/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: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q1(0(x1)) -> 0(q1(x1)) q1(1'(x1)) -> 1'(q1(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(1'(x1)) -> 1'(q3(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + 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) = [1] x1 + [0] p(0') = [1] x1 + [4] p(1) = [1] x1 + [0] p(1') = [1] x1 + [5] p(b) = [1] x1 + [0] p(q0) = [1] x1 + [3] p(q1) = [1] x1 + [2] p(q2) = [1] x1 + [0] p(q3) = [1] x1 + [1] p(q4) = [1] x1 + [0] Following rules are strictly oriented: q0(1'(x1)) = [1] x1 + [8] > [1] x1 + [6] = 1'(q3(x1)) q3(b(x1)) = [1] x1 + [1] > [1] x1 + [0] = b(q4(x1)) Following rules are (at-least) weakly oriented: 0(q1(1(x1))) = [1] x1 + [2] >= [1] x1 + [5] = q2(0(1'(x1))) 0(q2(0(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(0(0(x1))) 0(q2(1'(x1))) = [1] x1 + [5] >= [1] x1 + [5] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1] x1 + [6] >= [1] x1 + [9] = q2(0'(1'(x1))) 0'(q2(0(x1))) = [1] x1 + [4] >= [1] x1 + [4] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1] x1 + [9] >= [1] x1 + [9] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1] x1 + [7] >= [1] x1 + [10] = q2(1'(1'(x1))) 1'(q2(0(x1))) = [1] x1 + [5] >= [1] x1 + [5] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1] x1 + [10] >= [1] x1 + [10] = q2(1'(1'(x1))) q0(0(x1)) = [1] x1 + [3] >= [1] x1 + [6] = 0'(q1(x1)) q1(0(x1)) = [1] x1 + [2] >= [1] x1 + [2] = 0(q1(x1)) q1(1'(x1)) = [1] x1 + [7] >= [1] x1 + [7] = 1'(q1(x1)) q2(0'(x1)) = [1] x1 + [4] >= [1] x1 + [7] = 0'(q0(x1)) q3(1'(x1)) = [1] x1 + [6] >= [1] x1 + [6] = 1'(q3(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: 0(q1(1(x1))) -> q2(0(1'(x1))) 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q1(0(x1)) -> 0(q1(x1)) q1(1'(x1)) -> 1'(q1(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(1'(x1)) -> 1'(q3(x1)) - Weak TRS: q0(1'(x1)) -> 1'(q3(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + 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) = [1] x1 + [5] p(0') = [1] x1 + [0] p(1) = [1] x1 + [0] p(1') = [1] x1 + [0] p(b) = [1] x1 + [0] p(q0) = [1] x1 + [0] p(q1) = [1] x1 + [3] p(q2) = [1] x1 + [0] p(q3) = [1] x1 + [0] p(q4) = [1] x1 + [0] Following rules are strictly oriented: 0(q1(1(x1))) = [1] x1 + [8] > [1] x1 + [5] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1] x1 + [3] > [1] x1 + [0] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1] x1 + [3] > [1] x1 + [0] = q2(1'(1'(x1))) q0(0(x1)) = [1] x1 + [5] > [1] x1 + [3] = 0'(q1(x1)) Following rules are (at-least) weakly oriented: 0(q2(0(x1))) = [1] x1 + [10] >= [1] x1 + [10] = q2(0(0(x1))) 0(q2(1'(x1))) = [1] x1 + [5] >= [1] x1 + [5] = q2(0(1'(x1))) 0'(q2(0(x1))) = [1] x1 + [5] >= [1] x1 + [5] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(0'(1'(x1))) 1'(q2(0(x1))) = [1] x1 + [5] >= [1] x1 + [5] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1] x1 + [0] >= [1] x1 + [0] = q2(1'(1'(x1))) q0(1'(x1)) = [1] x1 + [0] >= [1] x1 + [0] = 1'(q3(x1)) q1(0(x1)) = [1] x1 + [8] >= [1] x1 + [8] = 0(q1(x1)) q1(1'(x1)) = [1] x1 + [3] >= [1] x1 + [3] = 1'(q1(x1)) q2(0'(x1)) = [1] x1 + [0] >= [1] x1 + [0] = 0'(q0(x1)) q3(1'(x1)) = [1] x1 + [0] >= [1] x1 + [0] = 1'(q3(x1)) q3(b(x1)) = [1] x1 + [0] >= [1] x1 + [0] = b(q4(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: 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q1(0(x1)) -> 0(q1(x1)) q1(1'(x1)) -> 1'(q1(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(1'(x1)) -> 1'(q3(x1)) - Weak TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + 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) = [1] x1 + [4] p(0') = [1] x1 + [1] p(1) = [1] x1 + [2] p(1') = [1] x1 + [1] p(b) = [1] x1 + [0] p(q0) = [1] x1 + [0] p(q1) = [1] x1 + [3] p(q2) = [1] x1 + [3] p(q3) = [1] x1 + [0] p(q4) = [1] x1 + [0] Following rules are strictly oriented: q2(0'(x1)) = [1] x1 + [4] > [1] x1 + [1] = 0'(q0(x1)) Following rules are (at-least) weakly oriented: 0(q1(1(x1))) = [1] x1 + [9] >= [1] x1 + [8] = q2(0(1'(x1))) 0(q2(0(x1))) = [1] x1 + [11] >= [1] x1 + [11] = q2(0(0(x1))) 0(q2(1'(x1))) = [1] x1 + [8] >= [1] x1 + [8] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1] x1 + [6] >= [1] x1 + [5] = q2(0'(1'(x1))) 0'(q2(0(x1))) = [1] x1 + [8] >= [1] x1 + [8] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1] x1 + [5] >= [1] x1 + [5] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1] x1 + [6] >= [1] x1 + [5] = q2(1'(1'(x1))) 1'(q2(0(x1))) = [1] x1 + [8] >= [1] x1 + [8] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1] x1 + [5] >= [1] x1 + [5] = q2(1'(1'(x1))) q0(0(x1)) = [1] x1 + [4] >= [1] x1 + [4] = 0'(q1(x1)) q0(1'(x1)) = [1] x1 + [1] >= [1] x1 + [1] = 1'(q3(x1)) q1(0(x1)) = [1] x1 + [7] >= [1] x1 + [7] = 0(q1(x1)) q1(1'(x1)) = [1] x1 + [4] >= [1] x1 + [4] = 1'(q1(x1)) q3(1'(x1)) = [1] x1 + [1] >= [1] x1 + [1] = 1'(q3(x1)) q3(b(x1)) = [1] x1 + [0] >= [1] x1 + [0] = b(q4(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: 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q1(0(x1)) -> 0(q1(x1)) q1(1'(x1)) -> 1'(q1(x1)) q3(1'(x1)) -> 1'(q3(x1)) - Weak TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + Applied Processor: NaturalMI {miDimension = 2, miDegree = 1, 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 1 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) = [1 0] x1 + [2] [0 0] [1] p(0') = [1 0] x1 + [2] [0 0] [0] p(1) = [1 0] x1 + [2] [0 0] [0] p(1') = [1 0] x1 + [0] [0 0] [1] p(b) = [1 1] x1 + [0] [0 0] [1] p(q0) = [1 0] x1 + [0] [0 0] [2] p(q1) = [1 0] x1 + [0] [0 0] [2] p(q2) = [1 2] x1 + [0] [0 0] [0] p(q3) = [1 0] x1 + [0] [0 0] [1] p(q4) = [1 1] x1 + [0] [0 0] [0] Following rules are strictly oriented: 0'(q2(0(x1))) = [1 0] x1 + [6] [0 0] [0] > [1 0] x1 + [4] [0 0] [0] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1 0] x1 + [4] [0 0] [0] > [1 0] x1 + [2] [0 0] [0] = q2(0'(1'(x1))) Following rules are (at-least) weakly oriented: 0(q1(1(x1))) = [1 0] x1 + [4] [0 0] [1] >= [1 0] x1 + [4] [0 0] [0] = q2(0(1'(x1))) 0(q2(0(x1))) = [1 0] x1 + [6] [0 0] [1] >= [1 0] x1 + [6] [0 