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[CompSci] Is the specification / definition of an automaton (like a turing machine) a type of algorithm? (lemy.lol)

submitted 7 months ago by [email protected] to c/[email protected]

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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)?

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[–] [email protected] 5 points 7 months ago* (last edited 7 months ago) (3 children)

Ermph, maybe there is a mathematical correspondence but the connotations are different. We usually think of an algorithm in terms of the relationship between its input and its output. But determining that by inspecting the state diagram of a Turing machine is literally impossible, by a result called Rice's theorem. Are you taking a class or reading a book? Or just curious, etc. (that is fine)? What direction do you want to take this in?

A "Turing machine" is a mathematical construction that can be given a formalized definition, but "algorithm" is a jargon word that is a bit less formal and has some wiggle room. But, if you say there is an algorithm to compute X, that means (except in some special contexts) that there is a Turing machine that does it, and vice versa.

Actually nowadays "algorithm" can include steps where you generate random numbers ("roll a 6 sided die and do X with the result"). Turing machines are deterministic and that difference is sometimes significant. There is additional jargon to make this precise, that I'll skip here.

[–] [email protected] 2 points 7 months ago (2 children)

Are you taking a class or reading a book?

Reading a book actually. A programmer by craft who never studied CS, so decided to do it on my own. I appreciate the depth of your answer, thank you! :)

So Turing machines cannot be considered equivalent to algorithms when we involve steps like random number generation? How does church-turing address this? Isn't that part of what's "computable"?

[–] [email protected] 2 points 7 months ago* (last edited 7 months ago)

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 → N that outputs different natural numbers on every invocation rand() and is hence impure. Output functions are also impure, e.g., a function write: N → 1 that 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.

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