Bi-infinite calculating automaton

Using the concept of extroversion, we designed and studied an abstract automaton that performs multiplication by $3_{(4)}$ in the quadratic number system; besides, it computes an infinite number of related operations. The multiplier by $3_{(4)}$ is used as an example for simplicity. The device is infinite, so the research is mostly theoretical. Nevertheless, it also has some practical value because it reveals the capabilities of real-life computational processes. In particular, it helps find the fastest possible calculations. The device design is unusual. It is a T-shaped cross of two infinities: the infinity of the states (“horizontal”) and the infinity of the input alphabet (“vertical”). That is why the name: bi-infinity automation. Similar bi-infinite devices are generated by many other critical computing devices. Therefore, the transition to bi-infinity helps better understand the essence of computational processes. B-technology can implement some finite slices of each bi-infinite device.