Question

I was wondering, is it possible to define an API and give it as an input to a TM turing machine and the output will be the code in c or any other natural/programming language?

I guess not but how do I show it formally by reduction etc. ?

Answer 1

Your question seems to be asking if it is possible to write a program (in other words, a Turing Machine) that takes in a program specification (what you are calling an “API”) and outputs a program in some programming language. The answer to this is “of course”. Let’s look at a few examples.

1. Let’s say I write my input specification as a C program and I want to output a program in C. I call this Turing Machine program “copy” and I can trivially implement it in any language (or use the cp command on the command line).

2. Let’s say I write my input specification as a C program, and I want to output a program in assembly code. I call this Turing Machine program “compiler” and I can implement it in any language (or you can download a C compiler that already exists).

3. Let’s say I write my input specification in English and I want to output a program in C. I call this Turing Machine program “software engineer” and I implement it in the brain of a human by sending that human to school (or you can hire an existing software engineer).

The moral here is that a Turing Machine can do anything that a human can do (and a human can do anything a Turing Machine can do). But it’s really, really, really hard to write a computer program that is as general-purpose as a human.