Module 28

1
Module 28

• Context Free Grammars
• Examples of real-life grammars
• Definition of a grammar G
• Deriving strings and defining L(G)
• Context-Free Language definition

2
Grammars

• Examples of real-life grammars

3
English Grammar

• The underdog Spartans beat the Wolverines.
• Diagram this sentence
• The underdog Spartans beat the Wolverines.
• ltnoun phrasegt ltverb phrasegt ltpctgt
• The underdog Spartans beat the Wolverines.
• ltarticlegt ltadjgt ltnoungt ltverbgtltobjectgtltpctgt
• Can we generalize these rules?
• ltsentencegt --gt ltnoun phrasegt ltverb phrasegt ltpctgt
  ...
• ltnoun phrasegt --gt ltarticlegt ltadjgt ltnoungt ltnoungt
  ...
• ltverb phrasegt --gt ltverbgt ltobjectgt ltverbgt
  ltadverbgt ...
• ltnoungt --gt Spartans man boy woman ...
• ltverbgt --gt shock run catch ...
• ...

4
Deriving a legal sentence

• Grammar rules
• ltsentencegt --gt ltnoun phrasegt ltverb phrasegt ltpctgt
  ...
• ltnoun phrasegt --gt ltarticlegt ltadjgt ltnoungt ltnoungt
  ...
• ltverb phrasegt --gt ltverbgt ltobjectgt ltverbgt
  ltadverbgt ...
• ltnoungt --gt Spartans man boy woman ...
• ltverbgt --gt shock run catch ...
• ...

ltsentencegt gt ltnoun phrasegt ltverb phrasegt
ltpctgt gt ltarticlegt ltadjgt ltnoungt ltverb phrasegt
ltpctgt gt The ltadjgt ltnoungt ltverb phrasegt
ltpctgt gt ...
5
C Grammar

• points td6 xp x2p2 fg3 safety 2
• Diagram this statement
• points td6 xp x2p2 fg3 safety 2
• Can we generalize these rules?
• ltstatementgt --gt ltassignment statementgt ltif-then
  statementgt ...
• ltassignment statementgt --gt ltvariablegt
  ltexpressiongt
• ltvariablegt --gt points td xp x2p fg
  safety ...
• ltexpressiongt --gt ltexpressiongt ltexpressiongt
  ltvariablegt ...
• ...

6
Deriving a legal statement

• Grammar Rules
• ltstatementgt --gt ltassignment statementgt ltif-then
  statementgt ...
• ltassignment statementgt --gt ltvariablegt
  ltexpressiongt
• ltvariablegt --gt points td xp x2p fg
  safety ...
• ltexpressiongt --gt ltexpressiongt ltexpressiongt
  ltvariablegt ...
• ...

ltstatementgt gt ltassignment statementgt gt
ltvariablegt ltexpressiongt gt points
ltexpressiongt gt points ltexpressiongt
ltexpressiongt ...
7
Context-Free Grammars

• Definition

8
Definition

• A context-free grammar G (V, S, S, P)
• V finite set of variables (nonterminals)
• S finite set of characters (terminals)
• S start variable
• element of V
• role is similar to that of q0 for an FSA or NFA
• P finite set of grammar rules or production
  rules
• Syntax of a production
• variable --gt string of variables and terminals

9
English Context-Free Grammar

• ECFG (V, S, S, P)
• V ltsentencegt, ltnoun phrasegt, ltverb phrasegt,
  ...
• people sometimes use lt gt to delimit variables
• In this course, we generally will use capital
  letters to denote variables
• S a, b, c, ..., z, , ,, ., ...
• S ltsentencegt
• P ltsentencegt --gt ltnoun phrasegt ltverb phrasegt
  ltpctgt, ltnoun phrasegt --gt ltarticlegt ltadjgt ltnoungt,
  ...

10
aibi igt0 CFG

• ABG (V, S, S, P)
• V S
• S a, b
• S S
• P S --gt aSb, S --gt ab or S --gt aSb ab
• second format saves some space

11
Context-Free Grammars

• Deriving strings, defining L(G), and defining
  context-free languages

12
Defining --gt, gt notation

• First --gt notation
• This is used to define the productions of a
  grammar
• S --gt aSb ab
• Second gtG notation
• This is used to denote the application of a
  production rule from a grammar G
• S gtABG aSb gtABG aaSbb gtABG aaabbb
• We say that string S derives string aSb (in one
  step)
• We say that string aSb derives string aaSbb (in
  one step)
• We say that string aaSbb derives string aaabbb
  (in one step)
• We often omit the grammar subscript when the
  intended grammar is unambiguous

13
Defining gt continued

• Third gtkG notation
• This is used to denote k applications of
  production rules from a grammar G
• S gt2ABG aaSbb
• We say that string S derives string aaSbb in two
  steps
• aSb gt2ABG aaabbb
• We say that string aSb derives string aaabbb in
  two steps
• We often omit the grammar subscript when the
  intended grammar is unambiguous

14
Defining gt continued

• Fourth gtG notation
• This is used to denote 0 or more applications of
  production rules from a grammar G
• S gtABG S
• We say that string S derives string S in 0 or
  more steps
• S gtABG aaSbb
• We say that string S derives string aaSbb in 0 or
  more steps
• aSb gtABG aaSbb
• We say that string aSb derives string aaSbb in 0
  or more steps
• aSb gtABG aaabbb
• We say that string aSb derives string aaabbb in 0
  or more steps
• We often omit the grammar subscript when the
  intended grammar is unambiguous

15
Defining derivations

• Derivation of a string x
• The complete step by step derivation of a string
  x from the start variable S
• Key fact each step in a derivation makes only
  one application of a production rule from G
• Example Derivation of string aaabbb using ABG
• S gtABG aSb gtABG aaSbb gtABG aaabbb
• Example 2 AG (V, S, S, P) where P S --gtSS a
• Deriving string aaa
• S gt SS gt Sa gt SSa gt aSa gt aaa

16
Defining L(G)

• Generating strings
• If S gtG x, then grammar G generates string x
• Note G generates strings which contain terminals
  and nonterminals
• aSb contains nonterminals and terminals
• S contains only nonterminals
• aaabbb contains only terminals
• L(G)
• The set of strings over S generated by grammar G
• Note we only consider terminal strings generated
  by G
• aibi i gt 0 L(ABG)
• ai i gt 0 L(AG)

17
Context-Free Languages

• Context-Free Languages
• A language L is a context-free language (CFL) iff
  there exists a CFG G such that L(G) L
• Results so far
• ai i gt 0 is a CFL
• One CFG G such that L(G) this language is AG
• Note this language is also regular
• aibi i gt 0 is a CFL
• One CFG G such that L(G) this language is ABG
• Note this language is NOT regular

18
Example

• Let BAL the set of strings over (,) in which
  the parentheses are balanced
• Prove that BAL is a CFL
• To prove this, you need to come up with a CFG
  BALG such that L(BALG) BAL
• BALG (V, S, S, P)
• V S
• S (, )
• S S
• P ?
• Give derivations of ((( ))) and ( )(( )) with
  your grammar