CSCI 2670 Introduction to Theory of Computing

1. CSCI 2670Introduction to Theory of Computing September 14, 2005

2. Agenda • Yesterday • Pumping lemma • Today • Introduce context-free grammars • Formally define CFG’s • Begin designing CFG’s • Tomorrow • More on designing CFG’s • Chomsky normal form

3. Announcement • Quiz tomorrow • Regular expressions • Pumping lemma

4. Context-free grammars • The shortcoming of finite automata is that each state has very limited meaning • You have no memory of where you’ve been – only knowledge of where you are • Context-free grammars are a more powerful method of describing languages • Example: {0n1n | n  0} is a CFG

5. Example CFG • Context-free grammars use substitution to maintain knowledge S  (S) S  SS S  () • All possible legal parenthesis pairings can be expressed by consecutive applications of these rules • Is this a regular language?

6. Example S  (S) | SS | () • (()())(()) • S  SS  (S)(S)  (SS)(())  (()())(()) • This sequence of substitutions is called a derivation

7. S S S S S S ( () () ) ( () ) Parse tree S  (S) | SS | () S

8. Example 2 S  Sb | Bb B  aBb | aCb C  ε • Derivation for aaabbbbb SSb Bbb aaBbbbb aaaεbbbbb aBbbb aaaCbbbbb

9. S B B B C a a a ε b b b b b Example 2 parse tree S

10. Example 2 S  Sb | Bb B  aBb | aCb C  ε Question 1: What language does this grammar accept? Answer: {anbm | m > n > 0} Question 2: Can this CFG be simplified? Answer: yes. Replace BaCb with Bab and remove Cε

11. Context-free grammar definition • A context-free grammar is a 4-tuple (V,,R,S), where • V is a finite set called the variables, •  is a finite set, disjoint from V, called the terminals, • R is a finite set of rules, with each rule being a variable and a string of variables and terminals, and • S  V is the start variable.

12. More definitions • If u, v, and w are strings of variables and terminals, and A  w is a rule of the grammar, we say uAv yields uwv • Denoted uAv  uwv

13. More definitions • If a sequence of rules leads from u to v – i.e., u  u1  u2  …  v, we denote this u * v • (I can’t do the actual notation in powerpoint – the * should be over the double bars) • The language of the grammar is {w  * | s * w}