A memristive stateful neural network allowing complete Boolean in-memory computing attracts high interest in future electronics. Various Boolean logic gates and functions demonstrated so far confirm their practical potential as an emerging computing device. However, spatio-temporal efficiency of the stateful logic is still too limited to replace conventional computing technologies. This study proposes a ternary-state memristor device (simply a ternary memristor) for application to ternary stateful logic. The ternary-state implementable memristor device is developed with bilayered tantalum oxide by precisely controlling the oxygen content in each oxide layer. The device can operate 157 ternary logic gates in one operational clock, which allows an experimental demonstration of a functionally complete three-valued Lukasiewicz logic system. An optimized logic cascading strategy with possible ternary gates is approximate to 20% more efficient than conventional binary stateful logic, suggesting it can be beneficial for higher performance in-memory computing.