Languages and Machines

1
Languages and Machines

• Unit two Regular languages and Finite State
  Automata

2
Review of week one

• A language is a set of strings (the set of
  different things you can say). May be infinite.
• A string is a sequence of symbols. Minimum length
  zero, maximum length some finite number.
• A symbol is just some mark on the page or screen.
  A language has a finite alphabet of symbols.

3
Review of week one

• In a context-dependent language, the meaning of a
  phrase depends on the context
• In a context-sensitive language, the structure of
  a phrase depends on the context
• Most natural languages are context-dependent but
  not context-sensitive
• A context-free language is one where the
  structure of a phrase is always the same,
  independent of context
• A regular language is a context-free language
  which has simple rules for forming valid strings
  (e.g. "94", "getWidth())

4
Classes of formal language
phrase structure
context-sensitive
context-free
regular
5
Regular languages

• Here are examples of strings from a regular
  language with alphabet a,b
• a
• b
• ab
• aaaaa
• ababab

6
Regular languages

1. the empty set is a regular language
2. the set consisting of the empty string (?) is a
   regular language
3. the set consisting of a one-symbol string is a
   regular language
4. a new regular language can be made by taking a
   string from a regular language and concatenating
   it with a string from a regular language
5. a new regular language can be made by taking the
   disjoint union of two regular languages

7
Recognizing regular languages

• regular languages can be recognized and
  interpreted by a finite-state machine
• for example, here is a machine to recognize a
  two-bit string

0
0
acceptor states
1
1
8
Regular expressions

• Wouldnt it be nice if we had a compact way of
  specifying a regular language?
• we have!
• its a special notation called a
• regular expression

9
Examples of regular languages

• the set of all two-symbol strings containing the
  letters a and b
• (ab)2
• the set of all two-bit strings
• (01)2
• the set of all possible words
• (a..z)
• the set of all decimal integers
• (0(1..9)(0..9))
• the set of Java identifiers
• JavaLetter JavaLetterOrDigit

10
More examples of regular languages

• all the possible three-bit strings
• (01)3
• all the single-digit decimal numbers
• (0123456789) (0..9)
• all the possible repetitions of the traffic-light
  sequence (red, amber, green, amber)
• (red amber green amber)

11
Activity

• Write down the regular expression denoting the
  following regular languages
• The language with two strings the cat and the
  mat
• Arithmetic expressions with two operands,
• e.g. 1 2, 3 4
• The allowed operator are , -, ,
• The allowed operands are single digit decimal
  numbers
• The language consisting of all possible binary
  strings
• The language of HTML tags such as ltHEADgt

12
Suggested Answers

• The language with two strings the cat and the
  mat
• the (cat mat) or (the (cm)at)
• Arithmetic expressions with two operands,
• e.g. 1 2, 3 4.
• (0..9) (-) (0..9)
• The language consisting of all possible binary
  strings
• (01)
• The language of HTML tags such as ltHEADgt
• lt (A..Z) gt

13
A cautionary note

• You have been using a metalanguage!
• The regular expression strings form a language
  having terminal symbols ( ) plus literal
  symbols e.g. a stands for the letter a
• this can cause problems when the metalanguage and
  the language get confused e.g. the language
  consisting of strings of one to three vertical
  bars

14
A cautionary note

• we can fix this by some ghastly escape
  convention, e.g. convert the above to
• "" "" ""
• now we have problems with the quote symbol!
• the best idea is to choose metalanguage symbols
  which are rarely encountered in the language
  being described, and use bold-face or color to
  distinguish

15
Regular languages and regular expressions

• Regular Expression
• a
• a b
• a b

• Regular Language
• the empty set
• the set consisting of the empty string (?)
• the set consisting of a one-symbol string (e.g.
  "a")
• a new regular language can be made by taking a
  string from a regular language and concatenating
  it with a string from a regular language
• a new regular language can be made by taking the
  union of two regular languages

16
Regular languages and regular expressions

• The other ways of forming regular expressions
  are just shorthand
• a0
• a1 a
• a2 aa
• a a aa aaa ...
• a a aa aaa ...

17
Regular languages and regular expressions

• Brackets are used to show precedence of the
  operations
• (a b ) ? a b
• default precedence is
• or or n
• concatenation

18
Activity

• Give examples of the following languages
• (x y z)3
• x y z
• a b2
• (a b)2

19
Suggested Answers

• Give examples of the following languages
• (x y z)3 xzy
• x y z
• a b2 abb
• (a b)2 abab

20
From Regular Expressions to Finite State Automata

• It is an amazing fact that any regular expression
  has an equivalent finite state automaton which
  recognizes it
• and every finite state automaton recognizes some
  regular expression
• we will prove these propositions later

21
Finite State Machines
transition
D
0
00
start state
B
1
0
end state
E
A
01
0
output symbol
1
C
1
input symbol
F
10

• an FSM to add two binary numbers

22
Finite state automata

• These are simple machines with no output symbols
• they can only recognize strings of input symbols
• acceptance is shown by a special state

23
NFAs

• The kind of finite state automata we shall be
  using are called nondeterministic finite automata
• "nondeterministic" means we can do naughty things
  like
• have a transition without a symbol
• label two exit transitions with the same symbol
• not show the paths which lead to failure

24
Example of an NFA
b
a
b
c
a
a
a

• what regular language does this NFA represent?
• a b a b c a

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Examples of conversion from REs to NFAs

• (a b)2
• a b2
• (a b)2
• (a b)

a
b
a
b
a
b
b
a
a
b
b
a
b
26
Activity

• Convert the following regular expressions to
  NFAs
• JavaLetter JavaLetterOrDigit
• (red amber green amber)
• Convert the following NFAs to REs

a
b
c
a
b
d
27
Suggested answer

3. (ab)
4. (acbd)

javaLetter
javaLetterOrDigit
amber
amber
red
green
28
Summary

• regular expressions give us a neat notation for
  describing regular languages
• nondeterministic finite automata (NFAs) provide a
  diagrammatic version of regular expressions
• these notations are equivalent
• finite automata theory is crucial in generating
  lexical analyzers from regular expressions