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Consider a finite state machine as usual, but every transition, it can also update an integer counter by adding or subtracting a number. Say, a transition function of the form $\delta(q,a) = (p,k)$ moves to the new state $p$, and add $k$ to the counter, where $k \in \mathbb{Z}$ (so $k$ can be positive, negative, or zero).

A string is accepted if the final state and the counter value is in $F$, where $F$ is a finite set of pairs of states and counter values.

Is this model known? I could not find any reference of this particular extension.

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Assuming $k$ can be any integer, then this can be formalized as a blind one-counter automaton. Usually these automata accept on final state when its counter is zero, but we can easily model your acceptance type if you allow $\epsilon$ transitions (that do not consume input). If I am not mistaken, like with finite state automata, one can get rid of the $\epsilon$, but that is a non-trivial result.

There are several types of one-counter automata. In the most general form they are allowed to test whether the value of the counter equals zero. The languages they accept are a strict subset of the context-free languages.

The model you are probably looking for is called blind, it cannot test for zero, except as the final test for acceptance at the end of the computation.

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  • $\begingroup$ "Counter" may be misleading, since in one-counter machines you can also branch the run according to the value of the counter (i.e. zero-tests), which makes the model very different (and much stronger). $\endgroup$ – Shaull Oct 2 '16 at 11:40
  • $\begingroup$ You are right. I add some words on that. Thanks. $\endgroup$ – Hendrik Jan Oct 2 '16 at 11:44
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This model is a variant of weighted automata, which are widely studied (although there are a lot of open questions about them). You can start here: Handbook of Weighted Automata.

Note that sometimes they are called "distance automata" (although this is becoming less common).

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