CSCI 2670 Introduction to Theory of Computing

1. CSCI 2670Introduction to Theory of Computing September 21, 2004

2. Agenda • Last week • Pumping lemma • Context-free grammars • Examples, definition, strategies for building • This week • More on CFG’s • Ambiguity, Pushdown automata, pumping lemma for CFG’s • Next week • Midterm (Chapters 1 & 2)

3. Announcement • Homework due next Tuesday (9/28) • 2.8, 2.9, 2.12, 2.13, 2.15, 2.18 d, 2.22

4. Ambiguity • Consider the CFG ({S},{0,1},R,S), where the rules of R are S  0 | 1 | S + S | S * S • Derive the string 0 * 1 + 1

5. S S + * S S 1 0 S S + * S S 1 0 1 1 Ambiguity S  0 | 1 | S + S | S * S • 0 * 1 + 1 S S Different parse trees!

6. Leftmost derivation & ambiguity • A derivation of a string w in a grammar G is a leftmost derivation if every step of the derivation replaced the leftmost variable • A string is derived ambiguously in CFG G if it has two or more different leftmost derivations • The grammar G is ambiguous if it generates some string ambiguously • Some grammars are inherently ambiguous

7. Chomsky normal form • Method of simplifying a CFG Definition: A context-free grammar is in Chomsky normal form if every rule is of one of the following forms A  BC A  a where a is any terminal and A is any variable, and B, and C are any variables or terminals other than the start variable the rule S  ε is permitted, where S is the start variable

8. CFG’s and Chomsky normal form Theorem: Any context-free language is generated by a context-free grammar in Chomsky normal form. Proof idea: Convert any CFG to one in Chomsky normal form by removing or replacing all rules in the wrong form • Add a new start symbol • Eliminate ε rules of the form A  ε • Eliminate unit rules of the form A  B • Convert remaining rules into proper form

9. Convert a CFG to Chomsky normal form • Add a new start symbol • Create the following new rule S0 S where S is the start symbol and S0 is not used in the CFG

10. Convert a CFG to Chomsky normal form • Eliminate all ε rules A  ε, where A is not the start variable • For each rule with an occurrence of A on the right-hand side, add a new rule with the A deleted R  uAv becomes R  uAv | uv R  uAvAw becomes R  uAvAw | uvAw | uAvw | uvw • If we have R  A, replace it with R  ε unless we had already removed R  ε

11. Convert a CFG to Chomsky normal form • Eliminate all unit rules of the form A  B • For each rule B  u, add a new rule A  u, where u is a string of terminals and variables, unless this rule had already been removed • Repeat until all unit rules have been replaced

12. Convert a CFG to Chomsky normal form • Convert remaining rules into proper form • What’s left? • Replace each rule A  u1u2…uk, where k  3 and ui is a variable or a terminal with k-1 rules A  u1A1 A1  u2A2 … Ak-2  uk-1uk

13. Example S  S1 | S2 S1 S1b | Ab | ε A  aAb | ab S2 S2a | Ba | ε B  bBa | ba Step 1: Add a new start symbol

14. Example S0  S S  S1 | S2 S1 S1b | Ab A  aAb | ab | ε S2 S2a | Ba B  bBa | ba | ε Step 2: Eliminate ε rules

15. Example S0  S S  S1 | S2 S1 S1b | Ab | b A  aAb | ab S2 S2a | Ba | a B  bBa | ba Step 3: Eliminate all unit rules

16. Example S0  S1b | Ab | b | S2a | Ba | a S  S1b | Ab | b | S2a | Ba | a S1 S1b | Ab | b A  aAb | ab S2 S2a | Ba | a B  bBa | ba Step 4: Convert remaining rules to proper form

17. Example S0  S1b | Ab | b | S2a | Ba | a S  S1b | Ab | b | S2a | Ba | a S1 S1b | Ab | b A  aA1 | ab A1  Ab S2 S2a | Ba | a B  bB1| ba B1  Ba

18. Tomorrow • Pushdown automata