Languages

1
Lecture 3

• Topics
• Languages
• Language classes
• Closure properties

2
In class reading assignment

• Read Section 1.5 (pages 27-30) and answer the
  following questions
• What is a language?
• What is L?
• What is aa,bbcc,dd?
• What is aa,bb2?
• What are the five shortest strings in aa,bb?
• What are the five shortest strings in aa,bb?
• What does it mean if a language is
  infinite/finite?
• Is there a difference between and L?

3
Questions

• What is the English Language?
• What is the C Language?

4
Describing a language L

• List all elements in L
• Doesnt work for infinite languages
• English language description of L
• many different descriptions possible, though some
  are better than others
• Formal mathematical description of L
• Give a method for recognizing all strings in L
• give a program for recognizing strings in L

5
Example

• Describing the language of Legal Identifiers of
  C
• English 1 Legal identifiers of C
• English 2 (CSE 231 textbook)
• In C, an identifier should begin with a
  letter, which may be followed by any number of
  letters, digits or underscores.
• Formal Definition
• Regular expression A(AD_)
• A a, b, , z, A, B, , Z
• D 0, 1, 2 , 3, 4, 5, 6, 7, 8, 9

6
Example Continued

• Program to recognize language
• bool main( )
• char a
• getc(cin, a)
• if (cin.eof( ) (!alpha(a))
• return false
• getc(cin, a)
• while (!cin.eof( ))
• if (!alpha(a) !digit(a) !! (a ! _))
• return false
• getc(cin,a)
• return true

7
Work on these examples

• anbm m,n gt 0
• shortest 3 strings?
• English language description?
• C program?
• anbn n gt 0
• shortest 3 strings?
• English language description?
• C program?

8
Language classes

• A language class is a set of languages
• an element of a language class is a language
  which is a set of strings

9
Example language classes

• the set of finite languages
• the set of languages with at most 3 strings
• CARD-3
• the set of languages whose l.r.p.s can be solved
• How are the above language classes related?

10
Closure Properties

• A set is closed under an operation if applying
  the operation to elements of the set produces
  another element of the set
• Example/Counterexample
• set of integers and addition
• set of integers and division

11
Integers and Addition
7
Integers
12
Integers and Division
.4
2
5
Integers
13
Closure Properties for LCs

• We will be interested in closure properties of
  language classes.
• The elements of a language class are languages
  which are sets themselves
• Remember that we apply the set operations to the
  languages (elements of the language classes)
  rather than the language classes themselves
• Example/Counterexample
• Finite languages and set union operator
• CARD-3 and set union operator

14
Finite Sets and Set Union
0,1,00,11
Finite Sets
15
CARD-3 and Set Union
0,1,00,11
CARD-3
16
Finite Sets and Set Complement
l,00,01,10,11,000,...
0,1
Finite Sets
17
Infinite Number of Facts

• A closure property can represent an infinite
  number of facts
• Example Finite languages closed under union
• union is a finite language
• union l is a finite language
• union 0 is a finite language
• ...
• l union is a finite language
• ...

18
First-order logic

• A way to formally write a closure property
• For all L1, ...,Lk in LC, L1 op ... op Lk in LC
• Only one expression is needed because of the for
  all quantifier
• Number of languages k is determined by arity of
  the operation
• complement is unary so only one language
• union is binary so two languages

19
Example F-O logic statements

• For all L1,L2 in FINITE, L1 union L2 in FINITE
• For all L1,L2 in LC3, L1 union L2 in LC3
• For all L in FINITE, Lc in FINITE
• For all L in LC3, Lc in LC3