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Let $A$ be a problem. I would like to solve $A$ in $NL$. So, I am trying to construct Turing machine working in $NL$:

Let $w$ be a input of length $n$. Firstly, we run machine $M_1$ that working in $LOGSPACE$. It produces a output $o$ of length $P(n)$ where $P$ is a polynomial. Secondly, a machine $M_2$ working in $NL$ reads the output $o$ written by $M_1$.

Does t work in $NL$? Why?

P.S. If machine $M_2$ worked in $L$ it would be obvious that problem is solvable in $L$. But, I am not sure if that algorithm solves $A$ in $NL$.

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$M_2$, while running on $w$, calls $M_1(w)$ as a subroutine. So if $M_1(w)$'s output is stored on $M_2$'s working tape then it requires polynomial space, not logarithmic space. So, $M_2$ does not work in logarithmic space no matter it is deterministic or nondeterministic.

Of course you could design your machine $M_2$ so that $M_1(w,i)$ would return $i$th bit on demand (on each call). In this case it would work in logarithmic space. But it depends on the problem you solve.

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