Deterministic Finite-State Machine (or Deterministic Finite Automaton)

Deterministic Finite-State Machine (or Deterministic Finite Automaton)
A DFA is a 5-tuple, (S, Σ, T, s, A), consisting of:
S: a finite set of states
Σ: a finite set of symbols called the alphabet
T: a transition function (T : S × Σ → S)
s: a start state (s  S)
A: a set of accept states (A  S)
Let M be a DFA such that M = (S, Σ, T, s, A), and X = x0x1 ... xn be a string over
the alphabet Σ. M accepts the string X if a sequence of states, r0,r1, ..., rn,
exists in S with the following conditions:
1. r0 = s
2. ri+1 = T(ri, xi), for i = 0, ..., n-1
3. rn  A.
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 Σ the following constants are defined:
– (empty set) ∅
– (empty string) ε
– (literal character) a in Σ
The following operations are defined:
– concatenation
– alternation: a|b
– Replication: R*
– Wildcard: . (exactly one occurrence)
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.