0] [0] = q2(0(0(x1))) 0(q2(1'(x1))) = [1 0] x1 + [4] [0 0] [1] >= [1 0] x1 + [4] [0 0] [0] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1 0] x1 + [4] [0 0] [0] >= [1 0] x1 + [2] [0 0] [0] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1 0] x1 + [2] [0 0] [1] >= [1 0] x1 + [2] [0 0] [0] = q2(1'(1'(x1))) 1'(q2(0(x1))) = [1 0] x1 + [4] [0 0] [1] >= [1 0] x1 + [4] [0 0] [0] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1 0] x1 + [2] [0 0] [1] >= [1 0] x1 + [2] [0 0] [0] = q2(1'(1'(x1))) q0(0(x1)) = [1 0] x1 + [2] [0 0] [2] >= [1 0] x1 + [2] [0 0] [0] = 0'(q1(x1)) q0(1'(x1)) = [1 0] x1 + [0] [0 0] [2] >= [1 0] x1 + [0] [0 0] [1] = 1'(q3(x1)) q1(0(x1)) = [1 0] x1 + [2] [0 0] [2] >= [1 0] x1 + [2] [0 0] [1] = 0(q1(x1)) q1(1'(x1)) = [1 0] x1 + [0] [0 0] [2] >= [1 0] x1 + [0] [0 0] [1] = 1'(q1(x1)) q2(0'(x1)) = [1 0] x1 + [2] [0 0] [0] >= [1 0] x1 + [2] [0 0] [0] = 0'(q0(x1)) q3(1'(x1)) = [1 0] x1 + [0] [0 0] [1] >= [1 0] x1 + [0] [0 0] [1] = 1'(q3(x1)) q3(b(x1)) = [1 1] x1 + [0] [0 0] [1] >= [1 1] x1 + [0] [0 0] [1] = b(q4(x1)) * Step 5: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q1(0(x1)) -> 0(q1(x1)) q1(1'(x1)) -> 1'(q1(x1)) q3(1'(x1)) -> 1'(q3(x1)) - Weak TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + 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) = [1 1] x1 + [0] [0 1] [0] p(0') = [1 0] x1 + [0] [0 0] [0] p(1) = [1 0] x1 + [0] [0 1] [2] p(1') = [1 0] x1 + [0] [0 1] [1] p(b) = [1 0] x1 + [0] [0 1] [0] p(q0) = [1 0] x1 + [0] [0 1] [1] p(q1) = [1 1] x1 + [0] [0 1] [0] p(q2) = [1 0] x1 + [0] [0 1] [0] p(q3) = [1 0] x1 + [0] [0 1] [1] p(q4) = [1 0] x1 + [0] [0 0] [1] Following rules are strictly oriented: q1(1'(x1)) = [1 1] x1 + [1] [0 1] [1] > [1 1] x1 + [0] [0 1] [1] = 1'(q1(x1)) Following rules are (at-least) weakly oriented: 0(q1(1(x1))) = [1 2] x1 + [4] [0 1] [2] >= [1 1] x1 + [1] [0 1] [1] = q2(0(1'(x1))) 0(q2(0(x1))) = [1 2] x1 + [0] [0 1] [0] >= [1 2] x1 + [0] [0 1] [0] = q2(0(0(x1))) 0(q2(1'(x1))) = [1 1] x1 + [1] [0 1] [1] >= [1 1] x1 + [1] [0 1] [1] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1 1] x1 + [2] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = q2(0'(1'(x1))) 0'(q2(0(x1))) = [1 1] x1 + [0] [0 0] [0] >= [1 1] x1 + [0] [0 0] [0] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1 1] x1 + [2] [0 1] [3] >= [1 0] x1 + [0] [0 1] [2] = q2(1'(1'(x1))) 1'(q2(0(x1))) = [1 1] x1 + [0] [0 1] [1] >= [1 1] x1 + [0] [0 1] [1] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1 0] x1 + [0] [0 1] [2] >= [1 0] x1 + [0] [0 1] [2] = q2(1'(1'(x1))) q0(0(x1)) = [1 1] x1 + [0] [0 1] [1] >= [1 1] x1 + [0] [0 0] [0] = 0'(q1(x1)) q0(1'(x1)) = [1 0] x1 + [0] [0 1] [2] >= [1 0] x1 + [0] [0 1] [2] = 1'(q3(x1)) q1(0(x1)) = [1 2] x1 + [0] [0 1] [0] >= [1 2] x1 + [0] [0 1] [0] = 0(q1(x1)) q2(0'(x1)) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 0'(q0(x1)) q3(1'(x1)) = [1 0] x1 + [0] [0 1] [2] >= [1 0] x1 + [0] [0 1] [2] = 1'(q3(x1)) q3(b(x1)) = [1 0] x1 + [0] [0 1] [1] >= [1 0] x1 + [0] [0 0] [1] = b(q4(x1)) * Step 