YES Problem: 2(7(x1)) -> 1(8(x1)) 2(8(1(x1))) -> 8(x1) 2(8(x1)) -> 4(x1) 5(9(x1)) -> 0(x1) 4(x1) -> 5(2(3(x1))) 5(3(x1)) -> 6(0(x1)) 2(8(x1)) -> 7(x1) 4(7(x1)) -> 1(3(x1)) 5(2(6(x1))) -> 6(2(4(x1))) 9(7(x1)) -> 7(5(x1)) 7(2(x1)) -> 4(x1) 7(0(x1)) -> 9(3(x1)) 6(9(x1)) -> 9(x1) 9(5(9(x1))) -> 5(7(x1)) 4(x1) -> 9(6(6(x1))) 9(x1) -> 6(7(x1)) 6(2(x1)) -> 7(7(x1)) 2(4(x1)) -> 0(7(x1)) 6(6(x1)) -> 3(x1) 0(3(x1)) -> 5(3(x1)) Proof: String Reversal Processor: 7(2(x1)) -> 8(1(x1)) 1(8(2(x1))) -> 8(x1) 8(2(x1)) -> 4(x1) 9(5(x1)) -> 0(x1) 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 8(2(x1)) -> 7(x1) 7(4(x1)) -> 3(1(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 7(9(x1)) -> 5(7(x1)) 2(7(x1)) -> 4(x1) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 9(5(9(x1))) -> 7(5(x1)) 4(x1) -> 6(6(9(x1))) 9(x1) -> 7(6(x1)) 2(6(x1)) -> 7(7(x1)) 4(2(x1)) -> 7(0(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [6](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [3](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [5](x0) = [0 0 0]x0 [0 1 0] , [1 1 0] [9](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [4](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0] [1](x0) = [0 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [8](x0) = [0 0 1]x0 [0 0 0] , [1 1 1] [0] [2](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [1 1 0] [7](x0) = [0 0 0]x0 [0 0 0] orientation: [1 1 2] [1 1 0] 7(2(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 8(1(x1)) [0 0 0] [0 0 0] [1 1 2] [1] [1 0 0] 1(8(2(x1))) = [0 0 1]x1 + [2] >= [0 0 1]x1 = 8(x1) [0 0 0] [0] [0 0 0] [1 1 1] [0] [1 1 0] 8(2(x1)) = [0 0 1]x1 + [1] >= [0 0 0]x1 = 4(x1) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] 9(5(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 0(x1) [0 0 0] [0 0 0] [1 1 0] [1 1 0] 4(x1) = [0 0 0]x1 >= [0 0 0]x1 = 3(2(5(x1))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] 3(5(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 0(6(x1)) [0 0 0] [0 0 0] [1 1 1] [0] [1 1 0] 8(2(x1)) = [0 0 1]x1 + [1] >= [0 0 0]x1 = 7(x1) [0 0 0] [0] [0 0 0] [1 1 0] [1 1 0] 7(4(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(1(x1)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] 6(2(5(x1))) = [0 1 0]x1 >= [0 0 0]x1 = 4(2(6(x1))) [0 0 0] [0 0 0] [1 1 0] [1 1 0] 7(9(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 5(7(x1)) [0 0 0] [0 0 0] [1 1 0] [0] [1 1 0] 2(7(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 4(x1) [0 0 0] [1] [0 0 0] [1 1 0] [1 1 0] 0(7(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(9(x1)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] 9(6(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 9(x1) [0 0 0] [0 0 0] [1 1 0] [1 0 0] 9(5(9(x1))) = [0 0 0]x1 >= [0 0 0]x1 = 7(5(x1)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] 4(x1) = [0 0 0]x1 >= [0 0 0]x1 = 6(6(9(x1))) [0 0 0] [0 0 0] [1 1 0] [1 1 0] 9(x1) = [0 0 0]x1 >= [0 0 0]x1 = 7(6(x1)) [0 0 0] [0 0 0] [1 1 