/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(2(0(x1))) -> 1(0(1(x1))) 0(2(1(x1))) -> 1(0(2(x1))) 0(2(R(x1))) -> 1(0(1(R(x1)))) 1(2(0(x1))) -> 2(0(1(x1))) 1(2(1(x1))) -> 2(0(2(x1))) 1(2(R(x1))) -> 2(0(1(R(x1)))) L(2(0(x1))) -> L(1(0(1(x1)))) L(2(1(x1))) -> L(1(0(2(x1)))) - Signature: {0/1,1/1,L/1} / {2/1,R/1} - Obligation: derivational complexity wrt. signature {0,1,2,L,R} + 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 + [8] p(1) = [1] x1 + [0] p(2) = [1] x1 + [9] p(L) = [1] x1 + [0] p(R) = [1] x1 + [0] Following rules are strictly oriented: 0(2(0(x1))) = [1] x1 + [25] > [1] x1 + [8] = 1(0(1(x1))) 0(2(R(x1))) = [1] x1 + [17] > [1] x1 + [8] = 1(0(1(R(x1)))) L(2(0(x1))) = [1] x1 + [17] > [1] x1 + [8] = L(1(0(1(x1)))) Following rules are (at-least) weakly oriented: 0(2(1(x1))) = [1] x1 + [17] >= [1] x1 + [17] = 1(0(2(x1))) 1(2(0(x1))) = [1] x1 + [17] >= [1] x1 + [17] = 2(0(1(x1))) 1(2(1(x1))) = [1] x1 + [9] >= [1] x1 + [26] = 2(0(2(x1))) 1(2(R(x1))) = [1] x1 + [9] >= [1] x1 + [17] = 2(0(1(R(x1)))) L(2(1(x1))) = [1] x1 + [9] >= [1] x1 + [17] = L(1(0(2(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: 0(2(1(x1))) -> 1(0(2(x1))) 1(2(0(x1))) -> 2(0(1(x1))) 1(2(1(x1))) -> 2(0(2(x1))) 1(2(R(x1))) -> 2(0(1(R(x1)))) L(2(1(x1))) -> L(1(0(2(x1)))) - Weak TRS: 0(2(0(x1))) -> 1(0(1(x1))) 0(2(R(x1))) -> 1(0(1(R(x1)))) L(2(0(x1))) -> L(1(0(1(x1)))) - Signature: {0/1,1/1,L/1} / {2/1,R/1} - Obligation: derivational complexity wrt. signature {0,1,2,L,R} + 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 0] [0] p(1) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(2) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(L) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] p(R) = [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] Following rules are strictly oriented: 1(2(R(x1))) = [1 0 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] > [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(1(R(x1)))) Following rules are (at-least) weakly oriented: 0(2(0(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(0(1(x1))) 0(2(1(x1))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 1(0(2(x1))) 0(2(R(x1))) = [1 0 0] [2] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 0] [1] [0 0 0] x1 + [0] [0 0 0] [0] = 1(0(1(R(x1)))) 1(2(0(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] = 2(0(1(x1))) 1(2(1(x1))) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 2(0(2(x1))) L(2(0(x1))) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] = L(1(0(1(x1)))) L(2(1(x1))) = [1 0 1] [0] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [1] [0 0 0] [1] = L(1(0(2(x1)))) * Step 3: NaturalMI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(2(1(x1))) -> 1(0(2(x1))) 1(2(0(x1))) -> 2(0(1(x1))) 1(2(1(x1))) -> 2(0(2(x1))) L(2(1(x1))) -> L(1(0(2(x1)))) - Weak TRS: 0(2(0(x1))) -> 1(0(1(x1))) 0(2(R(x1))) -> 1(0(1(R(x1)))) 1(2(R(x1))) -> 2(0(1(R(x1)))) L(2(0(x1))) -> L(1(0(1(x1)))) - Signature: {0/1,1/1,L/1} / {2/1,R/1} - Obligation: derivational complexity wrt. signature {0,1,2,L,R} + 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 1] x1 + [0] [0 0 0] [0] p(1) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [0] p(2) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 