Question

If you have a Turing machine with a tape that is not just linear at each move, instead you have the ability to move up, down, left, or right. There is a single read/write head. The transition function looks like.

The tape extends up and right infinitely. That is, you can never go left or down of where you start, but you can go infinitely up and to the right.

Is this machine equivalent to the standard Turing machine model? That is, does this type of machine and the standard model recognize the same type of languages?

Answer 1

In his 1948 essay, "Intelligent Machinery", Turing wrote that his machine consisted of:

...an unlimited memory capacity obtained in the form of an infinite tape marked out into squares, on each of which a symbol could be printed. At any moment there is one symbol in the machine; it is called the scanned symbol. The machine can alter the scanned symbol, and its behavior is in part determined by that symbol, but the symbols on the tape elsewhere do not affect the behavior of the machine. However, the tape can be moved back and forth through the machine, this being one of the elementary operations of the machine. Any symbol on the tape may therefore eventually have an innings.^[18]

— Turing 1948

Facts

Many machines that might be thought to have more computational capability than a simple universal Turing machine can be shown to have no more power (Hopcroft and Ullman p. 159, cf. Minsky (1967)). They might compute faster, perhaps, or use less memory, or their instruction set might be smaller, but they cannot compute more powerfully (i.e. more mathematical functions). (Recall that the Church–Turing thesis hypothesizes this to be true for any kind of machine: that anything that can be "computed" can be computed by some Turing machine.)

Conclusion

What do you understand power means?

Power means if your machine is capable to handle more complex language than your other machine that means your machine is power ful than other .

But

We all aware about that any Turing machine recognize Unrestricted language only.

So any Turing machine existing in this world having the same power as Standard Turing Machine