [0 1] [0] >= [1 1] X + [0] [0 1] [0] = a__f(mark(X)) mark(p(X)) = [1 1] X + [0] [0 1] [0] >= [1 1] X + [0] [0 1] [0] = a__p(mark(X)) * Step 5: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__p(s(0())) -> 0() mark(0()) -> 0() mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost derivational complexity wrt. signature {0,a__f,a__p,cons,f,mark,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] [2] p(a__f) = [1 1] x1 + [1] [0 1] [2] p(a__p) = [1 0] x1 + [0] [0 1] [0] p(cons) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [2] p(f) = [1 1] x1 + [0] [0 1] [2] p(mark) = [1 4] x1 + [0] [0 1] [0] p(p) = [1 0] x1 + [0] [0 1] [0] p(s) = [1 0] x1 + [1] [0 1] [0] Following rules are strictly oriented: mark(cons(X1,X2)) = [1 4] X1 + [1 0] X2 + [8] [0 1] [0 0] [2] > [1 4] X1 + [1 0] X2 + [0] [0 1] [0 0] [2] = cons(mark(X1),X2) mark(f(X)) = [1 5] X + [8] [0 1] [2] > [1 5] X + [1] [0 1] [2] = a__f(mark(X)) Following rules are (at-least) weakly oriented: a__f(X) = [1 1] X + [1] [0 1] [2] >= [1 1] X + [0] [0 1] [2] = f(X) a__f(0()) = [3] [4] >= [3] [4] = cons(0(),f(s(0()))) a__f(s(0())) = [4] [4] >= [4] [4] = a__f(a__p(s(0()))) a__p(X) = [1 0] X + [0] [0 1] [0] >= [1 0] X + [0] [0 1] [0] = p(X) a__p(s(0())) = [1] [2] >= [0] [2] = 0() mark(0()) = [8] [2] >= [0] [2] = 0() mark(p(X)) = [1 4] X + [0] [0 1] [0] >= [1 4] X + [0] [0 1] [0] = a__p(mark(X)) mark(s(X)) = [1 4] X + [1] [0 1] [0] >= [1 4] X + [1] [0 1] [0] = s(mark(X)) * Step 6: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__p(s(0())) -> 0() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost derivational complexity wrt. signature {0,a__f,a__p,cons,f,mark,p,s} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 2, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(a__f) = [1 2 0 2] [0] [0 0 0 0] x1 + [0] [0 0 1 2] [2] [0 0 0 0] [0] p(a__p) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 1 2] [0] [0 0 0 0] [0] p(cons) = [1 0 0 3] [1 0 0 0] [0] [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 1 1] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] p(f) = [1 0 0 2] [0] [0 0 0 0] x1 + [0] [0 0 1 2] [2] [0 0 0 0] [0] p(mark) = [1 0 2 1] [0] [0 0 0 0] x1 + [2] [0 0 1 1] [0] [0 0 0 0] [0] p(p) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 1 2] [0] [0 0 0 0] [0] p(s) = [1 0 0 0] [0] [0 0 0 0] x1 + [2] [0 0 1 2] [0] [0 0 0 0] [0] Following rules are strictly oriented: a__f(s(0())) = [4] [0] [2] [0] > [0] [0] [2] [0] = a__f(a__p(s(0()))) Following rules are (at-least) weakly oriented: a__f(X) = [1 2 0 2] [0] [0 0 0 0] X + [0] [0 0 1 2] [2] [0 0 0 0] [0] >= [1 0 0 2] [0] [0 0 0 0] X + [0] [0 0 1 2] [2] [0 0 0 0] [0] = f(X) a__f(0()) = [0] [0] [2] [0] >= [0] [0] [0] [0] = cons(0(),f(s(0()))) a__p(X) = [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 2] [0] [0 0 0 0] [0] >= [1 0 1 0] [0] [0 0 0 0] X + [0] [0 0 1 2] [0] [0 0 0 0] [0] = p(X) a__p(s(0())) = [0] [0] [0] [0] >= [0] [0] [0] [0] = 0() mark(0()) = [0] [2] [0] [0] >= [0] [0] [0] [0] = 0() mark(cons(X1,X2)) = [1 0 2 5] [1 0 0 2] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [2] [0 