YES(?,O(n^2)) 1172.00/295.48 YES(?,O(n^2)) 1172.00/295.49 1172.00/295.49 Problem: 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 1172.00/295.49 Proof: 1172.00/295.49 Complexity Transformation Processor: 1172.00/295.49 strict: 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 weak: 1172.00/295.49 1172.00/295.49 Matrix Interpretation Processor: dim=1 1172.00/295.49 1172.00/295.49 max_matrix: 1172.00/295.49 1 1172.00/295.49 interpretation: 1172.00/295.49 [q2](x0) = x0 + 131, 1172.00/295.49 1172.00/295.49 [0](x0) = x0 + 131, 1172.00/295.49 1172.00/295.49 [q1](x0) = x0 + 112, 1172.00/295.49 1172.00/295.49 [q0](x0) = x0 + 126, 1172.00/295.49 1172.00/295.49 [1](x0) = x0 + 17 1172.00/295.49 orientation: 1172.00/295.49 1(q0(1(x1))) = x1 + 160 >= x1 + 260 = 0(1(q1(x1))) 1172.00/295.49 1172.00/295.49 1(q0(0(x1))) = x1 + 274 >= x1 + 374 = 0(0(q1(x1))) 1172.00/295.49 1172.00/295.49 1(q1(1(x1))) = x1 + 146 >= x1 + 146 = 1(1(q1(x1))) 1172.00/295.49 1172.00/295.49 1(q1(0(x1))) = x1 + 260 >= x1 + 260 = 1(0(q1(x1))) 1172.00/295.49 1172.00/295.49 0(q1(x1)) = x1 + 243 >= x1 + 148 = q2(1(x1)) 1172.00/295.49 1172.00/295.49 1(q2(x1)) = x1 + 148 >= x1 + 148 = q2(1(x1)) 1172.00/295.49 1172.00/295.49 0(q2(x1)) = x1 + 262 >= x1 + 257 = 0(q0(x1)) 1172.00/295.49 problem: 1172.00/295.49 strict: 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 weak: 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 Splitting Processor: 1172.00/295.49 strict: 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 weak: 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 Matrix Interpretation Processor: dim=3 1172.00/295.49 1172.00/295.49 max_matrix: 1172.00/295.49 [1 1 1] 1172.00/295.49 [0 1 1] 1172.00/295.49 [0 0 0] 1172.00/295.49 interpretation: 1172.00/295.49 [1 0 0] [0] 1172.00/295.49 [q2](x0) = [0 1 1]x0 + [1] 1172.00/295.49 [0 0 0] [0], 1172.00/295.49 1172.00/295.49 [1 0 1] [0] 1172.00/295.49 [0](x0) = [0 1 1]x0 + [1] 1172.00/295.49 [0 0 0] [0], 1172.00/295.49 1172.00/295.49 [1 1 1] [0] 1172.00/295.49 [q1](x0) = [0 1 1]x0 + [0] 1172.00/295.49 [0 0 0] [1], 1172.00/295.49 1172.00/295.49 [1 0 0] [0] 1172.00/295.49 [q0](x0) = [0 1 0]x0 + [1] 1172.00/295.49 [0 0 0] [0], 1172.00/295.49 1172.00/295.49 [1 1 1] [0] 1172.00/295.49 [1](x0) = [0 1 1]x0 + [1] 1172.00/295.49 [0 0 0] [0] 1172.00/295.49 orientation: 1172.00/295.49 [1 2 2] [2] [1 2 2] [1] 1172.00/295.49 1(q0(1(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [3] = 0(1(q1(x1))) 1172.00/295.49 [0 0 0] [0] [0 0 0] [0] 1172.00/295.49 1172.00/295.49 [1 1 2] [2] [1 1 1] [1] 1172.00/295.49 1(q0(0(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [3] = 0(0(q1(x1))) 1172.00/295.49 [0 0 0] [0] [0 0 0] [0] 1172.00/295.49 1172.00/295.49 [1 1 1] [1] [1 1 1] [0] 1172.00/295.49 1(q2(x1)) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = q2(1(x1)) 1172.00/295.49 [0 0 0] [0] [0 0 0] [0] 1172.00/295.49 1172.00/295.49 [1 1 1] [1] [1 1 1] [0] 1172.00/295.49 0(q1(x1)) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = q2(1(x1)) 1172.00/295.49 [0 0 0] [0] [0 0 0] [0] 1172.00/295.49 1172.00/295.49 [1 0 0] [0] [1 0 0] [0] 1172.00/295.49 0(q2(x1)) = [0 1 1]x1 + [2] >= [0 1 0]x1 + [2] = 0(q0(x1)) 1172.00/295.49 [0 0 0] [0] [0 0 0] [0] 1172.00/295.49 1172.00/295.49 [1 3 3] [3] [1 3 3] [3] 1172.00/295.49 1(q1(1(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [3] = 1(1(q1(x1))) 1172.00/295.49 [0 0 0] [0] [0 0 0] [0] 1172.00/295.49 1172.00/295.49 [1 2 3] [3] [1 2 2] [3] 1172.00/295.49 1(q1(0(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [3] = 1(0(q1(x1))) 1172.00/295.49 [0 0 0] [0] [0 0 0] [0] 1172.00/295.49 problem: 1172.00/295.49 strict: 1172.00/295.49 1172.00/295.49 weak: 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 Qed 1172.00/295.49 1172.00/295.49 strict: 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 weak: 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 Splitting Processor: 1172.00/295.49 strict: 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 weak: 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 Matrix Interpretation Processor: dim=3 1172.00/295.49 1172.00/295.49 max_matrix: 1172.00/295.49 [1 1 0] 1172.00/295.49 [0 1 1] 1172.00/295.49 [0 0 0] 1172.00/295.49 interpretation: 1172.00/295.49 [1 0 0] [0] 1172.00/295.49 [q2](x0) = [0 1 1]x0 + [0] 1172.00/295.49 [0 0 0] [1], 1172.00/295.49 1172.00/295.49 [1 0 0] [0] 1172.00/295.49 [0](x0) = [0 1 1]x0 + [1] 1172.00/295.49 [0 0 0] [1], 1172.00/295.49 1172.00/295.49 [1 1 0] 1172.00/295.49 [q1](x0) = [0 1 1]x0 1172.00/295.49 [0 0 0] , 1172.00/295.49 1172.00/295.49 [1 0 0] [0] 1172.00/295.49 [q0](x0) = [0 1 1]x0 + [1] 1172.00/295.49 [0 0 0] [0], 1172.00/295.49 1172.00/295.49 [1 1 0] [0] 1172.00/295.49 [1](x0) = [0 1 1]x0 + [0] 1172.00/295.49 [0 0 0] [1] 1172.00/295.49 orientation: 1172.00/295.49 [1 3 2] [1] [1 3 2] [0] 1172.00/295.49 1(q1(1(x1))) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [1] = 1(1(q1(x1))) 1172.00/295.49 [0 0 0] [1] [0 0 0] [1] 1172.00/295.49 1172.00/295.49 [1 2 2] [3] [1 2 1] [1] 1172.00/295.49 1(q1(0(x1))) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = 1(0(q1(x1))) 1172.00/295.49 [0 0 0] [1] [0 0 0] [1] 1172.00/295.49 1172.00/295.49 [1 2 1] [2] [1 2 1] [0] 1172.00/295.49 1(q0(1(x1))) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = 0(1(q1(x1))) 1172.00/295.49 [0 0 0] [1] [0 0 0] [1] 1172.00/295.49 1172.00/295.49 [1 1 1] [3] [1 1 0] [0] 1172.00/295.49 1(q0(0(x1))) = [0 1 1]x1 + [3] >= [0 1 1]x1 + [3] = 0(0(q1(x1))) 1172.00/295.49 [0 0 0] [1] [0 0 0] [1] 1172.00/295.49 1172.00/295.49 [1 1 1] [0] [1 1 0] [0] 1172.00/295.49 1(q2(x1)) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [1] = q2(1(x1)) 1172.00/295.49 [0 0 0] [1] [0 0 0] [1] 1172.00/295.49 1172.00/295.49 [1 1 0] [0] [1 1 0] [0] 1172.00/295.49 0(q1(x1)) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [1] = q2(1(x1)) 1172.00/295.49 [0 0 0] [1] [0 0 0] [1] 1172.00/295.49 1172.00/295.49 [1 0 0] [0] [1 0 0] [0] 1172.00/295.49 0(q2(x1)) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = 0(q0(x1)) 1172.00/295.49 [0 0 0] [1] [0 