/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^2)) * Step 1: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: a(q1(b(x1))) -> q2(a(y(x1))) a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q1(a(x1)) -> a(q1(x1)) q1(y(x1)) -> y(q1(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) q3(y(x1)) -> y(q3(x1)) y(q1(b(x1))) -> q2(y(y(x1))) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + 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(a) = [1] x1 + [0] p(b) = [1] x1 + [2] p(bl) = [1] x1 + [8] p(q0) = [1] x1 + [3] p(q1) = [1] x1 + [0] p(q2) = [1] x1 + [0] p(q3) = [1] x1 + [0] p(q4) = [1] x1 + [0] p(x) = [1] x1 + [0] p(y) = [1] x1 + [1] Following rules are strictly oriented: a(q1(b(x1))) = [1] x1 + [2] > [1] x1 + [1] = q2(a(y(x1))) q0(a(x1)) = [1] x1 + [3] > [1] x1 + [0] = x(q1(x1)) q0(y(x1)) = [1] x1 + [4] > [1] x1 + [1] = y(q3(x1)) y(q1(b(x1))) = [1] x1 + [3] > [1] x1 + [2] = q2(y(y(x1))) Following rules are (at-least) weakly oriented: a(q2(a(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(a(a(x1))) a(q2(y(x1))) = [1] x1 + [1] >= [1] x1 + [1] = q2(a(y(x1))) q1(a(x1)) = [1] x1 + [0] >= [1] x1 + [0] = a(q1(x1)) q1(y(x1)) = [1] x1 + [1] >= [1] x1 + [1] = y(q1(x1)) q2(x(x1)) = [1] x1 + [0] >= [1] x1 + [3] = x(q0(x1)) q3(bl(x1)) = [1] x1 + [8] >= [1] x1 + [8] = bl(q4(x1)) q3(y(x1)) = [1] x1 + [1] >= [1] x1 + [1] = y(q3(x1)) y(q2(a(x1))) = [1] x1 + [1] >= [1] x1 + [1] = q2(y(a(x1))) y(q2(y(x1))) = [1] x1 + [2] >= [1] x1 + [2] = q2(y(y(x1))) 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: a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q1(a(x1)) -> a(q1(x1)) q1(y(x1)) -> y(q1(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) q3(y(x1)) -> y(q3(x1)) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) y(q1(b(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + 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(a) = [1] x1 + [0] p(b) = [1] x1 + [0] p(bl) = [1] x1 + [6] p(q0) = [1] x1 + [5] p(q1) = [1] x1 + [0] p(q2) = [1] x1 + [0] p(q3) = [1] x1 + [1] p(q4) = [1] x1 + [0] p(x) = [1] x1 + [0] p(y) = [1] x1 + [0] Following rules are strictly oriented: q3(bl(x1)) = [1] x1 + [7] > [1] x1 + [6] = bl(q4(x1)) Following rules are (at-least) weakly oriented: a(q1(b(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(a(y(x1))) a(q2(a(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(a(a(x1))) a(q2(y(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(a(y(x1))) q0(a(x1)) = [1] x1 + [5] >= [1] x1 + [0] = x(q1(x1)) q0(y(x1)) = [1] x1 + [5] >= [1] x1 + [1] = y(q3(x1)) q1(a(x1)) = [1] x1 + [0] >= [1] x1 + [0] = a(q1(x1)) q1(y(x1)) = [1] x1 + [0] >= [1] x1 + [0] = y(q1(x1)) q2(x(x1)) = [1] x1 + [0] >= [1] x1 + [5] = x(q0(x1)) q3(y(x1)) = [1] x1 + [1] >= [1] x1 + [1] = y(q3(x1)) y(q1(b(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(y(y(x1))) y(q2(a(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(y(a(x1))) y(q2(y(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(y(y(x1))) 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: a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q1(a(x1)) -> a(q1(x1)) q1(y(x1)) -> y(q1(x1)) q2(x(x1)) -> x(q0(x1)) q3(y(x1)) -> y(q3(x1)) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q3(bl(x1)) -> bl(q4(x1)) y(q1(b(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + 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(a) = [1] x1 + [0] p(b) = [1] x1 + [4] p(bl) = [1] x1 + [3] p(q0) = [1] x1 + [3] p(q1) = [1] x1 + [0] p(q2) = [1] x1 + [4] p(q3) = [1] x1 + [0] p(q4) = [1] x1 + [0] p(x) = [1] x1 + [1] p(y) = [1] x1 + [0] Following rules are strictly oriented: q2(x(x1)) = [1] x1 + [5] > [1] x1 + [4] = x(q0(x1)) Following rules are (at-least) weakly oriented: a(q1(b(x1))) = [1] x1 + [4] >= [1] x1 + [4] = q2(a(y(x1))) a(q2(a(x1))) = [1] x1 + [4] >= [1] x1 + [4] = q2(a(a(x1))) a(q2(y(x1))) = [1] x1 + [4] >= [1] x1 + [4] = q2(a(y(x1))) q0(a(x1)) = [1] x1 + [3] >= [1] x1 + [1] = x(q1(x1)) q0(y(x1)) = [1] x1 + [3] >= [1] x1 + [0] = y(q3(x1)) q1(a(x1)) = [1] x1 + [0] >= [1] x1 + [0] = a(q1(x1)) q1(y(x1)) = [1] x1 + [0] >= [1] x1 + [0] = y(q1(x1)) q3(bl(x1)) = [1] x1 + [3] >= [1] x1 + [3] = bl(q4(x1)) q3(y(x1)) = [1] x1 + [0] >= [1] x1 + [0] = y(q3(x1)) y(q1(b(x1))) = [1] x1 + [4] >= [1] x1 + [4] = q2(y(y(x1))) y(q2(a(x1))) = [1] x1 + [4] >= [1] x1 + [4] = q2(y(a(x1))) y(q2(y(x1))) = [1] x1 + [4] >= [1] x1 + [4] = q2(y(y(x1))) 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: a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q1(a(x1)) -> a(q1(x1)) q1(y(x1)) -> y(q1(x1)) q3(y(x1)) -> y(q3(x1)) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) y(q1(b(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + Applied Processor: NaturalMI {miDimension = 3, 