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Traditionally in the turnstile model of data streams, there is a state vector$x=(x_1,x_2,\ldots,x_n)$ which is updated through a stream of pairs $(i,k)$where $i\in [n]$ and $k\in \Z$. Upon receiving $(i,k)$, $x_i\gets x_i + k$. Adistinct count algorithm in the turnstile model takes one pass of the streamand then estimates $\norm{x}_0 = |\{i\in[n]\mid x_i\neq 0\}|$ (aka $L_0$, theHamming norm).  In this paper, we define a finite-field version of the turnstile model. Let$F$ be any finite field. Then in the $F$-turnstile model, for each $i\in [n]$,$x_i\in F$; for each update $(i,k)$, $k\in F$. The update $x_i\gets x_i+k$ isthen computed in the field $F$. A distinct count algorithm in the $F$-turnstilemodel takes one pass of the stream and estimates $\norm{x}_{0;F} =|\{i\in[n]\mid x_i\neq 0_F\}|$.  We present a simple distinct count algorithm, called $F$-\pcsa{}, in the$F$-turnstile model for any finite field $F$. The new $F$-\pcsa{} algorithmtakes $m\log(n)\log (|F|)$ bits of memory and estimates $\norm{x}_{0;F}$ with$O(\frac{1}{\sqrt{m}})$ relative error where the hidden constant depends on theorder of the field.  $F$-\pcsa{} is straightforward to implement and has several applications inthe real world with different choices of $F$. Most notably, it makes distinctcount with deletions as simple as distinct count without deletions.

Probabilistic Counting in Generalized Turnstile Models