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I am currently studying about polynomial time in Comp Theory. The definition for polynomial time given in my textbook as follows: $$P= \bigcup_{k}TIME(n^k)$$ I have not seen the Union over k notation before. K is a constant. A quick introduction to the meaning of this notation would be enormously helpful.

Thank you

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  • $\begingroup$ Simple: $k$ is not a constant here. It's a bound variable. $\endgroup$
    – Raphael ♦
    Apr 10, 2016 at 12:28
  • $\begingroup$ This is a good question! That notation is indeed confusing, especially if this is the first time you see it. As the other comments/answers say, it is a shorthand for $\cup_{k=1}^\infty$. $\endgroup$
    – Ran G.
    Apr 11, 2016 at 0:55

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Authors often leave out the domain of the union if it is clear within the context. Here, $k$ is a natural number, i.e. $P = \bigcup\limits_{k \in \mathbb{N}} TIME(n^k) $. This is what it means for a complexity function to be bounded by a polynomial: there is a constant $k$ such that the function is in $\mathcal{O}(n^k)$.

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  • $\begingroup$ (Also, the result will be the same in this case if k ranges over the integers, or the rationals, or the reals, or the positive rationals, or the positive reals. ​ (or, for that matter, the primes, etc.)) ​ ​ ​ ​ $\endgroup$
    – user12859
    Apr 9, 2016 at 22:49
  • $\begingroup$ So its the domain of the union. Got it. Thank you. $\endgroup$
    – Delete My Account
    Apr 10, 2016 at 6:38
  • $\begingroup$ @PenguinsAndApples No, not the domain; that's $\mathbb{N}$. It's a bound variable. $\endgroup$
    – Raphael ♦
    Apr 10, 2016 at 12:28

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