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(a) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(b) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(bl) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(q0) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(q1) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(q2) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(q3) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(q4) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(x) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(y) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: a(q2(a(x1))) = [1 0 2] [1] [0 0 1] x1 + [1] [0 0 1] [1] > [1 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(a(a(x1))) a(q2(y(x1))) = [1 0 1] [1] [0 0 1] x1 + [1] [0 0 1] [1] > [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(a(y(x1))) Following rules are (at-least) weakly oriented: a(q1(b(x1))) = [1 1 2] [1] [0 0 1] x1 + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(a(y(x1))) q0(a(x1)) = [1 0 2] [0] [0 0 1] x1 + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = x(q1(x1)) q0(y(x1)) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = y(q3(x1)) q1(a(x1)) = [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 1] x1 + [0] [0 0 1] [0] = a(q1(x1)) q1(y(x1)) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] = y(q1(x1)) q2(x(x1)) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = x(q0(x1)) q3(bl(x1)) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = bl(q4(x1)) q3(y(x1)) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = y(q3(x1)) y(q1(b(x1))) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(y(y(x1))) y(q2(a(x1))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(y(a(x1))) y(q2(y(x1))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(y(y(x1))) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: q1(a(x1)) -> a(q1(x1)) q1(y(x1)) -> y(q1(x1)) q3(y(x1)) -> y(q3(x1)) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) y(q1(b(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + Applied Processor: NaturalMI {miDimension = 3, 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(a) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(b) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(bl) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(q0) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(q1) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(q2) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(q3) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(q4) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(x) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(y) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: q1(a(x1)) = [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [1] > [1 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [1] = a(q1(x1)) Following rules are (at-least) weakly oriented: a(q1(b(x1))) = [1 0 3] [1] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] = q2(a(y(x1))) a(q2(a(x1))) = [1 0 2] [2] [0 0 0] x1 + [0] [0 0 1] [2] >= [1 0 2] [2] [0 0 0] x1 + [0] [0 0 1] [2] = q2(a(a(x1))) a(q2(y(x1))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] = q2(a(y(x1))) q0(a(x1)) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = x(q1(x1)) q0(y(x1)) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = y(q3(x1)) q1(y(x1)) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] = y(q1(x1)) q2(x(x1)) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = x(q0(x1)) q3(bl(x1)) = [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = bl(q4(x1)) q3(y(x1)) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = y(q3(x1)) y(q1(b(x1))) = [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] = q2(y(y(x1))) y(q2(a(x1))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] = q2(y(a(x1))) y(q2(y(x1))) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [0] = q2(y(y(x1))) * Step 6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: q1(y(x1)) -> y(q1(x1)) q3(y(x1)) -> y(q3(x1)) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q1(a(x1)) -> a(q1(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) y(q1(b(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + Applied Processor: NaturalMI {miDimension = 3, 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(a) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(b) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(bl) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(q0) = [1 1 0] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(q1) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(q2) = [1 1 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(q3) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(q4) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(x) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(y) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] Following rules are strictly