/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: 0(q1(x1)) -> q2(1(x1)) 0(q2(x1)) -> 0(q0(x1)) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 1(q2(x1)) -> q2(1(x1)) - Signature: {0/1,1/1} / {q0/1,q1/1,q2/1} - Obligation: innermost derivational complexity wrt. signature {0,1,q0,q1,q2} + 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 + [9] p(1) = [1] x1 + [6] p(q0) = [1] x1 + [0] p(q1) = [1] x1 + [8] p(q2) = [1] x1 + [4] Following rules are strictly oriented: 0(q1(x1)) = [1] x1 + [17] > [1] x1 + [10] = q2(1(x1)) 0(q2(x1)) = [1] x1 + [13] > [1] x1 + [9] = 0(q0(x1)) Following rules are (at-least) weakly oriented: 1(q0(0(x1))) = [1] x1 + [15] >= [1] x1 + [26] = 0(0(q1(x1))) 1(q0(1(x1))) = [1] x1 + [12] >= [1] x1 + [23] = 0(1(q1(x1))) 1(q1(0(x1))) = [1] x1 + [23] >= [1] x1 + [23] = 1(0(q1(x1))) 1(q1(1(x1))) = [1] x1 + [20] >= [1] x1 + [20] = 1(1(q1(x1))) 1(q2(x1)) = [1] x1 + [10] >= [1] x1 + [10] = q2(1(x1)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 1(q2(x1)) -> q2(1(x1)) - Weak TRS: 0(q1(x1)) -> q2(1(x1)) 0(q2(x1)) -> 0(q0(x1)) - Signature: {0/1,1/1} / {q0/1,q1/1,q2/1} - Obligation: innermost derivational complexity wrt. signature {0,1,q0,q1,q2} + 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 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(1) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(q0) = [1 0 1] [3] [0 0 1] x1 + [0] [0 0 1] [1] p(q1) = [1 0 1] [3] [0 0 0] x1 + [2] [0 0 1] [0] p(q2) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [1] Following rules are strictly oriented: 1(q0(0(x1))) = [1 0 1] [4] [0 0 0] x1 + [0] [0 0 1] [2] > [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [2] = 0(0(q1(x1))) 1(q1(0(x1))) = [1 0 1] [4] [0 0 0] x1 + [0] [0 0 1] [1] > [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [1] = 1(0(q1(x1))) Following rules are (at-least) weakly oriented: 0(q1(x1)) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [1] = q2(1(x1)) 0(q2(x1)) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [2] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [2] = 0(q0(x1)) 1(q0(1(x1))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [1] = 0(1(q1(x1))) 1(q1(1(x1))) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [0] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [0] = 1(1(q1(x1))) 1(q2(x1)) = [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [1] >= [1 0 1] [3] [0 0 0] x1 + [0] [0 0 1] [1] = q2(1(x1)) * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 1(q2(x1)) -> q2(1(x1)) - Weak TRS: 0(q1(x1)) -> q2(1(x1)) 0(q2(x1)) -> 0(q0(x1)) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) - Signature: {0/1,1/1} / {q0/1,q1/1,q2/1} - Obligation: innermost derivational complexity wrt. signature {0,1,q0,q1,q2} + 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 