Deterministic finite automaton

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I want to see transducers and discussions of how these work, ASAP! Alec (talk) 23:58, 12 January 2018 (UTC)
  • Source is "An Introduction to Formal Languages and Automata" - 4th edition, Peter Linz - great book.


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:

  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.


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"


See also


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