MC 306 Theory of Computation Tuesday, 92303

1
MC 306 Theory of ComputationTuesday, 9/23/03

• Last time
• Converting NFAs to DFAs
• Todays Class
• FA closure under language operations

• Assignments due
• HW 2 due Wednesday at noon
• Reading 2.4
• Exercises p. 60, parts a and b of 2.14-2.16
• New Hand-in problems (not due yet) p. 60, part c
  of 2.14-2.16, also 2.16h

2
Regular Languages

• Call a language L over ? regular, if it is the
  language L(M) accepted by some DFA M. We know
• Theorem 1. L is regular if and only if it is the
  language L(M) for some NFA M.
• Theorem 2. The set of regular languages is closed
  under the operation of union.
• Closed under an operation means that if I
  perform that operation on members of the set (of
  regular languages), I get another member of the
  set, i.e., another regular language

3
Closure under language operations

• Theorem 3. The set of regular languages is closed
  under the all the following operations. In other
  words if L1 and L2 are regular, then so is
• Union L1 ? L2 (already proved)
• Intersection L1 ? L2 (homework assignment)
• Complement L1c ? \ L1 --- proof?
• Difference L1 \ L2 --- proof?
• Concatenation L1L2 --- proof?
• Kleene star L1 --- proof?

4
Regular Expressions

• Let ? be an alphabet. A regular expression is a
  shorthand way to represent some languages over ?.
• Example ? 0, 1
• (0 ? 1) 0 denotes the language consisting of
  strings formed by concatenating either symbol 0
  or 1 with any number of 0s
• L((0 ? 1) 0 ) 0, 00, 000, ? 1, 10,
  100, 1000,
• Note Some texts use instead of ? (0 1) 0
• Also, some texts use a dot for concatenation (0
  1) 0

5
Exercises

• 1) List some the elements in the language L(r),
  where r is the following regular expression (ab
  ? aba)
• 2) Find regular expressions for each of the
  following languages
• Strings containing exactly two as
• Strings not containing exactly two as
• Strings in which each a is preceded by a b

6
Inductive Definition of Regular Expressions

• Base case ?, e, ?, and each symbol x ? ? are
  regular expressions
• Inductive case If r1 and r2 are both regular
  expressions, then so is
• (r1 ? r2)
• (r1r2)
• r1
• Hence a regular expression is a string over the
  alphabet ? ? ?, e, ?, , , (, )
• Not every string over this alphabet is regular
• ) ( a)))

7
Inductive definition of language of a regular
expression

• If r is a regular expression, we define L(r) as
  follows
• L(?) ?, L(?) ?, L(x) x for x ? ?
• If r1 and r2 are both regular expressions, then
• L(r1 ? r2) L(r1) ? L(r2)
• L(r1r2) L(r1)L(r2)
• L(r1) (L(r1))

8
Regular expressions versus regular languages

• We defined a regular language to be one that is
  accepted by some DFA (or NFA).
• We will prove that a language is regular by this
  definition if and only if it corresponds to some
  regular expression. Thus,
• 1) Given a DFA or NFA, there is some regular
  expression to describe the language it accepts
• 2) Given a regular expression, we can construct a
  DFA or NFA to accept the language it represents

9
Application Lexical-analyzers and
lexical-analyzer generators

• Lexical analyzer
• Input Character string comprising a computer
  program in some language
• Output A string of symbols representing tokens
  elements of that language
• Ex in C
• Input if (x 3) y 2
• Output (sort of) if-token, expression-token,
  variable-name-token, assignment-token,
  numeric-constant token, statement-separator-token.

10
Lexical-analyzer generators

• Input A list of tokens in a programming
  language, described as regular expressions
• Output A lexical analyzer for that language
• Technique Builds an NFA recognizing the language
  tokens, then converts to DFA.

11
Regular Expressions Applications and Limitations

• Regular expressions have many applications
• Specification of syntax in programming languages
• Design of lexical analyzers for compilers
• Representation of patterns to match in search
  engines
• Provide an abstract way to talk about programming
  problems (language correpsonds to inputs that
  produce output yes in a yes/no programming
  problem)
• Limitations
• There are lots of reasonable languages that are
  not regular hence lots of programming problems
  that cant be solved using power of a DFA or NFA