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Context-Free Languages: Design, Proof , Structure

MA/CSSE 474 Theory of Computation. Context-Free Languages: Design, Proof , Structure. Your Questions?. Previous class days' material Reading Assignments. HW 9 problems Exam 2 Anything else. Designing Context-Free Grammars. ● Generate related regions together. A n B n

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Context-Free Languages: Design, Proof , Structure

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  1. MA/CSSE 474 Theory of Computation Context-Free Languages:Design, Proof, Structure

  2. Your Questions? • Previous class days' material • Reading Assignments • HW 9 problems • Exam 2 • Anything else

  3. Designing Context-Free Grammars ● Generate related regions together. AnBn ● Generate concatenated regions: ABC ●Generate outside in: A aAb

  4. Concatenating Independent Languages Let L = {anbncm : n, m 0}. The cm portion of any string in L is completely independent of the anbn portion, so we should generate the two portions separately and concatenate them together. G = ({S, N, C, a, b, c}, {a, b, c}, R, S} where: R = { }.

  5. L = { : k 0 and i≤k (ni 0)} Examples of strings in L: , abab, aabbaaabbbabab Note that L = {anbn : n 0}*. G = ({S, M, a, b}, {a, b}, R, S} where: R = { }

  6. Another Example: Unequal a’s and b’s L = {anbm: nm} G = (V, , R, S), where V = {a, b, S, },  = {a, b}, R =

  7. Simplifying Context-Free Grammars Remove non-productive and unreachable non-terminals.

  8. Remove Unproductive Nonterminals removeunproductive(G: CFG) = • G = G. • Mark every nonterminal symbol in G as unproductive. • Mark every terminal symbol in G as productive. • Until one entire pass has been made without any new nonterminal symbol being marked do: For each rule X  in R do: If every symbol in  has been marked as productive and X has not yet been marked as productive then: Mark X as productive. • Remove from G every unproductive symbol. • Remove from G every rule that contains an unproductive symbol. • Return G.

  9. Remove Unreachable Nonterminals removeunreachable(G: CFG) = • G = G. • Mark S as reachable. • Mark every other nonterminal symbol as unreachable. • Until one entire pass has been made without any new symbol being marked do: For each rule X A (where A V - ) in R do: If X has been marked as reachable and A has not then: Mark A as reachable. • Remove from G every unreachable symbol. • Remove from G every rule with an unreachable symbol on the left-hand side. • Return G.

  10. Proving the Correctness of a Grammar L: AnBn= {anbn : n 0} G = ({S, a, b}, {a, b}, R, S), R = { S aSb S  } ● Prove that G generates only strings in L. ● Prove that G generates all the strings in L.

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