Closure Properties

1
Lecture 8

• Closure Properties
• Definition
• Language class definition
• set of languages
• Closure properties and first-order logic
  statements
• For all, there exists

2
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

3
Integers and Addition
7
Integers
4
Integers and Division
.4
2
5
Integers
5
Language Classes

• We will be interested in closure properties of
  language classes
• A language class is a set of languages
• Thus, the elements of a language class (set of
  languages) are languages which are sets
  themselves
• Crucial Observation
• When we say that a language class is closed under
  some set operation, we apply the set operation to
  the languages (elements of the language classes)
  rather than the language classes themselves

6
Example Language Classes

• In all these examples, we do not explicitly state
  what the underlying alphabet S is
• Finite languages
• Languages with a finite number of strings
• CARD-3
• Languages with at most 3 strings

7
Finite Sets and Set Union
0,1,00,11
Finite Sets
8
CARD-3 and Set Union
0,1,00,11
CARD-3
CARD-3 sets with at most 3 elements
9
Finite Sets and Set Complement
l,00,01,10,11,000,...
0,1
Finite Sets
10
Infinite Number of Facts

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

11
First-order logic and closure properties

• 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

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Example F-O logic statements

• For all L1,L2 in FINITE, L1 union L2 in FINITE
• For all L1,L2 in CARD-3, L1 union L2 in CARD-3
• For all L in FINITE, Lc in FINITE
• For all L in CARD-3, Lc in CARD-3
• Note, not all these statements are true

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Stating a closure property is false

• What is true if a set is not closed under some
  k-ary operator?
• There exist k elements of that set which, when
  combined together under the given operator,
  produce an element not in the set
• There exists L1, ...,Lk in LC, L1 op ... op Lk
  not in LC
• Example
• Finite sets and set complement

14
Complementing a F-O logic statement

• Complement For all L1,L2 in CARD-3, L1 union L2
  in CARD-3
• not (For all L1,L2 in CARD-3, L1 union L2 in
  CARD-3)
• There exists L1,L2 in CARD-3, not (L1 union L2 in
  CARD-3)
• There exists L1,L2 in CARD-3, L1 union L2 not in
  CARD-3

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Proving/Disproving

• Which is easier and why?
• Proving a closure property is true
• Proving a closure property is false