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​​A Quine is a (non-empty) program $P$ that takes no inputs and returns its own source code $\langle P\rangle$ as the only output. For a function $f$ (with appropriate domains and range) define an $f$-Quine as a program $P$ that takes an input $x$ and outputs $f(\langle P\rangle, x)$. Thus, with no inputs and when $f = id$ we recover the standard notion.

Given a source code $\langle f\rangle$ for $f$ in some programming language is it possible to efficiently compile it into an $f$-Quine in the same language? I think this would amount to making Kleene's recursion theorem very concrete.

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    $\begingroup$ The standard proof of the recursion theorem is perfectly constructive and explicit: let $Q$ be a program that takes as input the code $\langle R\rangle$ of a program with one input, and outputs the code of the program $f(\langle R(\langle R\rangle)\rangle,x)$ (that is, hardwire the input $\langle R\rangle$ into $R$, and hardwire the code of resulting program into $f$). Then put $P=Q(\langle Q\rangle)$. $\endgroup$ Commented Jun 10, 2023 at 10:17
  • $\begingroup$ Hi there! When I was a grad student, I implemented something like this for fun and it seems to still work. It makes self-referential webpages. If you're interested, see here: michaelwehar.com/quines/howto.html Maybe we could collaborate to make a more refined version of this in the future. :) $\endgroup$ Commented Jun 16, 2023 at 17:05
  • $\begingroup$ Here's the program that builds "f-Quines" for you: michaelwehar.com/quines/self_reference_made_easy.html $\endgroup$ Commented Jun 16, 2023 at 17:06
  • $\begingroup$ For more explanation, when you define G(x,y), x refers to the webpage's own source code and y refers to the input that will be put by the user into the generated webpage's textbox. It then outputs the code for a webpage with the textbox that takes in y as input and evaluates G(e,y) when you click the button where e is the webpage's own source code. $\endgroup$ Commented Jun 16, 2023 at 17:11
  • $\begingroup$ To better understand, as an example, you could type in "console.log(x);" and click generate. Then you could create a webpage with the generated code. This webpage will print its own source code to the console and ignore the input y. $\endgroup$ Commented Jun 16, 2023 at 17:14

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