oriented: y(q2(a(x1))) = [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [1] > [1 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(y(a(x1))) Following rules are (at-least) weakly oriented: a(q1(b(x1))) = [1 0 2] [3] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [1] = q2(a(y(x1))) a(q2(a(x1))) = [1 0 3] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 3] [1] [0 0 0] x1 + [0] [0 0 1] [1] = q2(a(a(x1))) a(q2(y(x1))) = [1 0 2] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [1] = q2(a(y(x1))) q0(a(x1)) = [1 0 1] [1] [0 0 1] x1 + [0] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] = x(q1(x1)) q0(y(x1)) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] = y(q3(x1)) q1(a(x1)) = [1 0 2] [0] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 2] [0] [0 0 0] x1 + [1] [0 0 1] [0] = a(q1(x1)) q1(y(x1)) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] = y(q1(x1)) q2(x(x1)) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 1] [2] >= [1 1 1] [0] [0 0 0] x1 + [0] [0 0 1] [2] = x(q0(x1)) q3(bl(x1)) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = bl(q4(x1)) q3(y(x1)) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] = y(q3(x1)) y(q1(b(x1))) = [1 0 1] [2] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(y(y(x1))) y(q2(y(x1))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] = q2(y(y(x1))) * Step 7: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: q1(y(x1)) -> y(q1(x1)) q3(y(x1)) -> y(q3(x1)) y(q2(y(x1))) -> q2(y(y(x1))) - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q1(a(x1)) -> a(q1(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) y(q1(b(x1))) -> q2(y(y(x1))) y(q2(a(x1))) -> q2(y(a(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + Applied Processor: NaturalMI {miDimension = 3, 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(a) = [1 1 1] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(b) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(bl) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(q0) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] p(q1) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(q2) = [1 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(q3) = [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(q4) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(x) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(y) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] Following rules are strictly oriented: y(q2(y(x1))) = [1 2 0] [1] [0 1 0] x1 + [1] [0 0 0] [1] > [1 2 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] = q2(y(y(x1))) Following rules are (at-least) weakly oriented: a(q1(b(x1))) = [1 3 0] [3] [0 1 0] x1 + [2] [0 0 0] [1] >= [1 2 0] [1] [0 1 0] x1 + [2] [0 0 0] [0] = q2(a(y(x1))) a(q2(a(x1))) = [1 2 1] [2] [0 1 0] x1 + [3] [0 0 0] [1] >= [1 2 1] [2] [0 1 0] x1 + [3] [0 0 0] [0] = q2(a(a(x1))) a(q2(y(x1))) = [1 2 0] [1] [0 1 0] x1 + [2] [0 0 0] [1] >= [1 2 0] [1] [0 1 0] x1 + [2] [0 0 0] [0] = q2(a(y(x1))) q0(a(x1)) = [1 1 1] [0] [0 1 0] x1 + [2] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = x(q1(x1)) q0(y(x1)) = [1 1 0] [0] [0 1 0] x1 + [1] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = y(q3(x1)) q1(a(x1)) = [1 2 1] [1] [0 1 0] x1 + [1] [0 0 0] [1] >= [1 2 0] [1] [0 1 0] x1 + [1] [0 0 0] [1] = a(q1(x1)) q1(y(x1)) = [1 2 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] >= [1 2 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] = y(q1(x1)) q2(x(x1)) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = x(q0(x1)) q3(bl(x1)) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = bl(q4(x1)) q3(y(x1)) = [1 2 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 1 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = y(q3(x1)) y(q1(b(x1))) = [1 3 0] [2] [0 1 0] x1 + [1] [0 0 0] [1] >= [1 2 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] = q2(y(y(x1))) y(q2(a(x1))) = [1 2 1] [2] [0 1 0] x1 + [2] [0 0 0] [1] >= [1 2 1] [1] [0 1 0] x1 + [2] [0 0 0] [0] = q2(y(a(x1))) * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: q1(y(x1)) -> y(q1(x1)) q3(y(x1)) -> y(q3(x1)) - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q1(a(x1)) -> a(q1(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) y(q1(b(x1))) -> q2(y(y(x1))) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + Applied Processor: NaturalMI {miDimension = 3, 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(a) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(b) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(bl) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(q0) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] p(q1) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] p(q2) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(q3) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [0] p(q4) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(x) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(y) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] Following