# Deterministic finite automaton

**Stub grade: A****

## Definition

A *deterministic finite automaton* or *DFA* is a tuple of 4 items:

- [ilmath]A:\eq(Q,\Sigma,\delta,q_0)[/ilmath] [ilmath]\sum[/ilmath] where:
- [ilmath]Q[/ilmath] is a finite set of "states",
- [ilmath]\Sigma[/ilmath] is the alphabet of the DFA, a finite set of symbols which may appear in "strings",
- [ilmath]\delta:Q\times\Sigma\rightarrow Q[/ilmath] is a function, called the "
*transition function*" and - [ilmath]q_0\in Q[/ilmath] is the "initial state" or "starting state" of the automaton

- in an "acceptor" type there's another item, [ilmath]F[/ilmath], for [ilmath]F\subseteq Q[/ilmath], the "final" or "accepting" states.

There are two "defined" types:

In the practice of working with formal languages these terms are rarely used and DFAs are usually implicitly acceptors. In the wider practice of "programming" DFAs are a very useful model and are neither "pure" acceptors or transducers. DFAs and regex certainly do go hand in hand conceptually but (almost^{[Note 1]}) always do not use DFAs as their implementation as this wouldn't allow the implementation of more powerful features which programmers have become accustomed to.

I hope to explore deviations from DFAs without sacrificing the performance that comes from their determinism, for example adding a counter along certain transitions to allow "depth" of an expression.

### Acceptor-type

A tuple of 5 items:

- [ilmath]A:\eq(Q,\Sigma,\delta,q_0,F)[/ilmath] (or perhaps as an ordered pair: [ilmath]A:\eq(A',F)[/ilmath] for [ilmath]A'[/ilmath] the underlying DFA as described above) with terms exactly as above but with an additional item [ilmath]F\in\mathcal{P}(Q)[/ilmath] which is:
- [ilmath]F\subseteq Q[/ilmath] is a set of "accepting states" or "final states"

### Transducer-type

## See also

## Notes

- ↑ As I hope to make one, and they probably exist out there somewhere!

## References