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I have two questions on Kaveh's answer to Definition of uniform boolean circuit :

  1. Kaveh mentions that the input is in unary encoding. In the definition it says the input is $1^n$, afaik $1^n$ is a sequence of 1's repeated n times, but unary encoding is a sequence of 1's and ends with 0 there is no power of 1 that gives 0... So how is $1^n$ a unary encoding?
  2. Why do we use a unary encoding and not the binary encoding of n? what happens if we use binary instead?
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Here is the binary encoding of 5: $$101$$ And here is its unary encoding: $$11111$$ Hopefully the difference is apparent.

Let's see what happens if we switch to binary encoding in the context of the linked question. Given the binary encoding of $n$, we need to produce a circuit, operating on $n$ bits, in logspace, and so in polynomial time. How much time do we have? The binary encoding of $n$ has length roughly $\log n$, so we have time $O(\log^C n)$, which isn't enough to produce a circuit which reads all input bits.

In contrast, if we use unary encoding, then the length of the input is $n$, and so we have time $O(n^C)$, which is more reasonable (in addition, we are only allowed to use $O(\log n)$ space).

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  • $\begingroup$ Oh, I was a bit confused because in the definition I know of unary encoding 5 would be 111110 (five 1's followed by a terminal 0) I had a doubt why the 0 is omitted here. Thanks for clarifying $\endgroup$
    – user206904
    Mar 14, 2021 at 16:41

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