0] [0] [1 1 0] 2(6(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 7(7(x1)) [0 0 0] [1] [0 0 0] [1 1 2] [1 0 0] 4(2(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 7(0(x1)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] 6(6(x1)) = [0 1 0]x1 >= [0 0 0]x1 = 3(x1) [0 0 0] [0 0 0] [1 0 0] [1 0 0] 3(0(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(5(x1)) [0 0 0] [0 0 0] problem: 7(2(x1)) -> 8(1(x1)) 8(2(x1)) -> 4(x1) 9(5(x1)) -> 0(x1) 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 8(2(x1)) -> 7(x1) 7(4(x1)) -> 3(1(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 7(9(x1)) -> 5(7(x1)) 2(7(x1)) -> 4(x1) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 9(5(9(x1))) -> 7(5(x1)) 4(x1) -> 6(6(9(x1))) 9(x1) -> 7(6(x1)) 2(6(x1)) -> 7(7(x1)) 4(2(x1)) -> 7(0(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) String Reversal Processor: 2(7(x1)) -> 1(8(x1)) 2(8(x1)) -> 4(x1) 5(9(x1)) -> 0(x1) 4(x1) -> 5(2(3(x1))) 5(3(x1)) -> 6(0(x1)) 2(8(x1)) -> 7(x1) 4(7(x1)) -> 1(3(x1)) 5(2(6(x1))) -> 6(2(4(x1))) 9(7(x1)) -> 7(5(x1)) 7(2(x1)) -> 4(x1) 7(0(x1)) -> 9(3(x1)) 6(9(x1)) -> 9(x1) 9(5(9(x1))) -> 5(7(x1)) 4(x1) -> 9(6(6(x1))) 9(x1) -> 6(7(x1)) 6(2(x1)) -> 7(7(x1)) 2(4(x1)) -> 0(7(x1)) 6(6(x1)) -> 3(x1) 0(3(x1)) -> 5(3(x1)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [6](x0) = [1 0 0]x0 + [1] [1 0 0] [1], [1 0 0] [3](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [0](x0) = [1 0 0]x0 + [1] [1 0 0] [1], [1 0 0] [0] [5](x0) = [1 0 0]x0 + [1] [1 0 0] [1], [1 0 0] [0] [9](x0) = [1 0 0]x0 + [1] [1 0 0] [1], [1 0 0] [0] [4](x0) = [1 0 0]x0 + [1] [1 0 0] [1], [1 0 0] [0] [1](x0) = [0 0 0]x0 + [0] [0 1 0] [1], [1 0 1] [0] [8](x0) = [1 0 0]x0 + [0] [1 1 1] [1], [1 1 1] [0] [2](x0) = [0 0 1]x0 + [0] [1 0 0] [1], [1 0 0] [7](x0) = [0 0 1]x0 [0 0 1] orientation: [1 0 2] [0] [1 0 1] [0] 2(7(x1)) = [0 0 1]x1 + [0] >= [0 0 0]x1 + [0] = 1(8(x1)) [1 0 0] [1] [1 0 0] [1] [3 1 2] [1] [1 0 0] [0] 2(8(x1)) = [1 1 1]x1 + [1] >= [1 0 0]x1 + [1] = 4(x1) [1 0 1] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 5(9(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 0(x1) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 4(x1) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 5(2(3(x1))) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 5(3(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 6(0(x1)) [1 0 0] [1] [1 0 0] [1] [3 1 2] [1] [1 0 0] 2(8(x1)) = [1 1 1]x1 + [1] >= [0 0 1]x1 = 7(x1) [1 0 1] [1] [0 0 1] [1 0 0] [0] [1 0 0] [0] 4(7(x1)) = [1 0 0]x1 + [1] >= [0 0 0]x1 + [0] = 1(3(x1)) [1 0 0] [1] [0 0 0] [1] [3 0 0] [2] [3 0 0] [2] 5(2(6(x1))) = [3 0 0]x1 + [3] >= [3 0 0]x1 + [3] = 6(2(4(x1))) [3 0 0] [3] [3 0 0] [3] [1 0 0] [0] [1 0 0] [0] 9(7(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 7(5(x1)) [1 0 0] [1] [1 0 0] [1] [1 1 1] [0] [1 0 0] [0] 7(2(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 4(x1) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 