Let $f$ be a function from the set $\{1,$dots,$n\}$ to itself. The pointwise representation of $f$ is a binary string of $nlogn$ bits which is the concatenation of the binary encodings of $f(1),f(2),$dots,$f(n).$ The cycle representation of $f$ is another string of $nlogn$ bits which is also a sequence of $n$ elements of $\{1,$dots,$n\}$ which we will now describe.

Any permutation of $f$ of a set divides its elements into cycles. For example, if $f$ is represented by the pointwise sequence $365412,f$ takes $1$ to $3$ to $5$ and back to $1,$ from $4$ to itself, and from $2$ to $6$ and back to $2.$ We write the three cycles as $(135),(4),$ and $(26).$ To write $f$ in cycle representation, we first:

Write the cycles starting with their largest element, so we have $(513),(4),$ and $(6,2).$

Place the cycles in relative order in increasing order of their largest elements $-$ in the example we put $(4)$ first, then $(513),$ and then $(62),$ giving us the sequence $451362.$

Now for your problems. In the context of circuit complexity, one of the conversions is very easy, and the other is harder:

$($a$)$ Prove that you can convert the cycle representation of a permutation into its pointwise representation in $A{C}^0.$ Specifically, let CYC$-$PW be the language is a permutation of $\{1,$dots,$n\}$ in cycle form, i and $j$ are elements of $\{1,$dots,$n\},$ and $f(i)=j.$ Prove that CYC$-$PW is in $A{C}^0.$

$($b$)$ Prove that there is a logspace transducer PW$-$CYC that takes the pointwise representation of $f$ and outputs the cycle representation of $f.($We are not equipped in this course to prove it$,$ but this problem is hard for log$-$space under the approriate reductions, giving us an example of a one$-$way function in this context.$)$