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Context Free Grammars

Context Free Grammars. CSCI 432 Computer Science Theory. Why Study CFGs. What is "grammar"? What does grammar have to do with computing? Examples of grammar in computing?. Terminology. S → bA A → aA A → e Those are three rules or productions.

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Context Free Grammars

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  1. Context Free Grammars CSCI 432 Computer Science Theory

  2. Why Study CFGs • What is "grammar"? • What does grammar have to do with computing? • Examples of grammar in computing?

  3. Terminology S → bA A → aA A → e Those are three rules or productions. The starting rule is usually the first rule listed. S and A are nonterminals. a and b are terminals. Only a single nonterminal appears on the left side of a rule.

  4. Example S → bA1 A → aA2 A → e 3 Suppose we want to generate the word "baaa" using this grammar. S bA1 baA2 baaA2 baaaA2 baaa3 textbook page 70

  5. Non Context Free Grammar aAb → acb How the rule for A is used depends on whether or not it is surrounded by an a and a b. Thus, that rule is not free of context.

  6. Nondeterminism B → bB B → b We cannot determine which version of B to use to generate the desired word. We have to try possibilities until we generate the desired word. Thus, using CFGs to generate words is nondeterministic. textbook page 72

  7. Example 1 So… Can a CFG represent a non-regular language? L(G) = {anbn} S → aSb S → e S aSb aaSbb aaaSbbb aaabbb All regular languages can be represented with a CFG. textbook example 3.1.1

  8. Example 2 These 3 rules could also be written as: S → aS | Sb | e a*b* S → aS S → Sb S → e So how would we generate the word aaabb? S S aS Sb aaSaSb aaaSaaSb aaaSbaaSbb aaaSbbaaaSbb aaabbaaabb textbook example 3.1.2

  9. Example 3 What does this grammar generate? S → aSa | bSb | e For example S aSa aaSaa aabSbaa aabaSabaa aabaabaa textbook example 3.1.3

  10. Example 4 Binary string with even number of 0's. 1S → 1S 2S → 0A0S 3S → e 4A → 1A 5A → e Try to generate the string 01110. S 0A0S 2 01A0S 4 011A0S 4 0111A0S 4 01110 3,5

  11. Practice Binary string with even number of 0's. 1S → 1S 2S → 0A0S 3S → e 4A → 1A 5A → e Generate the string 11010 S 1S 1 11S 1 110A0S 2 1101A0S 4 11010S 3 11010 5

  12. Practice Use the following grammar to produce the string "the dog eats the bone". 1S → NP VP 2NP → the N 3VP → V NP 4V→ eats | buries 5N→ dog | bone

  13. Grammatically Correct ≠Meaningful We could also use this grammar to generate the string "the bone buries the dog". 1S → NP VP 2NP → the N 3VP → V NP 4V→ eats | buries 5N→ dog | bone

  14. Using Grammars • How do we use a grammar to check the validity of data or a source code file? • Bad Answer : generate lots of possibilities until you generate a match. Then you know the data matches the grammar. • Good Answer : Parsing.

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