Deterministic Finite-State Machine (or Deterministic Finite Automaton)

1
Deterministic Finite-State Machine (or
Deterministic Finite Automaton) A DFA is a
5-tuple, (S, S, T, s, A), consisting of S
a finite set of states S a finite set of
symbols called the alphabet T a transition
function (T S S ? S) s a start state (s
? S) A a set of accept states (A ? S) Let M
be a DFA such that M (S, S, T, s, A), and X
x0x1 ... xn be a string over the alphabet S. M
accepts the string X if a sequence of states,
r0,r1, ..., rn, exists in S with the following
conditions 1. r0 s 2. ri1 T(ri, xi),
for i 0, ..., n-1 3. rn ? A.
2
Every Regular Expression can be Recognized by
some DFA

• Regular expressions consist of constants and
  operators that denote sets of strings and
  operations over these sets, respectively.
• Given a finite alphabet S the following constants
  are defined
• (empty set) Ø
• (empty string) e
• (literal character) a in S
• The following operations are defined
• concatenation
• alternation ab
• Replication R
• Wildcard . (exactly one occurrence)

3
Classical Limitation of DFA

• no DFA can recognize is bracket language, that is
  language that consists of properly paired
  brackets, such as (()()).
• More formally the language consisting of strings
  of the form anbn some finite number of a's,
  followed by an equal number of b's. It can be
  shown that no DFA can have enough states to
  recognize such a language.