7(0(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 9(3(x1)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 6(9(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 9(x1) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 9(5(9(x1))) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 5(7(x1)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 4(x1) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 9(6(6(x1))) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 9(x1) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 6(7(x1)) [1 0 0] [1] [1 0 0] [1] [1 1 1] [0] [1 0 0] 6(2(x1)) = [1 1 1]x1 + [1] >= [0 0 1]x1 = 7(7(x1)) [1 1 1] [1] [0 0 1] [3 0 0] [2] [1 0 0] [0] 2(4(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 0(7(x1)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] 6(6(x1)) = [1 0 0]x1 + [1] >= [0 0 0]x1 = 3(x1) [1 0 0] [1] [0 0 0] [1 0 0] [0] [1 0 0] [0] 0(3(x1)) = [1 0 0]x1 + [1] >= [1 0 0]x1 + [1] = 5(3(x1)) [1 0 0] [1] [1 0 0] [1] problem: 2(7(x1)) -> 1(8(x1)) 5(9(x1)) -> 0(x1) 4(x1) -> 5(2(3(x1))) 5(3(x1)) -> 6(0(x1)) 4(7(x1)) -> 1(3(x1)) 5(2(6(x1))) -> 6(2(4(x1))) 9(7(x1)) -> 7(5(x1)) 7(2(x1)) -> 4(x1) 7(0(x1)) -> 9(3(x1)) 6(9(x1)) -> 9(x1) 9(5(9(x1))) -> 5(7(x1)) 4(x1) -> 9(6(6(x1))) 9(x1) -> 6(7(x1)) 6(2(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 0(3(x1)) -> 5(3(x1)) String Reversal Processor: 7(2(x1)) -> 8(1(x1)) 9(5(x1)) -> 0(x1) 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 7(4(x1)) -> 3(1(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 7(9(x1)) -> 5(7(x1)) 2(7(x1)) -> 4(x1) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 9(5(9(x1))) -> 7(5(x1)) 4(x1) -> 6(6(9(x1))) 9(x1) -> 7(6(x1)) 2(6(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [6](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [3](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [5](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [9](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1] [4](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [0] [1](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [8](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [0] [2](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [0] [7](x0) = [0 0 0]x0 + [0] [0 0 0] [1] orientation: [1 0 1] [0] [1 0 0] [0] 7(2(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 8(1(x1)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] 9(5(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 0(x1) [0 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1] 4(x1) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 3(2(5(x1))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] 3(5(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 0(6(x1)) [0 0 0] [0 0 0] [1 0 0] [1] [1 0 0] 7(4(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 3(1(x1)) [0 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1] 6(2(5(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 