All-optical logic gates including AND, XOR, and NOT gates, as well as a half-adder, are realized based on two-dimensional lithium niobate photonic crystal (PhC) circuits with PhC micro-cavities. The proposed all-optical devices have an extinction ratio as high as 23 dB due to the effective all-optical switch function induced by two-missing-hole micro-cavities. These proposed devices can have potential implementation of complex integrated optical functionalities including all-optical computing in a lithium niobate slab or thin film.