A multi-acceptance deterministic finite state automaton is a tuple $(Q,\Sigma,\delta,q_0,T,f)$ where

  • $Q$, $\Sigma$, $\delta: Q \times \Sigma \to Q$, and $q_0 \in Q$ are defined in the standard way of DFAs;
  • $T$ is a finite set of labels;
  • $f: Q \to 2^L$ is a function that associates a set of labels to each state.

$\delta^{\\*}: Q \times \Sigma^{\\*} \to Q$ is defined in the standard way.

Such automaton recognizes $|T|$ languages, one for each label in $T$. More specifically, given a label $t \in T$ $$ L(t) = \{x \mid f(\delta^{\\*}(q_0, x)) \ni t \} $$ or if you prefer $$ L(t) = \{x \mid t \in f(\delta^{\\*}(q_0, x)) \} $$

Does this concept exists in literature? Googling "multi-acceptance DFA" and similar queries did not yield anything useful.

The reason I want to use this kind of automata is the following.

I have several classes of regular languages , e.g. $L_1, L_2: T \to 2^{\Sigma^{\\*}}$, which I already need to represent as DFAs, and I need to compute their union $L: T \to 2^{\Sigma^{\\*}}$ defined as $L(t) = \{L_1(t) \cup L_2(t)\}$. I think operations such as the union can be more efficient if I have a single automaton for each class of languages, instead of an automaton for each $L(t)$. Moreover, given the context in which I'm working, one can expect that many languages $L(t)$ and $L(t')$ are the same or have common aspects.

Do you suggest other approaches?


1 Answer 1


There have been studies on something called colored finite automata, which seem very close to your definition. Other colored models, e.g. regular expressions, also exist.

I haven't work on this personally so I'm not informed on the literature, but they are studied e.g. in https://doi.org/10.1587/transinf.2021FCP0012


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