Difference between revisions of "Deterministic finite automaton"

Revision as of 21:26, 17 January 2018

Definition

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

• [ilmath]A:\eq(Q,\Sigma,\delta,q_0,F)[/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

There are two "defined" types:

1. Acceptor-type automaton and
2. Transducer-type automaton

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

• NFAs
  • Equivalence of DFA-acceptors and NFA-acceptors

Notes

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

References