YES(?,O(n^2)) 33.96/15.96 YES(?,O(n^2)) 33.96/15.97 33.96/15.97 Problem: 33.96/15.97 a(b(x1)) -> b(a(x1)) 33.96/15.97 b(a(x1)) -> a(c(b(x1))) 33.96/15.97 33.96/15.97 Proof: 33.96/15.97 Complexity Transformation Processor: 33.96/15.97 strict: 33.96/15.97 a(b(x1)) -> b(a(x1)) 33.96/15.97 b(a(x1)) -> a(c(b(x1))) 33.96/15.97 weak: 33.96/15.97 33.96/15.97 Matrix Interpretation Processor: dim=4 33.96/15.97 33.96/15.97 max_matrix: 33.96/15.97 [1 1 1 0] 33.96/15.97 [0 0 0 1] 33.96/15.97 [0 0 0 1] 33.96/15.97 [0 0 0 0] 33.96/15.97 interpretation: 33.96/15.97 [1 0 0 0] 33.96/15.97 [0 0 0 0] 33.96/15.97 [c](x0) = [0 0 0 0]x0 33.96/15.97 [0 0 0 0] , 33.96/15.97 33.96/15.97 [1 1 1 0] [1] 33.96/15.97 [0 0 0 1] [0] 33.96/15.97 [a](x0) = [0 0 0 1]x0 + [1] 33.96/15.97 [0 0 0 0] [1], 33.96/15.97 33.96/15.97 [1 1 1 0] [1] 33.96/15.97 [0 0 0 1] [0] 33.96/15.97 [b](x0) = [0 0 0 1]x0 + [1] 33.96/15.97 [0 0 0 0] [1] 33.96/15.97 orientation: 33.96/15.97 [1 1 1 2] [3] [1 1 1 2] [3] 33.96/15.97 [0 0 0 0] [1] [0 0 0 0] [1] 33.96/15.97 a(b(x1)) = [0 0 0 0]x1 + [2] >= [0 0 0 0]x1 + [2] = b(a(x1)) 33.96/15.97 [0 0 0 0] [1] [0 0 0 0] [1] 33.96/15.97 33.96/15.97 [1 1 1 2] [3] [1 1 1 0] [2] 33.96/15.97 [0 0 0 0] [1] [0 0 0 0] [0] 33.96/15.97 b(a(x1)) = [0 0 0 0]x1 + [2] >= [0 0 0 0]x1 + [1] = a(c(b(x1))) 33.96/15.97 [0 0 0 0] [1] [0 0 0 0] [1] 33.96/15.97 problem: 33.96/15.97 strict: 33.96/15.97 a(b(x1)) -> b(a(x1)) 33.96/15.97 weak: 33.96/15.97 b(a(x1)) -> a(c(b(x1))) 33.96/15.97 Matrix Interpretation Processor: dim=2 33.96/15.97 33.96/15.97 max_matrix: 33.96/15.97 [1 2] 33.96/15.97 [0 1] 33.96/15.97 interpretation: 33.96/15.97 [1 0] 33.96/15.97 [c](x0) = [0 0]x0, 33.96/15.97 33.96/15.97 [1 2] [0] 33.96/15.97 [a](x0) = [0 1]x0 + [2], 33.96/15.97 33.96/15.97 [0] 33.96/15.97 [b](x0) = x0 + [1] 33.96/15.97 orientation: 33.96/15.97 [1 2] [2] [1 2] [0] 33.96/15.97 a(b(x1)) = [0 1]x1 + [3] >= [0 1]x1 + [3] = b(a(x1)) 33.96/15.97 33.96/15.97 [1 2] [0] [1 0] [0] 33.96/15.97 b(a(x1)) = [0 1]x1 + [3] >= [0 0]x1 + [2] = a(c(b(x1))) 33.96/15.97 problem: 33.96/15.97 strict: 33.96/15.97 33.96/15.97 weak: 33.96/15.97 a(b(x1)) -> b(a(x1)) 33.96/15.97 b(a(x1)) -> a(c(b(x1))) 33.96/15.97 Qed 33.96/15.97 EOF