A new measure of complexity for finite symbol sequences, named as lattice comple xity, is presented, based on LempelZiv complexity and the symbolic dynamics of onedimensional iterated maps system. To make lattice complexity distinguished from LempelZiv measure, an approach called finegraining method is also prop osed. When finegraining order is small enough, the two measures are almost equ al. When finegraining order goes to large, the differentiation between them be comes apparent. Applying these measures to studies of logistic map, we find thos e be regarded as complex sequences by lattice complexity are clearly generated a t the edge of the chaotic region. The derived properties of the measures are als o discussed.