2.1 Context-Free Grammars

















What's a Context-Free Grammar?

	S -> aSb
	S -> SS
	S -> X
	X -> c

What'a a more compact notation for a grammar?

	S -> aSb | SS | X
	X -> c

Grammar Terminology

What's a Rule or Production?

What's a Variable or Nonterminal?

What's a Terminal?

What's the Start Symbol or Start Variable or Start Nonterminal?

















Derivation

How do you use a grammar to produce strings of symbols?
What's a Derivation?

	1. write down the start nonterminal
	2. repeatedly apply productions until only terminals remain

How do you apply a single production?
What symbol is used to show a single production application?

	find a variable that matches left of production
	replace variable with right of production

Derive aabaa using the grammar.
Derive bbabbbb using the grammar.
Derive ab using the grammar.

	S -> X | Y
	X -> b | a X a
	Y -> a | b Y bb

















Classwork
(You may work with a partner.)

Derive aabbb using the grammar.

	S -> a A
	A -> a A | b B
	B -> b B | b

















What's the Language of a Grammar?

What's a Context-Free Language?

How do Context-Free Languages compare to Regular Languages?
How do Context-Free Grammars compare to DFAs, NFAs, and Regular Expressions?

	CFLs are a larger class than RLs
	CFGs are more powerful than DFA, NFA, regular expression

What are some applications of Context-Free Grammars?

	specify/parse programming languages
	study human languages (noun, verb, preposition)

















Designing Context-Free Grammars

How do you design a Context-Free Grammar?

	there is no cookbook algorithm
	use union of simpler CFGs
	use recursive productions

Design a grammar that generates the language.
(w is a string of 0's and 1's)

	{ w | the number of 0s in w is equal to the number of 1s }

















Classwork
(You may work with a partner.)

Design a grammar that generates the language.
(w is a string of 0's and 1's)


	{ w | the length of w is odd and 
	      the first symbol is the same as the middle symbol }

















Formal Definition

What's a Formal Definition of a Context-Free Grammar?

	a context-free grammar is a 4-tuple (V, Z, R, S)
	1. V is a finite set of Variables
	2. Z is a finite set of Terminals (Z should be Sigma)
	3. R is a finite set of Rules
	   rule is a variable and a string of variables and terminals
	4. S is a member of V and is the start symbol

Write the grammar in formal notation.

	S -> aSb
	S -> SS
	S -> X
	X -> c


How do you write a grammar informally?

How do you know which symbols are variables?

How do you know which symbols are terminals?

How do you know which symbol is the start symbol?