1.3 regular expressions

Properties of Regular Language

Unions, intersections, differences and complements of regular languages are regular.

Formal Def of a Regular Expression

Say that R is a regular expression if R is: - for some in the alphabet - - - () - () - ()

() are regualr expressions(inductive def)

some conclusions

  • L(R): the language of R
  • concatenating the empty set to any set yields the empty set, so

Equivalence with Finite Automata

  • hint: regular language is one that is recognized by some finite automation
  • a language is regular iff some regular expression describes it
  • prove see textbook p67 p70
  • GNFA: generalized nondeterministic finite automation, its transition arrows may have any regular expression as labels

DFA to regular expression

Write a table out.

Colum: k=0,1,...n-1

Row: all combination of two states

Base Case: when k=0, if there is a transition(including this state and itself, in this case is ), then is the combination(or+transitions if more than one); if there is not, then it is empty.

Then fill the table using formula.

The result should be

Pumping Lemma

Suppose it is regular. Let p be its pumping constant.

Consider the string w=xxx, which is a string in this language with length greater than p.

There must be a pumping decomposition of w: w=xyz, where and .

Prove this compositions variants is not in the language.

Then this violates the pumping lemma.

So it is not regular.