rules are strictly oriented: q1(y(x1)) = [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] > [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] = y(q1(x1)) Following rules are (at-least) weakly oriented: a(q1(b(x1))) = [1 0 3] [3] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] = q2(a(y(x1))) a(q2(a(x1))) = [1 0 2] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 2] [1] [0 0 0] x1 + [1] [0 0 1] [0] = q2(a(a(x1))) a(q2(y(x1))) = [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 1] [2] [0 0 0] x1 + [1] [0 0 1] [1] = q2(a(y(x1))) q0(a(x1)) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [0] = x(q1(x1)) q0(y(x1)) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = y(q3(x1)) q1(a(x1)) = [1 0 2] [1] [0 0 0] x1 + [1] [0 0 1] [0] >= [1 0 2] [1] [0 0 0] x1 + [1] [0 0 1] [0] = a(q1(x1)) q2(x(x1)) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] = x(q0(x1)) q3(bl(x1)) = [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] >= [1 1 0] [1] [0 0 0] x1 + [1] [0 0 0] [0] = bl(q4(x1)) q3(y(x1)) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [1] = y(q3(x1)) y(q1(b(x1))) = [1 0 2] [2] [0 0 0] x1 + [1] [0 0 1] [2] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [2] = q2(y(y(x1))) y(q2(a(x1))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] = q2(y(a(x1))) y(q2(y(x1))) = [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [2] >= [1 0 0] [1] [0 0 0] x1 + [1] [0 0 1] [2] = q2(y(y(x1))) * Step 9: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: q3(y(x1)) -> y(q3(x1)) - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q1(a(x1)) -> a(q1(x1)) q1(y(x1)) -> y(q1(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) y(q1(b(x1))) -> q2(y(y(x1))) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + Applied Processor: NaturalMI {miDimension = 3, 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(a) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(b) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(bl) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(q0) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(q1) = [1 0 1] [0] [0 0 1] x1 + [1] [0 0 1] [1] p(q2) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(q3) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] p(q4) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(x) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(y) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: q3(y(x1)) = [1 0 2] [2] [0 0 0] x1 + [0] [0 0 1] [1] > [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [1] = y(q3(x1)) Following rules are (at-least) weakly oriented: a(q1(b(x1))) = [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] >= [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] = q2(a(y(x1))) a(q2(a(x1))) = [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] >= [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] = q2(a(a(x1))) a(q2(y(x1))) = [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] >= [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] = q2(a(y(x1))) q0(a(x1)) = [1 0 2] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [1] = x(q1(x1)) q0(y(x1)) = [1 0 2] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [1] = y(q3(x1)) q1(a(x1)) = [1 0 2] [1] [0 0 1] x1 + [2] [0 0 1] [2] >= [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [2] = a(q1(x1)) q1(y(x1)) = [1 0 2] [1] [0 0 1] x1 + [2] [0 0 1] [2] >= [1 0 2] [1] [0 0 0] x1 + [0] [0 0 1] [2] = y(q1(x1)) q2(x(x1)) = [1 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 2] [0] [0 0 0] x1 + [0] [0 0 1] [0] = x(q0(x1)) q3(bl(x1)) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = bl(q4(x1)) y(q1(b(x1))) = [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] >= [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] = q2(y(y(x1))) y(q2(a(x1))) = [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] >= [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] = q2(y(a(x1))) y(q2(y(x1))) = [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] >= [1 0 3] [3] [0 0 0] x1 + [0] [0 0 1] [3] = q2(y(y(x1))) * Step 10: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: a(q1(b(x1))) -> q2(a(y(x1))) a(q2(a(x1))) -> q2(a(a(x1))) a(q2(y(x1))) -> q2(a(y(x1))) q0(a(x1)) -> x(q1(x1)) q0(y(x1)) -> y(q3(x1)) q1(a(x1)) -> a(q1(x1)) q1(y(x1)) -> y(q1(x1)) q2(x(x1)) -> x(q0(x1)) q3(bl(x1)) -> bl(q4(x1)) q3(y(x1)) -> y(q3(x1)) y(q1(b(x1))) -> q2(y(y(x1))) y(q2(a(x1))) -> q2(y(a(x1))) y(q2(y(x1))) -> q2(y(y(x1))) - Signature: {a/1,q0/1,q1/1,q2/1,q3/1,y/1} / {b/1,bl/1,q4/1,x/1} - Obligation: innermost derivational complexity wrt. signature {a,b,bl,q0,q1,q2,q3,q4,x,y} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))