4(2(6(x1))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] 7(9(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 5(7(x1)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] [1] 2(7(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 4(x1) [0 0 0] [1] [0 0 0] [0] [1 0 0] [1 0 0] 0(7(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(9(x1)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] 9(6(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 9(x1) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] 9(5(9(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 7(5(x1)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1] [1 0 0] 4(x1) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 6(6(9(x1))) [0 0 0] [0] [0 0 0] [1 0 0] [0] [1 0 0] [0] 9(x1) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 7(6(x1)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] 2(6(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 7(7(x1)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [1 0 0] 6(6(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(x1) [0 0 0] [0 0 0] [1 0 0] [1 0 0] 3(0(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(5(x1)) [0 0 0] [0 0 0] problem: 7(2(x1)) -> 8(1(x1)) 9(5(x1)) -> 0(x1) 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 7(9(x1)) -> 5(7(x1)) 2(7(x1)) -> 4(x1) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 9(5(9(x1))) -> 7(5(x1)) 9(x1) -> 7(6(x1)) 2(6(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [6](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [3](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [0](x0) = [0 0 0]x0 [0 0 0] , [1 0 1] [5](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [9](x0) = [0 0 1]x0 + [0] [0 0 1] [1], [1 0 1] [4](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [8](x0) = [0 1 0]x0 [0 0 0] , [1 1 0] [2](x0) = [0 0 0]x0 [0 1 0] , [1 0 0] [7](x0) = [0 0 1]x0 [0 0 0] orientation: [1 1 0] [1 0 0] 7(2(x1)) = [0 1 0]x1 >= [0 1 0]x1 = 8(1(x1)) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] 9(5(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 0(x1) [0 0 0] [1] [0 0 0] [1 0 1] [1 0 1] 4(x1) = [0 0 0]x1 >= [0 0 0]x1 = 3(2(5(x1))) [0 0 0] [0 0 0] [1 0 1] [1 0 1] 3(5(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 0(6(x1)) [0 0 0] [0 0 0] [1 0 1] [1 0 0] 6(2(5(x1))) = [0 0 0]x1 >= [0 0 0]x1 = 4(2(6(x1))) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] 7(9(x1)) = [0 0 1]x1 + [1] >= [0 0 0]x1 = 5(7(x1)) [0 0 0] [0] [0 0 0] [1 0 1] [1 0 1] 2(7(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 4(x1) [0 0 1] [0 0 0] [1 0 0] [1 0 0] 0(7(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(9(x1)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] 9(6(x1)) = [0 0 1]x1 + [0] >= [0 0 1]x1 + [0] = 9(x1) [0 0 1] [1] [0 0 1] [1] [1 0 1] [1] [1 0 1] 9(5(9(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 7(5(x1)) [0 0 0] [1] [0 0 0] [1 0 0] [0] [1 0 0] 9(x1) = [0 0 1]x1 + [0] >= [0 0 1]x1 = 7(6(x1)) [0 0 1] [1] [0 0 