CSI 3104 /Winter 2006: Introduction to Formal Languages Chapter 18: Decidability

1. CSI 3104 /Winter 2006: Introduction to Formal Languages Chapter 18: Decidability Chapter 18: Decidability I. Theory of Automata II. Theory of Formal Languages III. Theory of Turing Machines … Zaguia/Stojmenovic

2. Chapter 18: Decidability Examples of undecidable problems: • Are the languages generated by two context-free grammars the same language? • Is a particular context-free grammar ambiguous? • Given a context-free grammar that is ambiguous, is there another context-free grammar that generates the same language that is not ambiguous? Zaguia/Stojmenovic

3. More examples • Is the complement of a context-free language also context-free? • Is the intersection of two context-free languages also context-free? • Is the intersection of two context-free languages empty? • Given a context-free grammar, are there any words that are not in the language generated by the grammar? Zaguia/Stojmenovic

4. Chapter 18: Decidability Some Decidable Problems • Given a context-free grammar, is the language that it generates empty? Emptiness problem. • Given a context-free grammar, is the language that it generates finite or infinite? Finiteness problem. • Given a context-free grammar and a word w, is w a word that can be generated by the grammar? Membership problem. Zaguia/Stojmenovic

5. Chapter 18: Decidability Theorem: Given a context-free grammar, there is an algorithm to determine if the language it generates is empty. Proof. • S  Λ? (by the constructive algorithm in Chapter 13 for deciding if a nonterminal is nullable) • If not, convert the grammar to Chomsky normal form. • Is there a production of the form: S  terminal? • If not, repeat the steps: • Choose a nonterminal N such that: N  t (sequence of terminals). Replace N on the right sides of productions with t. Remove all other productions forN. • Stop if there is a production S  t or there are no more nonterminals to choose. Zaguia/Stojmenovic

6. Example 1 • Example 1: SXY XAX XAA Aa YBY YBB Bb • Step 1: replace all A’s by a, all B’s by b: • SXY XaX Xaa YbY Ybb • Step 1: Replace all X’s by aa and all Y’s by bb: • S aabb • Step 1: replace all S’sby aabb • Step 2: terminate step 2, S has been eliminated. CFG produces at least one word. Zaguia/Stojmenovic

7. Example 2 • XXY XAX Aa YBY YBB Bb • Replace A by a, B by b: SXY XaX YbY Ybb • Replace Y’s by bb: SXbb XaX • Terminate step 1, S still there, CFG generates no word. Zaguia/Stojmenovic

8. Chapter 18: Decidability An algorithm that can determine if a nonterminal X can generate a sequence of terminals • Reverse S and X and use the algorithm from the previous theorem. If X cannot generate a sequence of terminals, X is unproductive. Zaguia/Stojmenovic

9. Chapter 18: Decidability Theorem: There is an algorithm to decide whether or not a given nonterminal X in a context-free grammar can ever be used in generating words. Proof. • Determine if X is unproductive. • If not, Zaguia/Stojmenovic

10. Chapter 18: Decidability Find all unproductive nonterminals. • Remove all productions containing unproductive nonterminals. • Paint all X’s blue (on the right and left in the grammar). • If any nonterminal on the right of a production is blue, paint the nonterminal on the left blue. Then paint blue all occurrences of this nonterminal in the grammar. • Repeat step 4 until no new nonterminals are painted blue. • If S is blue, there is a derivation that uses X. If X cannot be used in a derivation, X is useless. Zaguia/Stojmenovic

11. Example 1 • S  ABa | bAZ | b A  Xb | bZa B  bAA • X  aZa | aaa ZZAbA • X terminates, Z is useless (why?) • A is blue, B is blue, S is blue • A  Xb B  bAA • A  aaab, B  bAaaab  bXbaaab • S  Aba  aaabBa  aaabbXbaaaba • X productive Zaguia/Stojmenovic

12. Example • SEC EAC ABC Cc Bb • A blue  EEC • E blue SEC • S blue A usefull Zaguia/Stojmenovic

13. Chapter 18: Decidability An algorithm that can decide if a nonterminal X is self-embedded. • Replace X’s on the left in productions with Ж. • Paint all X’s blue, and use the blue paint algorithm (steps 3 to 5 of the previous algorithm). • If Жis blue, then X is self-embedded. Zaguia/Stojmenovic

14. Chapter 18: Decidability Theorem: There is an algorithm to decide if a context-free grammar generates a language that is finite or infinite. • Find all useless nonterminals. • Test all nonterminals to see if any is self-embedded using the above algorithm. • If step 2 found a self-embedded nonterminal, then the language is infinite. Otherwise it is finite. Zaguia/Stojmenovic

15. Example • S  Aba | bAZ | b X aZa | bA | aaa • AXb | bZa ZZAbA BbAA • Z useless (it appears in all its own productions) • S  Aba | b X bA | aaa AXb BbAA • S  Aba | b Ж bA | aaa AXb BbAA • X blue; AXb  A blue; ЖbA Ж blue; BA  B blue; SAba  S blue  • Language generated by CFG is infinite Zaguia/Stojmenovic

16. Chapter 18: Decidability Membership problem Given a context-free grammar G and a word w = w1w2…wn, is w a word that can be generated by the grammar? CYK Algorithm (Cocke, Kasami(1965),Younger(1967) • Transform G into a Chomski Normal Form. • At each step i of the algorithm, 1≤i≤n, find all sub words of w with length i that can be generated starting from any of the non terminals. • Details … • Parsing not covered Zaguia/Stojmenovic