2 1] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(1) = [1 2 0] [1] [0 1 0] x1 + [0] [0 0 0] [0] p(q0) = [1 2 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(q1) = [1 2 0] [2] [0 1 0] x1 + [0] [0 0 0] [1] p(q2) = [1 2 0] [2] [0 1 0] x1 + [1] [0 0 0] [0] Following rules are strictly oriented: 1(q0(1(x1))) = [1 6 0] [4] [0 1 0] x1 + [1] [0 0 0] [0] > [1 6 0] [3] [0 1 0] x1 + [1] [0 0 0] [0] = 0(1(q1(x1))) 1(q2(x1)) = [1 4 0] [5] [0 1 0] x1 + [1] [0 0 0] [0] > [1 4 0] [3] [0 1 0] x1 + [1] [0 0 0] [0] = q2(1(x1)) Following rules are (at-least) weakly oriented: 0(q1(x1)) = [1 4 0] [3] [0 1 0] x1 + [1] [0 0 0] [0] >= [1 4 0] [3] [0 1 0] x1 + [1] [0 0 0] [0] = q2(1(x1)) 0(q2(x1)) = [1 4 0] [4] [0 1 0] x1 + [2] [0 0 0] [0] >= [1 4 0] [2] [0 1 0] x1 + [2] [0 0 0] [0] = 0(q0(x1)) 1(q0(0(x1))) = [1 6 1] [7] [0 1 0] x1 + [2] [0 0 0] [0] >= [1 6 0] [5] [0 1 0] x1 + [2] [0 0 0] [0] = 0(0(q1(x1))) 1(q1(0(x1))) = [1 6 1] [7] [0 1 0] x1 + [1] [0 0 0] [0] >= [1 6 0] [6] [0 1 0] x1 + [1] [0 0 0] [0] = 1(0(q1(x1))) 1(q1(1(x1))) = [1 6 0] [4] [0 1 0] x1 + [0] [0 0 0] [0] >= [1 6 0] [4] [0 1 0] x1 + [0] [0 0 0] [0] = 1(1(q1(x1))) * Step 4: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 1(q1(1(x1))) -> 1(1(q1(x1))) - Weak TRS: 0(q1(x1)) -> q2(1(x1)) 0(q2(x1)) -> 0(q0(x1)) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q2(x1)) -> q2(1(x1)) - Signature: {0/1,1/1} / {q0/1,q1/1,q2/1} - Obligation: innermost derivational complexity wrt. signature {0,1,q0,q1,q2} + 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 0] x1 + [2] [0 1] [1] p(1) = [1 4] x1 + [0] [0 1] [1] p(q0) = [1 0] x1 + [3] [0 1] [0] p(q1) = [1 4] x1 + [4] [0 1] [0] p(q2) = [1 0] x1 + [4] [0 1] [0] Following rules are strictly oriented: 1(q1(1(x1))) = [1 12] x1 + [12] [0 1] [2] > [1 12] x1 + [8] [0 1] [2] = 1(1(q1(x1))) Following rules are (at-least) weakly oriented: 0(q1(x1)) = [1 4] x1 + [6] [0 1] [1] >= [1 4] x1 + [4] [0 1] [1] = q2(1(x1)) 0(q2(x1)) = [1 0] x1 + [6] [0 1] [1] >= [1 0] x1 + [5] [0 1] [1] = 0(q0(x1)) 1(q0(0(x1))) = [1 4] x1 + [9] [0 1] [2] >= [1 4] x1 + [8] [0 1] [2] = 0(0(q1(x1))) 1(q0(1(x1))) = [1 8] x1 + [7] [0 1] [2] >= [1 8] x1 + [6] [0 1] [2] = 0(1(q1(x1))) 1(q1(0(x1))) = [1 8] x1 + [14] [0 1] [2] >= [1 8] x1 + [10] [0 1] [2] = 1(0(q1(x1))) 1(q2(x1)) = [1 4] x1 + [4] [0 1] [1] >= [1 4] x1 + [4] [0 1] [1] = q2(1(x1)) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0(q1(x1)) -> q2(1(x1)) 0(q2(x1)) -> 0(q0(x1)) 1(q0(0(x1))) -> 0(0(q1(x1))) 1(q0(1(x1))) -> 0(1(q1(x1))) 1(q1(0(x1))) -> 1(0(q1(x1))) 1(q1(1(x1))) -> 1(1(q1(x1))) 1(q2(x1)) -> q2(1(x1)) - Signature: {0/1,1/1} / {q0/1,q1/1,q2/1} - Obligation: innermost derivational complexity wrt. signature {0,1,q0,q1,q2} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))