0] [1 0 0] [1 0 0] 2(6(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 7(7(x1)) [0 0 0] [0 0 0] [1 0 0] [1 0 0] 6(6(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(x1) [0 0 1] [0 0 0] [1 0 1] [1 0 1] 3(0(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(5(x1)) [0 0 0] [0 0 0] problem: 7(2(x1)) -> 8(1(x1)) 9(5(x1)) -> 0(x1) 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 7(9(x1)) -> 5(7(x1)) 2(7(x1)) -> 4(x1) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 9(x1) -> 7(6(x1)) 2(6(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [6](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [3](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [5](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [0] [9](x0) = [0 0 0]x0 + [0] [1 0 0] [1], [1 1 0] [4](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [8](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 1 1] [0] [2](x0) = [0 1 0]x0 + [0] [1 0 0] [1], [1 0 0] [0] [7](x0) = [0 0 0]x0 + [0] [1 1 0] [1] orientation: [1 1 1] [0] [1 0 0] [0] 7(2(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 8(1(x1)) [1 2 1] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] 9(5(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 0(x1) [1 0 0] [1] [0 0 0] [1 1 0] [1 1 0] 4(x1) = [0 0 0]x1 >= [0 0 0]x1 = 3(2(5(x1))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] 3(5(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 0(6(x1)) [0 0 0] [0 0 0] [1 1 0] [1 0 0] 6(2(5(x1))) = [0 0 0]x1 >= [0 0 0]x1 = 4(2(6(x1))) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] 7(9(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 5(7(x1)) [1 0 0] [1] [0 0 0] [2 1 0] [1] [1 1 0] 2(7(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 4(x1) [1 0 0] [1] [0 0 0] [1 0 0] [1 0 0] 0(7(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(9(x1)) [0 0 0] [0 0 0] [1 0 0] [0] [1 0 0] [0] 9(6(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 9(x1) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 9(x1) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 7(6(x1)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [0] [1 0 0] [0] 2(6(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 7(7(x1)) [1 0 0] [1] [1 0 0] [1] [1 0 0] [1 0 0] 6(6(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(x1) [0 0 0] [0 0 0] [1 0 0] [1 0 0] 3(0(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(5(x1)) [0 0 0] [0 0 0] problem: 7(2(x1)) -> 8(1(x1)) 9(5(x1)) -> 0(x1) 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 7(9(x1)) -> 5(7(x1)) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 9(x1) -> 7(6(x1)) 2(6(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [6](x0) = x0, [3](x0) = x0, [0](x0) = 1x0, [5](x0) = 1x0, [9](x0) = 1x0, [4](x0) = 1x0, [1](x0) = x0, [8](x0) = x0, [2](x0) = x0, [7](x0) = x0 orientation: 7(2(x1)) = x1 >= x1 = 8(1(x1)) 9(5(x1)) = 2x1 >= 1x1 = 0(x1) 4(x1) = 1x1 >= 1x1 = 3(2(5(x1))) 3(5(x1)) = 1x1 >= 1x1 = 0(6(x1)) 6(2(5(x1))) = 