Apologies if the title is confusing, but I couldn’t think of better phrasing in short text.
Whenever we define / specify a certain automaton (such as a finite state machine or a turing machine) by defining all of its States, transition function, etc., this feels awfully similar to defining an algorithm. For example, I can define a machine that can tell if a number is divisible by 3. It is very similar to writing an algorithm and the steps to solving the problem.
Now I understand that the two aren’t exactly equivalent. But would it be incorrect to say that the specification of a machine is a type of algorithm, since we’re defining the steps it takes to solve a problem (how to respond to a specific state or input to solve a specific problem)?
In applied CS, it’s common to talk about pure and impure functions instead of Turing machines.
Pure functions are, broadly speaking, equivalent to Turing machines. A pure function may only depend on its inputs (like a Turing machine) and has no outputs besides the return value (like the end state of a Turing machine’s tape).
Impure functions cover algorithms that aren’t Turing machines. For example, you might have a random number generator
rand: 1 → Nthat outputs different natural numbers on every invocationrand()and is hence impure. Output functions are also impure, e.g., a functionwrite: N → 1that writes a number into a file can’t be a Turing machine, because Turing machines have no concept of files.Computer programs that consist entirely of pure functions aren’t very useful, because they can’t interact with the user or reality. The typical approach to solving this is to put the core program logic inside pure functions that interact with impure parts through a limited interface. That way, you can apply CS concepts to the important parts of the program and neatly separate out the impure parts.
Edit: Changed ∅ to 1 (singleton set) in function definitions. A function can’t return ∅ and a function taking ∅ can’t be called.