nLab
deterministic automaton

Definitions

(For the moment, mostly as an example of a coalgebra for an endofunctor.)

This will give the classical definition as a state-based system and then show how to turn that form into the coalgebraic form. The point is that other coalgebras should then be easier to interpret.

Definition

A deterministic automaton consists of the following data:

• a set of states, $Q$;

• a set, $\Sigma$, of inputs ;

• a function, $\delta :Q\times \Sigma \to Q$, called the next state function,

and

• a predicate, $\mathrm{final}:Q\to \mathrm{bool}=\{\perp ,\top \}$, the boolean domain.

In the usual interpretation if the automaton is in state $q$, and is ‘given’ the input $\sigma$, then it changes to being in state $\delta (q,\sigma )$.

The ‘final’ predicate, of course, returns $\top$ if a state is a final state and $\perp$ otherwise. (There will, in general, be a set of final or ‘accept’ states.)

(For the moment we will not look in any detail at the links between automata and languages.)

Currying $\delta$

The first step in transforming this to the coalgebraic form is to curry $\delta$, so as to obtain it in the form $\delta :Q\to Q^{\Sigma }$. We thus have for a state, $q\in Q$, $\delta (q,):\Sigma \to Q$. We then also have a product function

$$\alpha =\delta \times (\mathrm{final}):Q\to Q^{\Sigma }\times \mathrm{bool}.$$\alpha = \delta \times (final) : Q \to Q^\Sigma\times bool.

If we now write $\mathrm{HQ}=Q^{\Sigma }\times \mathrm{bool}$, we get a functor (for you to check) $H:\mathrm{Set}\to \mathrm{Set}$ and the deterministic automaton corresponded precisely to a coalgebra, $(Q,\alpha )$, for $H$.

… and the morphisms?

Rather than start with a classically defined morphism of automata, we will take a morphism $f:(Q,\alpha )\to (Q\prime ,\alpha \prime )$ of coalgebras. We thus have that $f:Q\to Q\prime$ is a mapping of states, and that $\alpha \prime \circ f=H(f)\circ \alpha$. Now let $q\in Q$ and $\sigma \in \Sigma$, then the equation tells us that $\delta \prime (f(q),\sigma )=f(\delta (q,\sigma )$ and that the final states of $Q$ are mapped onto those of $Q\prime$. In other words the concept of morphism of these $H$-coalgebras is precisely that of functional simulation of deterministic automata.

Terminal coalgebra

The terminal coalgebra, $(T,{\alpha }_T)$, for this endofunctor is neat. It has state space, $T=𝒫({\Sigma }^{*})$, the power set of the free monoid/ Kleene star of the set of ‘symbol labels’. This power set is precisely the set of all languages over the alphabet $\Sigma$, as a language? is defined to be a subset of ${\Sigma }^{*}$.

The next state function for $(T,{\alpha }_T)$ is defined by ${\delta }_T(L,s)=\{w\in {\Sigma }^{*}\mid \mathrm{sw}\in L\}$.

References

For a summary of automata theory , look at the Wikipedia.

For a thorough treatment, see

• M. V. Lawson, Finite Automata, Google books

or other texts on the subject.

For the coalgebraic treatment, this is discussed in:

• A. Kurz: Coalgebras and Modal Logic. Course Notes for ESSLLI 2001, Version of October 2001. Appeared on the CD-Rom ESSLLI’01, Department of Philosophy, University of Helsinki, Finland, available from site.

category: computer science