6: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q1(0(x1)) -> 0(q1(x1)) q3(1'(x1)) -> 1'(q3(x1)) - Weak TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q1(1'(x1)) -> 1'(q1(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + 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(0) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(0') = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(1) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 1] [1] p(1') = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(b) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] p(q0) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] p(q1) = [1 0 0] [0] [0 0 1] x1 + [1] [0 0 1] [0] p(q2) = [1 0 0] [0] [0 0 0] x1 + [1] [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] Following rules are strictly oriented: 0(q2(0(x1))) = [1 0 2] [1] [0 0 0] x1 + [1] [0 0 1] [1] > [1 0 2] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(0(0(x1))) 0(q2(1'(x1))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] > [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(0(1'(x1))) Following rules are (at-least) weakly oriented: 0(q1(1(x1))) = [1 0 2] [2] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(0'(1'(x1))) 0'(q2(0(x1))) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1 0 1] [1] [0 0 1] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(1'(1'(x1))) 1'(q2(0(x1))) = [1 0 1] [0] [0 0 1] x1 + [1] [0 0 1] [1] >= [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1 0 0] [0] [0 0 1] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = q2(1'(1'(x1))) q0(0(x1)) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] = 0'(q1(x1)) q0(1'(x1)) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 1'(q3(x1)) q1(0(x1)) = [1 0 1] [0] [0 0 1] x1 + [1] [0 0 1] [0] >= [1 0 1] [0] [0 0 0] x1 + [1] [0 0 1] [0] = 0(q1(x1)) q1(1'(x1)) = [1 0 0] [0] [0 0 1] x1 + [1] [0 0 1] [0] >= [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] = 1'(q1(x1)) q2(0'(x1)) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [1] = 0'(q0(x1)) q3(1'(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] = 1'(q3(x1)) q3(b(x1)) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = b(q4(x1)) * Step 7: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q1(0(x1)) -> 0(q1(x1)) q3(1'(x1)) -> 1'(q3(x1)) - Weak TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q1(1'(x1)) -> 1'(q1(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + 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) = [1 1] x1 + [0] [0 1] [1] p(0') = [1 2] x1 + [0] [0 0] [0] p(1) = [1 0] x1 + [2] [0 1] [1] p(1') = [1 0] x1 + [0] [0 1] [0] p(b) = [1 2] x1 + [0] [0 0] [0] p(q0) = [1 2] x1 + [0] [0 0] [0] p(q1) = [1 1] x1 + [2] [0 1] [0] p(q2) = [1 0] x1 + [0] [0 1] [0] p(q3) = [1 0] x1 + [0] [0 0] [0] p(q4) = [1 0] x1 + [0] [0 0] [0] Following rules are strictly oriented: q1(0(x1)) = [1 2] x1 + [3] [0 1] [1] > [1 2] x1 + [2] [0 1] [1] = 0(q1(x1)) Following rules are (at-least) weakly oriented: 0(q1(1(x1))) = [1 2] x1 + [6] [0 1] [2] >= [1 1] x1 + [0] [0 1] [1] = q2(0(1'(x1))) 