count = 0;
rules = {
   ({
      {0, 0},
      {0, 0}
     }) -> ({
      {0, 0},
      {0, 0}
     }), ({
      {1, 0},
      {0, 0}
     }) -> ({
      {0, 0},
      {1, 0}
     }), ({
      {0, 1},
      {0, 0}
     }) -> ({
      {0, 0},
      {0, 1}
     }), ({
      {1, 1},
      {0, 0}
     }) -> ({
      {0, 0},
      {1, 1}
     }),
   ({
      {0, 0},
      {1, 0}
     }) -> ({
      {0, 0},
      {1, 0}
     }), ({
      {1, 0},
      {1, 0}
     }) -> ({
      {0, 0},
      {1, 1}
     }), ({
      {0, 1},
      {1, 0}
     }) -> ({
      {0, 0},
      {1, 1}
     }), ({
      {1, 1},
      {1, 0}
     }) -> ({
      {1, 0},
      {1, 1}
     }),
   ({
      {0, 0},
      {0, 1}
     }) -> ({
      {0, 0},
      {0, 1}
     }), ({
      {1, 0},
      {0, 1}
     }) -> ({
      {0, 0},
      {1, 1}
     }), ({
      {0, 1},
      {0, 1}
     }) -> ({
      {0, 0},
      {1, 1}
     }), ({
      {1, 1},
      {0, 1}
     }) -> ({
      {0, 1},
      {1, 1}
     }),
   ({
      {0, 0},
      {1, 1}
     }) -> ({
      {0, 0},
      {1, 1}
     }), ({
      {1, 0},
      {1, 1}
     }) -> ({
      {1, 0},
      {1, 1}
     }), ({
      {0, 1},
      {1, 1}
     }) -> ({
      {0, 1},
      {1, 1}
     }), ({
      {1, 1},
      {1, 1}
     }) -> ({
      {1, 1},
      {1, 1}
     })
   };
nextgen0[m_] := Block[{a, c}, c = Flatten@Table[
      a = (({
           {m[[i, j]], m[[i, j + 1]]},
           {m[[i + 1, j]], m[[i + 1, j + 1]]}
          }) /. rules); {{i, j} -> a[[1, 1]], {i, j + 1} -> 
        a[[1, 2]], {i + 1, j} -> a[[2, 1]], {i + 1, j + 1} -> 
        a[[2, 2]]},
      {i, 1, Dimensions[m][[1]] - 1, 2}, {j, 1, 
       Dimensions[m][[2]] - 1, 2}
      ];
   Table[{i, j} /. c
    , {i, Dimensions[m][[1]]}
    , {j, Dimensions[m][[2]]}
    ]
   ];
nextgen[m_] := If[count++; Mod[count, 2] == 0,
   nextgen0[m]
   ,
   Block[{last, mt},
    last = Last[m];
    mt = m[[;; -2]];
    Join[
     Map[#[[3 ;; -3]] &,
      (moveforward[nextgen0[movebackward[
           
           PadRight[PadLeft[mt, Dimensions[mt] + {2, 2}], 
            Dimensions[mt] + {4, 4}]
           ]]]
        )[[3 ;; -3]]]
     ,
     {last}
     ]
    ]
   ];
moveforward[m_] := Transpose[RotateRight[Transpose[RotateRight[m]]]];
movebackward[m_] := Transpose[RotateLeft[Transpose[RotateLeft[m]]]];

m = Table[
   If[((i - 14)^2 + (j - 10)^2 < 4^2) || 
     i < 20 - 16 Exp[-(j - 10)^2/(800/25)], 0, 1], {i, 20}, {j, 20}];

gens = NestList[nextgen, m, 30];
ListAnimate[ArrayPlot /@ gens]