0 1 1] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] >= [1 0 2 1] [1 0 0 0] [0] [0 0 0 0] X1 + [0 0 0 0] X2 + [0] [0 0 1 1] [0 0 0 1] [0] [0 0 0 0] [0 0 0 0] [0] = cons(mark(X1),X2) mark(f(X)) = [1 0 2 6] [4] [0 0 0 0] X + [2] [0 0 1 2] [2] [0 0 0 0] [0] >= [1 0 2 1] [4] [0 0 0 0] X + [0] [0 0 1 1] [2] [0 0 0 0] [0] = a__f(mark(X)) mark(p(X)) = [1 0 3 4] [0] [0 0 0 0] X + [2] [0 0 1 2] [0] [0 0 0 0] [0] >= [1 0 3 2] [0] [0 0 0 0] X + [0] [0 0 1 1] [0] [0 0 0 0] [0] = a__p(mark(X)) mark(s(X)) = [1 0 2 4] [0] [0 0 0 0] X + [2] [0 0 1 2] [0] [0 0 0 0] [0] >= [1 0 2 1] [0] [0 0 0 0] X + [2] [0 0 1 1] [0] [0 0 0 0] [0] = s(mark(X)) * Step 7: NaturalMI. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__p(X) -> p(X) - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__f(s(0())) -> a__f(a__p(s(0()))) a__p(s(0())) -> 0() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost derivational complexity wrt. signature {0,a__f,a__p,cons,f,mark,p,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 3, 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] [1] p(a__f) = [1 0 5] [0] [0 1 6] x1 + [0] [0 0 0] [3] p(a__p) = [1 0 2] [1] [0 1 0] x1 + [1] [0 0 0] [1] p(cons) = [1 0 1] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [6] [0 0 0] [0 0 0] [2] p(f) = [1 0 0] [0] [0 1 6] x1 + [0] [0 0 0] [3] p(mark) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 1] [0] p(p) = [1 0 2] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(s) = [1 0 0] [4] [0 1 0] x1 + [0] [0 0 0] [2] Following rules are strictly oriented: a__p(X) = [1 0 2] [1] [0 1 0] X + [1] [0 0 0] [1] > [1 0 2] [0] [0 1 0] X + [1] [0 0 0] [1] = p(X) Following rules are (at-least) weakly oriented: a__f(X) = [1 0 5] [0] [0 1 6] X + [0] [0 0 0] [3] >= [1 0 0] [0] [0 1 6] X + [0] [0 0 0] [3] = f(X) a__f(0()) = [5] [6] [3] >= [5] [6] [2] = cons(0(),f(s(0()))) a__f(s(0())) = [14] [12] [3] >= [14] [7] [3] = a__f(a__p(s(0()))) a__p(s(0())) = [9] [1] [1] >= [0] [0] [1] = 0() mark(0()) = [0] [1] [1] >= [0] [0] [1] = 0() mark(cons(X1,X2)) = [1 1 1] [1 0 0] [6] [0 1 0] X1 + [0 0 0] X2 + [7] [0 0 0] [0 0 0] [2] >= [1 1 1] [1 0 0] [0] [0 1 0] X1 + [0 0 0] X2 + [7] [0 0 0] [0 0 0] [2] = cons(mark(X1),X2) mark(f(X)) = [1 1 6] [0] [0 1 6] X + [1] [0 0 0] [3] >= [1 1 5] [0] [0 1 6] X + [1] [0 0 0] [3] = a__f(mark(X)) mark(p(X)) = [1 1 2] [1] [0 1 0] X + [2] [0 0 0] [1] >= [1 1 2] [1] [0 1 0] X + [2] [0 0 0] [1] = a__p(mark(X)) mark(s(X)) = [1 1 0] [4] [0 1 0] X + [1] [0 0 0] [2] >= [1 1 0] [4] [0 1 0] X + [1] [0 0 0] [2] = s(mark(X)) * Step 8: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a__f(X) -> f(X) a__f(0()) -> cons(0(),f(s(0()))) a__f(s(0())) -> a__f(a__p(s(0()))) a__p(X) -> p(X) a__p(s(0())) -> 0() mark(0()) -> 0() mark(cons(X1,X2)) -> cons(mark(X1),X2) mark(f(X)) -> a__f(mark(X)) mark(p(X)) -> a__p(mark(X)) mark(s(X)) -> s(mark(X)) - Signature: {a__f/1,a__p/1,mark/1} / {0/0,cons/2,f/1,p/1,s/1} - Obligation: innermost derivational complexity wrt. signature {0,a__f,a__p,cons,f,mark,p,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))