1x1 >= 1x1 = 4(2(6(x1))) 7(9(x1)) = 1x1 >= 1x1 = 5(7(x1)) 0(7(x1)) = 1x1 >= 1x1 = 3(9(x1)) 9(6(x1)) = 1x1 >= 1x1 = 9(x1) 9(x1) = 1x1 >= x1 = 7(6(x1)) 2(6(x1)) = x1 >= x1 = 7(7(x1)) 6(6(x1)) = x1 >= x1 = 3(x1) 3(0(x1)) = 1x1 >= 1x1 = 3(5(x1)) problem: 7(2(x1)) -> 8(1(x1)) 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 7(9(x1)) -> 5(7(x1)) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 2(6(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) String Reversal Processor: 2(7(x1)) -> 1(8(x1)) 4(x1) -> 5(2(3(x1))) 5(3(x1)) -> 6(0(x1)) 5(2(6(x1))) -> 6(2(4(x1))) 9(7(x1)) -> 7(5(x1)) 7(0(x1)) -> 9(3(x1)) 6(9(x1)) -> 9(x1) 6(2(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 0(3(x1)) -> 5(3(x1)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [6](x0) = [0 0 0]x0 + [1] [0 1 0] [1], [1 0 0] [0] [3](x0) = [0 0 0]x0 + [1] [0 0 0] [0], [1 0 0] [0] [0](x0) = [0 0 0]x0 + [1] [0 1 0] [1], [1 0 0] [0] [5](x0) = [0 1 1]x0 + [0] [0 1 0] [1], [1 0 0] [0] [9](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1] [4](x0) = [0 0 0]x0 + [0] [0 0 0] [1], [1 0 0] [1](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [8](x0) = [0 0 0]x0 [0 0 0] , [1 1 1] [2](x0) = [0 0 0]x0 [0 0 1] , [1 0 0] [0] [7](x0) = [0 0 0]x0 + [0] [0 0 0] [1] orientation: [1 0 0] [1] [1 0 0] 2(7(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 1(8(x1)) [0 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1] 4(x1) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 5(2(3(x1))) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] 5(3(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = 6(0(x1)) [0 0 0] [2] [0 0 0] [2] [1 1 0] [2] [1 0 0] [2] 5(2(6(x1))) = [0 1 0]x1 + [1] >= [0 0 0]x1 + [1] = 6(2(4(x1))) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] 9(7(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 7(5(x1)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] 7(0(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 9(3(x1)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] 6(9(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [0] = 9(x1) [0 0 0] [1] [0 0 0] [1] [1 1 1] [0] [1 0 0] [0] 6(2(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [0] = 7(7(x1)) [0 0 0] [1] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] 6(6(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = 3(x1) [0 0 0] [2] [0 0 0] [0] [1 0 0] [0] [1 0 0] [0] 0(3(x1)) = [0 0 0]x1 + [1] >= [0 0 0]x1 + [1] = 5(3(x1)) [0 0 0] [2] [0 0 0] [2] problem: 4(x1) -> 5(2(3(x1))) 5(3(x1)) -> 6(0(x1)) 5(2(6(x1))) -> 6(2(4(x1))) 9(7(x1)) -> 7(5(x1)) 7(0(x1)) -> 9(3(x1)) 6(9(x1)) -> 9(x1) 6(2(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 0(3(x1)) -> 5(3(x1)) String Reversal Processor: 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 7(9(x1)) -> 5(7(x1)) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 