1] [1] p(L) = [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(R) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: L(2(1(x1))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [0] > [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = L(1(0(2(x1)))) Following rules are (at-least) weakly oriented: 0(2(0(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] = 1(0(1(x1))) 0(2(1(x1))) = [1 0 0] [0] [0 0 1] x1 + [1] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] = 1(0(2(x1))) 0(2(R(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] = 1(0(1(R(x1)))) 1(2(0(x1))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 2(0(1(x1))) 1(2(1(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 0] [1] = 2(0(2(x1))) 1(2(R(x1))) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = 2(0(1(R(x1)))) L(2(0(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] = L(1(0(1(x1)))) * Step 4: MI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: 0(2(1(x1))) -> 1(0(2(x1))) 1(2(0(x1))) -> 2(0(1(x1))) 1(2(1(x1))) -> 2(0(2(x1))) - Weak TRS: 0(2(0(x1))) -> 1(0(1(x1))) 0(2(R(x1))) -> 1(0(1(R(x1)))) 1(2(R(x1))) -> 2(0(1(R(x1)))) L(2(0(x1))) -> L(1(0(1(x1)))) L(2(1(x1))) -> L(1(0(2(x1)))) - Signature: {0/1,1/1,L/1} / {2/1,R/1} - Obligation: derivational complexity wrt. signature {0,1,2,L,R} + Applied Processor: MI {miKind = Automaton Nothing, miDimension = 3, miUArgs = NoUArgs, miURules = NoURules, miSelector = Just any strict-rules} + Details: We apply a matrix interpretation of kind Automaton Nothing: Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1 1 0] [0] [0 0 0] x_1 + [0] [0 1 0] [1] p(1) = [1 1 0] [0] [0 1 0] x_1 + [0] [0 0 1] [1] p(2) = [1 0 0] [0] [0 0 1] x_1 + [0] [0 0 0] [0] p(L) = [1 1 0] [0] [0 0 0] x_1 + [0] [0 0 0] [1] p(R) = [1 0 1] [0] [0 0 0] x_1 + [0] [0 0 0] [1] Following rules are strictly oriented: 0(2(1(x1))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 1] [2] > [1 0 1] [0] [0 0 0] x1 + [0] [0 0 1] [2] = 1(0(2(x1))) 1(2(0(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] = 2(0(1(x1))) 1(2(1(x1))) = [1 1 1] [1] [0 0 1] x1 + [1] [0 0 0] [1] > [1 0 1] [0] [0 0 1] x1 + [1] [0 0 0] [0] = 2(0(2(x1))) Following rules are (at-least) weakly oriented: 0(2(0(x1))) = [1 2 0] [1] [0 0 0] x1 + [0] [0 1 0] [2] >= [1 2 0] [0] [0 0 0] x1 + [0] [0 1 0] [2] = 1(0(1(x1))) 0(2(R(x1))) = [1 0 1] [1] [0 0 0] x1 + [0] [0 0 0] [2] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [2] = 1(0(1(R(x1)))) 1(2(R(x1))) = [1 0 1] [1] [0 0 0] x1 + [1] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [1] [0 0 0] [0] = 2(0(1(R(x1)))) L(2(0(x1))) = [1 2 0] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 2 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] = L(1(0(1(x1)))) L(2(1(x1))) = [1 1 1] [1] [0 0 0] x1 + [0] [0 0 0] [1] >= [1 0 1] [0] [0 0 0] x1 + [0] [0 0 0] [1] = L(1(0(2(x1)))) * Step 5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: 0(2(0(x1))) -> 1(0(1(x1))) 0(2(1(x1))) -> 1(0(2(x1))) 0(2(R(x1))) -> 1(0(1(R(x1)))) 1(2(0(x1))) -> 2(0(1(x1))) 1(2(1(x1))) -> 2(0(2(x1))) 1(2(R(x1))) -> 2(0(1(R(x1)))) L(2(0(x1))) -> L(1(0(1(x1)))) L(2(1(x1))) -> L(1(0(2(x1)))) - Signature: {0/1,1/1,L/1} / {2/1,R/1} - Obligation: derivational complexity wrt. signature {0,1,2,L,R} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))