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I am learning complexity theory with a background in mathematics, and I want to better understand why certain reductions are polynomial-time computable. Let me give two examples of my worries.

Example 1: The Cook–Levin reduction. In the proof of the Cook–Levin theorem (say in Arora–Barak or Sipser), given a language $L\in\textbf{NP}$ and a polynomial-time verifier $M$ for $L$, we construct a reduction $f\colon x\mapsto\varphi_x$ mapping a string $x\in\{0,1\}^*$ to (an encoding of) a CNF formula that is satisfiable iff $x\in L$. It is then proven that the size of this formula is polynomial in the running time $T(n)$ of $M$. I am comfortable up to this point. But then now many authors claim that this immediately implies that the reduction $f$ is polynomial-time computable. Let me quote three books:

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach section 2.3.4, page 49:

Moreover, this CNF formula can be computed in time polynomial in the running time of $M$.

Michael Sipser, Introduction to The Theory of Computation 3e, section 7.4, pages 309–310:

To see that we can generate the formula in polynomial time, observe its highly repetitive nature. Each component of the formula is composed of many nearly identical fragments, which differ only at the indices in a simple way. Therefore, we may easily construct a reduction that produces $\phi$ in polynomial time from the input $w$.

Michael Garey and David Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, section 2.6, page 44:

All that remains to be shown is that, for any fixed language $L$, $f_L(x)$ can be constructed from $x$ in time bounded by a polynomial function of $n=|x|$. … The polynomial boundedness of this computation will follow immediately once we show that $\text{Length}[f_L(x)]$ is bounded above by a polynomial function of $n$, where $\text{Length}[I]$ reflects the length of a string encoding the instance $I$ under a reasonable encoding scheme, as discussed in Section 2.1.

On an intuitive level, it feels right, and I am almost willing to accept such arguments. But nonetheless I feel uneasy about it, since we are arguing at a very high-level, with no discussion of the details of a TM that computes $f$.

Example 2: The halting problem is $\textbf{NP}$-hard. We may argue that the halting problem $\textsf{HALT}$ is $\textbf{NP}$-hard by reducing from $L\in\textbf{NP}$ as follows: since $\textbf{NP}\subset\textbf{EXP}$, there exists an exponential-time TM $M$ deciding $L$ (doing so by brute-force searching for a certificate); we may modify $M$ to enter an infinite loop if it rejects a string $x\in\{0,1\}^*$. This then gives us a reduction $x\mapsto\langle\llcorner M\lrcorner,x\rangle$ such that $x\in L$ iff $\langle \llcorner M\lrcorner,x\rangle\in\textsf{HALT}$, and intuitively this reduction is polynomial-time computable because “the modification of $M$ is quite minor”, whatever that means.

Summary. How do I be more comfortable with high-level arguments concerning whether certain functions are polynomial-time computable? Are there authors that write with careful attention to such details, or should I just learn to stop worrying about it? (This MathOverflow post may be relevant: https://mathoverflow.net/q/76558/)

Thank you.

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    $\begingroup$ How people are writing these proofs is a great example that shows math/computer science is a human/social activity. If you want to learn something, just make sure you have convinced yourself. If you want to convince others, just make sure you believe they will believe your proof, which might be just a word or two for an expert or many pages in case you are required to convince a complete beginner. $\endgroup$
    – John L.
    May 30, 2022 at 7:26

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The way you convince yourself of those claims is that you figure out how to write out an explicit proof of them. This same issue occurs with every mathematical proof. Every mathematical proof has some aspects that are described in detail in the proof and some claims or statements that are not explained in further detail (because it is assumed that they are obvious enough that the reader will be able to work out for themselves why they are true). So, the way you convince yourself is to spend the time to work out those details and convincing yourself that they are true. As you get more experienced, you will become adept at doing that, to the point where it is so easy that you do indeed find those statements obvious.

Example 1: How to prove that it is a polynomial-time reduction? You write an algorithm to implement $f$, and convince yourself that its running time is polynomial. It gets a bit tedious to write out the details, and once you understand the main idea for how to do the reduction, then creating the algorithm is pretty straightforward (it's more akin to writing a program than to mathematics), so many authors don't bother to go through those tedious details, but you can do it.

Example 2: Same.

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  • $\begingroup$ Example 2 is a great example! $\endgroup$
    – John L.
    May 30, 2022 at 7:34

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