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Regular Grammars

Regular Grammars. Reading: 3.3. What we know so far…. FSA = Regular Language Regular Expression describes a Regular Language Every Regular Language has a Regular Expression Now, what about grammars?. Right Linear Grammars.

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Regular Grammars

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  1. Regular Grammars Reading: 3.3

  2. What we know so far… • FSA = Regular Language • Regular Expression describes a Regular Language • Every Regular Language has a Regular Expression • Now, what about grammars?

  3. Right Linear Grammars • Right-linear: at most one variable on the right side of the production, must be the right-most symbol. A -> xBA -> x x is any string of terminals.

  4. Left Linear Grammars • Left-linear: at most one variable on the right side of the production, must be the left-most symbol. A -> BxA -> x x is any string of terminals.

  5. Regular Grammars • A grammar is regular if it is either left- or right-linear. • A linear grammar may have a mix of left and right productions. Its only restriction is that there is at most one symbol on the right. • All regular grammars are linear, but not all linear grammars are regular!

  6. Prove: All right-linear grammars describe regular languages • Idea: Find a way to convert any right-linear grammar into a FSA. • Terminals before a non-terminal represent arcs • Non-terminals represent non-final states. • Terminals without non-terminals following represent final states.

  7. Reg Grammar to NFA Step 1: Start variable is the initial node. S -> aD D -> abS D -> b S

  8. Reg Grammar to NFA Step 2: Each rule ending in a non-terminal is a chain of arcs followed by a non-final node. S -> aD D -> abS D -> b a D S

  9. Reg Grammar to NFA Step 2: Each rule ending in a non-terminal is a chain of arcs followed by a non-final node. S -> aD D -> abS D -> b a D S b a

  10. Reg Grammar to NFA Step 3: Each rule ending in a terminal leads to a final state. S -> aD D -> abS D -> b b a D S b a

  11. Example: Convert to a FSA • S -> aBS -> bSS -> λB -> aSB -> bB

  12. Does every regular language have a regular grammar that describes it? • Yes! Reverse the algorithm for constructing the FSA from the grammar • Every rule in δis a line in the grammar.

  13. Write a regular grammar

  14. Regular Language Wrap-up Regular expressions DFA or NFA Regular grammar

  15. Examples: • Find the grammar for the expression: aab*a • Find the regular expression for: S -> abS | bA A -> aA | b

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