2(6(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [6](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [3](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [0](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [5](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [9](x0) = [1 1 0]x0 + [1] [1 0 0] [0], [1 1 0] [4](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [0] [2](x0) = [1 0 0]x0 + [1] [1 0 0] [0], [1 1 0] [7](x0) = [0 0 0]x0 [0 0 0] orientation: [1 1 0] [1 1 0] 4(x1) = [0 0 0]x1 >= [0 0 0]x1 = 3(2(5(x1))) [0 0 0] [0 0 0] [1 1 0] [1 1 0] 3(5(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 0(6(x1)) [0 0 0] [0 0 0] [2 2 0] [1] [2 2 0] [1] 6(2(5(x1))) = [0 0 0]x1 + [0] >= [0 0 0]x1 + [0] = 4(2(6(x1))) [0 0 0] [0] [0 0 0] [0] [2 1 0] [1] [1 1 0] 7(9(x1)) = [0 0 0]x1 + [0] >= [0 0 0]x1 = 5(7(x1)) [0 0 0] [0] [0 0 0] [1 1 0] [1 0 0] 0(7(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(9(x1)) [0 0 0] [0 0 0] [1 1 0] [0] [1 0 0] [0] 9(6(x1)) = [1 1 0]x1 + [1] >= [1 1 0]x1 + [1] = 9(x1) [1 1 0] [0] [1 0 0] [0] [1 1 0] [0] [1 1 0] 2(6(x1)) = [1 1 0]x1 + [1] >= [0 0 0]x1 = 7(7(x1)) [1 1 0] [0] [0 0 0] [1 1 0] [1 0 0] 6(6(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(x1) [0 0 0] [0 0 0] [1 1 0] [1 1 0] 3(0(x1)) = [0 0 0]x1 >= [0 0 0]x1 = 3(5(x1)) [0 0 0] [0 0 0] problem: 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 0(7(x1)) -> 3(9(x1)) 9(6(x1)) -> 9(x1) 2(6(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Arctic Interpretation Processor: dimension: 1 interpretation: [6](x0) = x0, [3](x0) = x0, [0](x0) = 1x0, [5](x0) = 1x0, [9](x0) = x0, [4](x0) = 1x0, [2](x0) = x0, [7](x0) = x0 orientation: 4(x1) = 1x1 >= 1x1 = 3(2(5(x1))) 3(5(x1)) = 1x1 >= 1x1 = 0(6(x1)) 6(2(5(x1))) = 1x1 >= 1x1 = 4(2(6(x1))) 0(7(x1)) = 1x1 >= x1 = 3(9(x1)) 9(6(x1)) = x1 >= x1 = 9(x1) 2(6(x1)) = x1 >= x1 = 7(7(x1)) 6(6(x1)) = x1 >= x1 = 3(x1) 3(0(x1)) = 1x1 >= 1x1 = 3(5(x1)) problem: 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 9(6(x1)) -> 9(x1) 2(6(x1)) -> 7(7(x1)) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Arctic Interpretation Processor: dimension: 2 interpretation: [0 0] [6](x0) = [0 0]x0, [0 -&] [3](x0) = [-& -&]x0, [0 0 ] [0](x0) = [-& -&]x0, [0 0] [5](x0) = [2 2]x0, [0 0] [9](x0) = [0 0]x0, [1 1 ] [4](x0) = [-& 0 ]x0, [1 -&] [2](x0) = [-& 0 ]x0, [0 0 ] [7](x0) = [-& -&]x0 orientation: [1 1 ] [1 1 ] 4(x1) = [-& 0 ]x1 >= [-& -&]x1 = 3(2(5(x1))) [0 0 ] [0 0 ] 3(5(x1)) = [-& -&]x1 >= [-& -&]x1 = 0(6(x1)) [2 2] [2 2] 6(2(5(x1))) = [2 2]x1 >= [0 0]x1 = 4(2(6(x1))) [0 0] [0 0] 9(6(x1)) = [0 0]x1 >= [0 0]x1 = 9(x1) [1 1] [0 0 ] 2(6(x1)) = [0 0]x1 >= [-& -&]x1 = 7(7(x1)) [0 0] [0 -&] 6(6(x1)) = [0 0]x1 >= [-& -&]x1 = 3(x1) [0 0 ] [0 0 ] 3(0(x1)) = [-& -&]x1 >= [-& -&]x1 = 3(5(x1)) problem: 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 6(2(5(x1))) -> 4(2(6(x1))) 9(6(x1)) -> 9(x1) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Arctic Interpretation Processor: dimension: 2 interpretation: [0 0 ] [6](x0) = [0 -&]x0, [0 -&] [3](x0) = [-& -&]x0, [0 0 ] [0](x0) = [-& -&]x0, [0 0] [5](x0) = [1 0]x0, [0 -&] [9](x0) = [3 3 ]x0, [1 0 ] [4](x0) = [-& -&]x0, [0 0] [2](x0) = [2 3]x0 orientation: [1 0 ] [1 0 ] 4(x1) = [-& -&]x1 >= [-& -&]x1 = 3(2(5(x1))) [0 0 ] [0 0 ] 3(5(x1)) = [-& -&]x1 >= [-& -&]x1 = 0(6(x1)) [4 3] [3 2 ] 6(2(5(x1))) = [1 0]x1 >= [-& -&]x1 = 4(2(6(x1))) [0 0] [0 -&] 9(6(x1)) = [3 3]x1 >= [3 3 ]x1 = 9(x1) [0 0] [0 -&] 6(6(x1)) = [0 0]x1 >= [-& -&]x1 = 3(x1) [0 0 ] [0 0 ] 3(0(x1)) = [-& -&]x1 >= [-& -&]x1 = 3(5(x1)) problem: 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 9(6(x1)) -> 9(x1) 6(6(x1)) -> 3(x1) 3(0(x1)) -> 3(5(x1)) Arctic Interpretation Processor: dimension: 2 interpretation: [0 2] [6](x0) = [2 3]x0, [3 3 ] [3](x0) = [-& -&]x0, [0 1 ] [0](x0) = [-& -&]x0, [0 0] [5](x0) = [0 1]x0, [0 -&] [9](x0) = [-& 2 ]x0, [3 3] [4](x0) = [0 0]x0, [0 -&] [2](x0) = [0 -&]x0 orientation: [3 3] [3 3 ] 4(x1) = [0 0]x1 >= [-& -&]x1 = 3(2(5(x1))) [3 4 ] [3 4 ] 3(5(x1)) = [-& -&]x1 >= [-& -&]x1 = 0(6(x1)) [0 2] [0 -&] 9(6(x1)) = [4 5]x1 >= [-& 2 ]x1 = 9(x1) [4 5] [3 3 ] 6(6(x1)) = [5 6]x1 >= [-& -&]x1 = 3(x1) [3 4 ] [3 4 ] 3(0(x1)) = [-& -&]x1 >= [-& -&]x1 = 3(5(x1)) problem: 4(x1) -> 3(2(5(x1))) 3(5(x1)) -> 0(6(x1)) 9(6(x1)) -> 9(x1) 3(0(x1)) -> 3(5(x1)) Arctic Interpretation Processor: dimension: 2 interpretation: [0 0 ] [6](x0) = [0 -&]x0, [0 0] [3](x0) = [0 0]x0, [0 0 ] [0](x0) = [-& 0 ]x0, [0 0 ] [5](x0) = [0 -&]x0, [1 0] [9](x0) = [1 0]x0, [3 3] [4](x0) = [3 3]x0, [2 -&] [2](x0) = [0 2 ]x0 orientation: [3 3] [2 2] 4(x1) = [3 3]x1 >= [2 2]x1 = 3(2(5(x1))) [0 0] [0 0 ] 3(5(x1)) = [0 0]x1 >= [0 -&]x1 = 0(6(x1)) [1 1] [1 0] 9(6(x1)) = [1 1]x1 >= [1 0]x1 = 9(x1) [0 0] [0 0] 3(0(x1)) = [0 0]x1 >= [0 0]x1 = 3(5(x1)) problem: 3(5(x1)) -> 0(6(x1)) 9(6(x1)) -> 9(x1) 3(0(x1)) -> 3(5(x1)) Arctic Interpretation Processor: dimension: 2 interpretation: [0 -&] [6](x0) = [0 -&]x0, [1 2 ] [3](x0) = [-& -&]x0, [2 0 ] [0](x0) = [-& -&]x0, [0 -&] [5](x0) = [0 -&]x0, [1 -&] [9](x0) = [3 -&]x0 orientation: [2 -&] [2 -&] 3(5(x1)) = [-& -&]x1 >= [-& -&]x1 = 0(6(x1)) [1 -&] [1 -&] 9(6(x1)) = [3 -&]x1 >= [3 -&]x1 = 9(x1) [3 1 ] [2 -&] 3(0(x1)) = [-& -&]x1 >= [-& -&]x1 = 3(5(x1)) problem: 3(5(x1)) -> 0(6(x1)) 9(6(x1)) -> 9(x1) Arctic Interpretation Processor: dimension: 3 interpretation: [0 0 1] [6](x0) = [0 1 0]x0 [1 1 0] , [1 0 -&] [3](x0) = [-& -& -&]x0 [0 1 -&] , [0 0 -&] [0](x0) = [-& -& -&]x0 [-& 0 1 ] , [0 0 0] [5](x0) = [1 1 0]x0 [0 1 1] , [0 0 0 ] [9](x0) = [-& -& -&]x0 [-& 0 -&] orientation: [1 1 1 ] [0 1 1 ] 3(5(x1)) = [-& -& -&]x1 >= [-& -& -&]x1 = 0(6(x1)) [2 2 1 ] [2 2 1 ] [1 1 1 ] [0 0 0 ] 9(6(x1)) = [-& -& -&]x1 >= [-& -& -&]x1 = 9(x1) [0 1 0 ] [-& 0 -&] problem: 3(5(x1)) -> 0(6(x1)) Arctic Interpretation Processor: dimension: 3 interpretation: [0 0 0] [6](x0) = [2 0 0]x0 [0 0 0] , [2 1 0 ] [3](x0) = [-& 1 0 ]x0 [2 0 -&] , [2 0 2] [0](x0) = [0 0 2]x0 [0 0 1] , [1 -& -&] [5](x0) = [1 2 2 ]x0 [3 2 2 ] orientation: [3 3 3] [2 2 2] 3(5(x1)) = [3 3 3]x1 >= [2 2 2]x1 = 0(6(x1)) [3 2 2] [2 1 1] problem: Qed