Is length uniform AC0 computable? Consider the following problem:

Input: A binary string $w$.
Output: $|w|$ as a binary number.

Is it possible to compute this 
in $\mathsf{DLogTime}$-uniform $\mathsf{AC}^0$ 
(or equivalently in $\mathsf{AltTime}(O(1),O(\lg n))$ or $\mathsf{FO}$)?
Like Emil says, the answer is yes, because the length (in binary) of the input string is actually computable by a deterministic logtime Turing machine, so is a fortiori in $\mathsf{FO}$.

You may find this stated as Lemma 13 of Buss's The Boolean formula value problem is in ALOGTIME.  The idea, attributed to Dowd, is to use the query tape to do binary search (under the convention that the machine gets an "out of range" response if the query tape contains a binary number exceeding the length of the input).  I cut-and-paste the proof for quick reference (the number $n$ referenced in the proof is the length of the input, i.e., the desired output; the "index tape" is what Emil and I called query tape):


  The [...] Turing machine proceeds as follows. First determine the least value of $i$ such that $n &lt; 2^i$: this is easily done in $O(\log n)$ time since we assume that the index tape and its tape head are unaffected when the input tape is accessed. Once the value $2^i$ is obtained, written on the [index] tape, it is now easy to do a binary search to determine the value of $n$. Finally the index tape is copied to the output tape.


(NB: the paper actually says "Once the value $2^i$ is obtained, written on the input tape", etc., but it must be a typo, the computation described is obviously done by writing on the index/query tape. Besides, the input tape is usually read-only in this setting).