Trinity was my first test of making a computer in Minecraft. It was the result of several iterations of a design for an ALU, and took a long time to make, as at the time I was not particularly knowledgeable about the lowest level of computer internals. It is about as minimal as you can get, and is absolutely not turing-complete. It was more of just a test of how the theory could work.
Features:
ADD, JMP
The ALU is very simple in Trinity, as it only has an add function. This is made with the binary adders, shown in purple.
The memory uses D type flip-flops, which are made with redstone torches and repeaters.
The program counter is a 4 bit D flip-flop, used to store the current program location. The circuit behind it increments the program counter to advance the program.
I am under no illusions that this was a terrible computer in many ways, but it was just a first attempt to try and help familiarise myself with concepts