0 0] [1] 1172.00/295.49 problem: 1172.00/295.49 strict: 1172.00/295.49 1172.00/295.49 weak: 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 Qed 1172.00/295.49 1172.00/295.49 strict: 1172.00/295.49 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.00/295.49 weak: 1172.00/295.49 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.00/295.49 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.00/295.49 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.00/295.49 1(q2(x1)) -> q2(1(x1)) 1172.00/295.49 0(q1(x1)) -> q2(1(x1)) 1172.00/295.49 0(q2(x1)) -> 0(q0(x1)) 1172.00/295.49 Matrix Interpretation Processor: dim=3 1172.00/295.49 1172.00/295.49 max_matrix: 1172.00/295.49 [1 1 0] 1172.00/295.49 [0 1 1] 1172.00/295.49 [0 0 0] 1172.00/295.49 interpretation: 1172.00/295.49 [1 0 0] [1] 1172.00/295.49 [q2](x0) = [0 1 1]x0 + [0] 1172.00/295.49 [0 0 0] [1], 1172.00/295.49 1172.00/295.49 [1 0 0] [1] 1172.00/295.49 [0](x0) = [0 1 1]x0 + [0] 1172.00/295.49 [0 0 0] [1], 1172.00/295.49 1172.00/295.49 [1 1 0] [0] 1172.00/295.49 [q1](x0) = [0 1 1]x0 + [1] 1172.00/295.49 [0 0 0] [0], 1172.00/295.49 1172.00/295.49 [1 0 0] [1] 1172.00/295.49 [q0](x0) = [0 1 1]x0 + [0] 1172.00/295.49 [0 0 0] [1], 1172.00/295.49 1172.00/295.49 [1 1 0] [0] 1172.00/295.49 [1](x0) = [0 1 1]x0 + [0] 1172.00/295.49 [0 0 0] [1] 1172.00/295.49 orientation: 1172.00/295.49 [1 2 2] [3] [1 2 1] [2] 1172.00/295.49 1(q1(0(x1))) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = 1(0(q1(x1))) 1172.00/295.49 [0 0 0] [1] [0 0 0] [1] 1172.09/295.50 1172.09/295.50 [1 2 1] [2] [1 2 1] [2] 1172.09/295.50 1(q0(1(x1))) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = 0(1(q1(x1))) 1172.09/295.50 [0 0 0] [1] [0 0 0] [1] 1172.09/295.50 1172.09/295.50 [1 1 1] [3] [1 1 0] [2] 1172.09/295.50 1(q0(0(x1))) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = 0(0(q1(x1))) 1172.09/295.50 [0 0 0] [1] [0 0 0] [1] 1172.09/295.50 1172.09/295.50 [1 1 1] [1] [1 1 0] [1] 1172.09/295.50 1(q2(x1)) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [1] = q2(1(x1)) 1172.09/295.50 [0 0 0] [1] [0 0 0] [1] 1172.09/295.50 1172.09/295.50 [1 1 0] [1] [1 1 0] [1] 1172.09/295.50 0(q1(x1)) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [1] = q2(1(x1)) 1172.09/295.50 [0 0 0] [1] [0 0 0] [1] 1172.09/295.50 1172.09/295.50 [1 0 0] [2] [1 0 0] [2] 1172.09/295.50 0(q2(x1)) = [0 1 1]x1 + [1] >= [0 1 1]x1 + [1] = 0(q0(x1)) 1172.09/295.50 [0 0 0] [1] [0 0 0] [1] 1172.09/295.50 1172.09/295.50 [1 3 2] [2] [1 3 2] [2] 1172.09/295.50 1(q1(1(x1))) = [0 1 1]x1 + [2] >= [0 1 1]x1 + [2] = 1(1(q1(x1))) 1172.09/295.50 [0 0 0] [1] [0 0 0] [1] 1172.09/295.50 problem: 1172.09/295.50 strict: 1172.09/295.50 1172.09/295.50 weak: 1172.09/295.50 1(q1(0(x1))) -> 1(0(q1(x1))) 1172.09/295.50 1(q0(1(x1))) -> 0(1(q1(x1))) 1172.09/295.50 1(q0(0(x1))) -> 0(0(q1(x1))) 1172.09/295.50 1(q2(x1)) -> q2(1(x1)) 1172.09/295.50 0(q1(x1)) -> q2(1(x1)) 1172.09/295.50 0(q2(x1)) -> 0(q0(x1)) 1172.09/295.50 1(q1(1(x1))) -> 1(1(q1(x1))) 1172.09/295.50 Qed 1172.09/295.50 EOF