The Parikh finite word automaton (PA) was introduced and studied in 2003 by Klaedtke andRueß. Natural variants of the PA arise from viewing a PA equivalently as an automaton thatkeeps a count of its transitions and semilinearly constrains their numbers. Here we adoptthis view and define the affine PA, that extends the PA by having eachtransition induce an affine transformation on the PA registers, and the PA onletters, that restricts the PA by forcing any two transitions on the sameletter to affect the registers equally. Then we report on the expressiveness, closure, anddecidability properties of such PA variants. We note that deterministic PA are strictlyweaker than deterministic reversal-bounded counter machines.
