Resistive switching devices are considered as one of the most promising candidates for the next generation memories and nonvolatile logic applications. In this paper, BiFeO3:Ti/BiFeO3 (BFTO/BFO) bilayer structures with optimized BFTO/BFO thickness ratio which show symmetric, bipolar, and nonvolatile resistive switching with good retention and endurance performance, are presented. The resistive switching mechanism is understood by a model of flexible top and bottom Schottky-like barrier heights in the BFTO/BFO bilayer structures. The resistive switching at both positive and negative bias make it possible to use both polarities of reading bias to simultaneously program and store all 16 Boolean logic functions into a single cell of a BFTO/BFO bilayer structure in three logic cycles.