The Parikh automaton model equips a finite automaton with integer registers
and imposes a semilinear constraint on the set of their final settings.
Here the theories of typed monoids and of rational series are used to
characterize the language classes that arise algebraically. Complexity
bounds are derived, such as containment of the unambiguous Parikh automata
languages in NC$^1$. Affine Parikh automata, where each transition applies
an affine transformation on the registers, are also considered. Relying on
these characterizations, the landscape of relationships and closure
properties of the classes at hand is completed, in particular over unary
languages.

Brush up, a few more results.

The Parikh automaton model equips a finite automaton with integer registers
and imposes a semilinear constraint on the set of their final settings.
Here the theories of typed monoids and of rational series are used to
characterize the language classes that arise algebraically. Complexity
bounds are derived, such as containment of the unambiguous Parikh automata
languages in NC$^1$. Affine Parikh automata, where each transition applies
an affine transformation on the registers, are also considered. Relying on
these characterizations, the landscape of relationships and closure
properties of the classes at hand is completed, in particular over unary
languages.

Long version, including new results.

The Parikh automaton model equips a finite automaton with integer registers and imposes a semilinear constraint on the set of their final settings. Here the theory of typed monoids is used to characterize the language classes that arise algebraically. Complexity bounds are derived, such as containment of the unambiguous Parikh automata languages in NC$^1$. Noting that DetAPA languages are positive supports of rational $\mathbb{Z}$-series, DetAPA are further shown stronger than Parikh automata on unary langages. This suggests unary DetAPA languages as candidates for separating the two better known variants of uniform NC$^1$.