0(q2(0(x1))) = [1 2] x1 + [1] [0 1] [2] >= [1 2] x1 + [1] [0 1] [2] = q2(0(0(x1))) 0(q2(1'(x1))) = [1 1] x1 + [0] [0 1] [1] >= [1 1] x1 + [0] [0 1] [1] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1 3] x1 + [7] [0 0] [0] >= [1 2] x1 + [0] [0 0] [0] = q2(0'(1'(x1))) 0'(q2(0(x1))) = [1 3] x1 + [2] [0 0] [0] >= [1 3] x1 + [2] [0 0] [0] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1 2] x1 + [0] [0 0] [0] >= [1 2] x1 + [0] [0 0] [0] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1 1] x1 + [5] [0 1] [1] >= [1 0] x1 + [0] [0 1] [0] = q2(1'(1'(x1))) 1'(q2(0(x1))) = [1 1] x1 + [0] [0 1] [1] >= [1 1] x1 + [0] [0 1] [1] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1 0] x1 + [0] [0 1] [0] >= [1 0] x1 + [0] [0 1] [0] = q2(1'(1'(x1))) q0(0(x1)) = [1 3] x1 + [2] [0 0] [0] >= [1 3] x1 + [2] [0 0] [0] = 0'(q1(x1)) q0(1'(x1)) = [1 2] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1'(q3(x1)) q1(1'(x1)) = [1 1] x1 + [2] [0 1] [0] >= [1 1] x1 + [2] [0 1] [0] = 1'(q1(x1)) q2(0'(x1)) = [1 2] x1 + [0] [0 0] [0] >= [1 2] x1 + [0] [0 0] [0] = 0'(q0(x1)) q3(1'(x1)) = [1 0] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1'(q3(x1)) q3(b(x1)) = [1 2] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = b(q4(x1)) * Step 8: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q3(1'(x1)) -> 1'(q3(x1)) - Weak TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q1(0(x1)) -> 0(q1(x1)) q1(1'(x1)) -> 1'(q1(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + 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) = [1 2] x1 + [0] [0 1] [1] p(0') = [1 0] x1 + [0] [0 1] [0] p(1) = [1 3] x1 + [3] [0 1] [1] p(1') = [1 2] x1 + [0] [0 1] [0] p(b) = [1 3] x1 + [1] [0 0] [0] p(q0) = [1 2] x1 + [0] [0 1] [1] p(q1) = [1 1] x1 + [1] [0 1] [0] p(q2) = [1 2] x1 + [0] [0 1] [1] p(q3) = [1 0] x1 + [0] [0 0] [0] p(q4) = [1 0] x1 + [0] [0 0] [0] Following rules are strictly oriented: 1'(q2(0(x1))) = [1 6] x1 + [6] [0 1] [2] > [1 6] x1 + [4] [0 1] [2] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1 6] x1 + [2] [0 1] [1] > [1 6] x1 + [0] [0 1] [1] = q2(1'(1'(x1))) Following rules are (at-least) weakly oriented: 0(q1(1(x1))) = [1 6] x1 + [7] [0 1] [2] >= [1 6] x1 + [2] [0 1] [2] = q2(0(1'(x1))) 0(q2(0(x1))) = [1 6] x1 + [6] [0 1] [3] >= [1 6] x1 + [6] [0 1] [3] = q2(0(0(x1))) 0(q2(1'(x1))) = [1 6] x1 + [2] [0 1] [2] >= [1 6] x1 + [2] [0 1] [2] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1 4] x1 + [5] [0 1] [1] >= [1 4] x1 + [0] [0 1] [1] = q2(0'(1'(x1))) 0'(q2(0(x1))) = [1 4] x1 + [2] [0 1] [2] >= [1 4] x1 + [2] [0 1] [2] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1 4] x1 + [0] [0 1] [1] >= [1 4] x1 + [0] [0 1] [1] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1 6] x1 + [7] [0 1] [1] >= [1 6] x1 + [0] [0 1] [1] = q2(1'(1'(x1))) q0(0(x1)) = [1 4] x1 + [2] [0 1] [2] >= [1 1] x1 + [1] [0 1] [0] = 0'(q1(x1)) q0(1'(x1)) = [1 4] x1 + [0] [0 1] [1] >= [1 0] x1 + [0] [0 0] [0] = 1'(q3(x1)) q1(0(x1)) = [1 3] x1 + [2] [0 1] [1] >= [1 3] x1 + [1] [0 1] [1] = 0(q1(x1)) q1(1'(x1)) = [1 3] x1 + [1] [0 1] [0] >= [1 3] x1 + [1] [0 1] [0] = 1'(q1(x1)) q2(0'(x1)) = [1 2] x1 + [0] [0 1] [1] >= [1 2] x1 + [0] [0 1] [1] = 0'(q0(x1)) q3(1'(x1)) = [1 2] x1 + [0] [0 0] [0] >= [1 0] x1 + [0] [0 0] [0] = 1'(q3(x1)) q3(b(x1)) = [1 3] x1 + [1] [0 0] [0] >= [1 0] x1 + [1] [0 0] [0] = b(q4(x1)) * Step 9: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: q3(1'(x1)) -> 1'(q3(x1)) - Weak TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q1(0(x1)) -> 0(q1(x1)) q1(1'(x1)) -> 1'(q1(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + 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) = [1 1] x1 + [0] [0 1] [2] p(0') = [1 0] x1 + [0] [0 1] [0] p(1) = [1 0] x1 + [0] [0 1] [1] p(1') = [1 1] x1 + [0] [0 1] [2] p(b) = [1 2] x1 + [1] [0 1] [2] p(q0) = [1 1] x1 + [0] [0 1] [2] p(q1) = [1 2] x1 + [1] [0 1] [3] p(q2) = [1 1] x1 + [0] [0 1] [2] p(q3) = [1 1] x1 + [0] [0 1] [0] p(q4) = [1 1] x1 + [1] [0 1] [0] Following rules are strictly oriented: q3(1'(x1)) = [1 2] x1 + [2] [0 1] [2] > [1 2] x1 + [0] [0 1] [2] = 1'(q3(x1)) Following rules are (at-least) weakly oriented: 0(q1(1(x1))) = [1 3] x1 + [7] [0 1] [6] >= [1 3] x1 + [6] [0 1] [6] = q2(0(1'(x1))) 0(q2(0(x1))) = [1 3] x1 + [6] [0 1] [6] >= [1 3] x1 + [6] [0 1] [6] = q2(0(0(x1))) 0(q2(1'(x1))) = [1 3] x1 + [6] [0 1] [6] >= [1 3] x1 + [6] [0 1] [6] = q2(0(1'(x1))) 0'(q1(1(x1))) = [1 2] x1 + [3] [0 1] [4] >= [1 2] x1 + [2] [0 1] [4] = q2(0'(1'(x1))) 0'(q2(0(x1))) = [1 2] x1 + [2] [0 1] [4] >= [1 2] x1 + [2] [0 1] [4] = q2(0'(0(x1))) 0'(q2(1'(x1))) = [1 2] x1 + [2] [0 1] [4] >= [1 2] x1 + [2] [0 1] [4] = q2(0'(1'(x1))) 1'(q1(1(x1))) = [1 3] x1 + [7] [0 1] [6] >= [1 3] x1 + [6] [0 1] [6] = q2(1'(1'(x1))) 1'(q2(0(x1))) = [1 3] x1 + [6] [0 1] [6] >= [1 3] x1 + [6] [0 1] [6] = q2(1'(0(x1))) 1'(q2(1'(x1))) = [1 3] x1 + [6] [0 1] [6] >= [1 3] x1 + [6] [0 1] [6] = q2(1'(1'(x1))) q0(0(x1)) = [1 2] x1 + [2] [0 1] [4] >= [1 2] x1 + [1] [0 1] [3] = 0'(q1(x1)) q0(1'(x1)) = [1 2] x1 + [2] [0 1] [4] >= [1 2] x1 + [0] [0 1] [2] = 1'(q3(x1)) q1(0(x1)) = [1 3] x1 + [5] [0 1] [5] >= [1 3] x1 + [4] [0 1] [5] = 0(q1(x1)) q1(1'(x1)) = [1 3] x1 + [5] [0 1] [5] >= [1 3] x1 + [4] [0 1] [5] = 1'(q1(x1)) q2(0'(x1)) = [1 1] x1 + [0] [0 1] [2] >= [1 1] x1 + [0] [0 1] [2] = 0'(q0(x1)) q3(b(x1)) = [1 3] x1 + [3] [0 1] [2] >= [1 3] x1 + [2] [0 1] [2] = b(q4(x1)) * Step 10: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0(q1(1(x1))) -> q2(0(1'(x1))) 0(q2(0(x1))) -> q2(0(0(x1))) 0(q2(1'(x1))) -> q2(0(1'(x1))) 0'(q1(1(x1))) -> q2(0'(1'(x1))) 0'(q2(0(x1))) -> q2(0'(0(x1))) 0'(q2(1'(x1))) -> q2(0'(1'(x1))) 1'(q1(1(x1))) -> q2(1'(1'(x1))) 1'(q2(0(x1))) -> q2(1'(0(x1))) 1'(q2(1'(x1))) -> q2(1'(1'(x1))) q0(0(x1)) -> 0'(q1(x1)) q0(1'(x1)) -> 1'(q3(x1)) q1(0(x1)) -> 0(q1(x1)) q1(1'(x1)) -> 1'(q1(x1)) q2(0'(x1)) -> 0'(q0(x1)) q3(1'(x1)) -> 1'(q3(x1)) q3(b(x1)) -> b(q4(x1)) - Signature: {0/1,0'/1,1'/1,q0/1,q1/1,q2/1,q3/1} / {1/1,b/1,q4/1} - Obligation: derivational complexity wrt. signature {0,0',1,1',b,q0,q1,q2,q3,q4} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))